Pumping Ratio of the Na + /K + Pump

Abstract

Our current knowledge about the Na⁺/K⁺ pump is mainly of the structure and function of the pump itself. The knowledge of the relationship between the function of the pump and other cellular functions is inadequate. This is because we are unaware of what the pumping process is and why the pump takes 3 Na⁺ to 2 K⁺ as the pumping ratio. In recent years, I proposed a dipole-charging model to simulate the pumping process of the Na⁺/K⁺ pump and showed by this model that this pumping ratio of the Na⁺/K⁺ pump is determined by the ratio of the ionic mobilities of K⁺ to Na⁺ ions. In this article, I further propose an osmotic balance condition that the potassium ion concentration inside the cell and the sodium ion concentration outside the cell are equal. With this condition, the dipole-charging model is further extended to compare different pumping ratios and it shows that only the pumping ratio 3 Na⁺ to 2 K⁺ can be selected during evolution. This result also reveals the relationships between the 3 Na⁺ to 2 K⁺ pumping ratio, the membrane resting potential, and the osmotic balance condition of the cell. Furthermore, I discuss the osmotic balance condition, the stability of the dipole-charging model and the dipole-charging process, the electrogenic contribution by the Na⁺/K⁺ pump to the resting membrane potential, and the effectiveness and the prospect of the dipole-charging model. The dipole-charging model is capable of explaining all the phenomena related to the pumping process of the Na⁺/K⁺ pump and is especially suitable for study in situations under pathologic conditions, such as ischemia or hypoxia. Finally, I present analyses of the free energy change of the coupled overall reaction for the Na⁺/K⁺ pump and of some reports on different pumping ratios of the Na⁺/K⁺ pump. The exception of the pumping ratio in the brine shrimp Artemia is also discussed.

Introduction

There are many ion transporters in animal cells, such as the Na⁺/K⁺ pump, the sarco(endo)plasmic reticulum Ca²⁺ pump, and the plasma membrane Ca²⁺ pump, which are primary active transporters; and the Na⁺–glucose cotransporter and the Na⁺/Ca²⁺ exchanger as a countertransporter, which are secondary active transporters.¹ All these transporters take part in different physiological functions.² For example, for each adenosine triphosphate (ATP) hydrolyzed in one pumping cycle, the Na⁺/K⁺ pump transports 3 Na⁺ ions out of the cell in exchange for 2 K⁺ ions transported into the cell and maintains large-concentration gradients of Na⁺ and K⁺ ions.³ These concentration gradients are critical for maintaining resting membrane potential, the secondary active transport, and the osmotic balance of the cell.⁴

However, unlike other transporters, the Na⁺/K⁺ pump does not work at the equilibrium state.⁵ That is, the ATP provides more energy than required for pumping 3 Na⁺ and 2 K⁺ ions. So the pumping ratio of the Na⁺/K⁺ pump remains a question.⁶

In recent years, I proposed a dipole-charging model to simulate the pumping process of the Na⁺/K⁺ pump and showed that this pumping ratio of the Na⁺/K⁺ pump is determined by the ratio of the ionic mobilities of K⁺ to Na⁺ ions.⁶ In this model, the pumping of the Na⁺/K⁺ pump is considered to be a dipole-charging process, in which the pumping of 3 Na⁺ ions out and 2 K⁺ ions in creates two electric dipoles, which subsequently charge the cell membrane. We know that, for neurons, the resting potential is −70 mV and the cell membrane has an electric capacitance of 1 μF/cm². The amount of charge required for producing this membrane potential is about 4350 monovalent ions/μm².⁶ This means that 4350 monovalent ions/μm² corresponds to −70 mV, provided that the membrane capacitance is 1 μF/cm², regardless of cell type. However, the pumping rate of the Na⁺/K⁺ pump can be 100 times per second,⁷ and the density of the Na⁺/K⁺ pump can be several thousand pumps/μm² on the membrane fragments isolated from kidney outer medulla.^8–10 So any accumulation of charges on the membrane caused by charging of the two dipoles at this pumping rate and this pump density could significantly change the membrane potential if those charges were not discharged instantly and in time. For this reason, the pumping process of the Na⁺/K⁺ pump should be precisely regulated, including the intracellular and the extracellular concentrations of sodium and potassium ions, because these concentrations contribute to the electrolytic conductivities of the intracellular and the extracellular solutions. This is why we should develop a dipole-charging model to simulate the pumping process of the Na⁺/K⁺ pump.

By this dipole-charging model, we already showed that when the pumping ratio is taken as 3 Na⁺ to 2 K⁺, the concentration ratios C_K,i/C_Na,o ≈ 1.0 and C_K,o/C_Na,i ≈ 0.30 can be obtained,⁶ where C_Na,i and C_K,i are the sodium and potassium ion concentrations inside the cell, and C_Na,o and C_K,o are the sodium and potassium ion concentrations outside the cell. The observation values of C_K,i/C_Na,o and C_K,o/C_Na,i for different cell types are shown to be consistent with the two predicted values (see Table 1 in Ref.⁶).

Table 1.

Calculated C_K,o/C_Na,i and C_K,i/C_Na,o According to the Pumping Ratio n_na/n_k

n_na/n _k	C_K,o/C_Na,i	C_K,i/C_Na,o
1/1	0.68	0.68
2/1	0.17	1.4
3/1	0.08	2.0
1/2	2.7	0.34
2/2	0.68	0.68
3/2	0.30	1.0
1/3	6.1	0.23
2/3	1.5	0.45
3/3	0.68	0.68

n_na, n_k are assumed to take 1 to 3.

In this study, we take a further view of this question. First, we assume that the pumping ratio of the Na⁺/K⁺ pump can take different values and the corresponding concentration ratios are calculated according to the dipole-charging model. Then a suitable concentration ratio is selected according to the osmotic balance condition of the cell. With this concentration ratio, the pumping ratio of 3 Na⁺ to 2 K⁺ can be uniquely determined by the ratio of the ionic mobilities of K⁺ to Na⁺ ions. Finally, I discuss the fact that the pumping ratio of the Na⁺/K⁺ pump is not due to the energy requirement, and I also discuss some reports on different pumping ratios of the Na⁺/K⁺ pump.

Methods

The dipole-charging model

Detailed analysis of the dipole-charging model can be found in Ref.⁶ In this study, I outline the main features of the model.

The pumping process of the Na⁺/K⁺ pump involves two parts. In the first or the sodium part, 3 Na⁺ ions inside the cell are transported to the outside while leaving three negative charges inside. This will create an electric dipole that has charge +3e outside the cell membrane and charge −3e inside the cell membrane, where e is the elementary charge. This electric dipole will subsequently charge the cell membrane, which has a certain capacitance C_m. In the second or the potassium part, 2 K⁺ ions outside the cell are transported to the inside while leaving two negative charges outside. This will create an electric dipole that has charge +2e inside the cell membrane and charge −2e outside the cell membrane. This electric dipole will subsequently charge the cell membrane with the same capacitance C_m.

The dipole charging process can be simulated by an electric circuit, which I call the dipole-charging model. A group of equations that describe the circuit are as follows (see Figure 1):

FIG. 1.

Dipole-charging model of the Na⁺/K⁺ pump shown by an electric circuit. Adapted from Fig. 1 in Ref.⁶

where U_d is the electric potential difference produced by the pump dipole alone and has the unit of V;

C_m is the electric capacitance per unit area of the cell membrane and has the units of F/m²;

U_c is the electric potential difference across the capacitance C_m and has the unit of V;

R₁ = r_i + r_o, where r_i is the electric resistance from the inner end of the Na⁺/K⁺ pump to the inner surface of the cell membrane and has the unit of Ω; r_o is the electric resistance from the outer end of the Na⁺/K⁺ pump to the outer surface of the cell membrane and has the unit of Ω. r_o and r_i are a measure of the resistances of the extracellular and intracellular fluids, respectively;

R₂ = r_m is the electric resistance of the cell membrane and has the unit of Ω. r_m is a measure of the resistance by ion channels of the cell membrane;

i₁ is the current through r_i and r_o and has the unit of A;

i₂ is the current through r_m and has the unit of A;

i_c is the current through the capacitance C_m and has the unit of A.

Solving this group of equations, we have the following:⁶ $i_{c} = (U_{d} ∕ R_{1}) e x p (- t ∕ R_{∕ ∕} C_{m}) = (U_{d} ∕ R_{1}) e x p (- t ∕ τ)$ (5)

where R_// = R₁R₂/(R₁ + R₂), τ = R_//C_m is the capacitive time constant of the circuit.

The time–average value I_c of i_c over 3τ is as follows: $I_{c} = [{_{0}}^{3 τ} (U_{d} ∕ R_{1}) e x p (- t ∕ τ) d t] ∕ 3 τ \approx U_{d} ∕ 3 R_{1}$ (6)

I_c or i_c is the part of the charging current flow that directly charges the membrane capacitor.

The membrane charging by the pump dipole is asymmetric,⁶ that is, in the sodium part, the electric resistance r_i can be taken as zero, so R₁ = r_o; in the potassium part, the electric resistance r_o would be zero, and R₁ = r_i. According to Equation (6), $I_{c} = U_{d} ∕ 3 R_{1} = U_{d} ∕ 3 (r_{i} + r_{o})$

So in the sodium part, ${I_{c}}_{, 1} = U_{d} ∕ 3 r_{o}$ (7)

In the potassium part, ${I_{c}}_{, 2} = U_{d} ∕ 3 r_{i}$ (7′)

where I_c_,1 is the first or the sodium part of the charging current flow that directly charges the membrane, and I_c_,2 is the second or the potassium part of the charging current flow that directly charges the membrane.

The other difference between the two parts is that the potential difference U_d is proportional to the transported charge, or the dipole charge. We assume that the pumping ratio is n_na Na⁺ to n_k K⁺, meaning that in each pumping cycle, n_na Na⁺ ions are pumped out of the cell and n_k K⁺ ions are pumped into the cell, respectively. Then, in the sodium part, U_d = n_na ψ, and in the potassium part, U_d = n_k ψ, where ψ is defined as the proportional coefficient.

So, in the sodium part, ${I_{c}}_{, 1} = n_{n a} ψ ∕ 3 r_{o}$ (8)

In the potassium part, ${I_{c}}_{, 2} = n_{k} ψ ∕ 3 r_{i}$ (8′)

The main contributions to r_i and r_o are made by the sodium and potassium ions,⁶ so we have the following: $(1 ∕ r_{o}) = (1 ∕ r_{N a, o}) + (1 ∕ r_{K, o})$ (9) $(1 ∕ r_{i}) = (1 ∕ r_{N a, i}) + (1 ∕ r_{K, i})$ (9′)

where r_Na,o and r_K,o are the sodium and potassium resistances outside the cell, respectively; and r_Na,i and r_K,i are the sodium and potassium resistances inside the cell, respectively.

In case of the Na⁺/K⁺ pump, the ion flow is produced by a single ion. The corresponding resistance can be calculated as follows:⁶ $r = 1 0^{10} ∕ 4 π κ_{s}$ (10)

where κ_s is the electrolytic conductivity of the solution.

Combining Equations (8), (8′), (9), (9′), and (10), we have ${I_{c}}_{, 1} = (n_{n a} ∕ 3) 4 π ψ 1 0^{- 10} (κ_{N a, o} + κ_{K, o})$ (11)

{I_{c}}_{, 2} = (n_{k} ∕ 3) 4 π ψ 1 0^{- 10} (κ_{N a, i} + κ_{K, i})

(11′)

where κ_Na,o and κ_K,o are the electrolytic conductivities contributed by the sodium and potassium ions in the extracellular fluid, respectively; and κ_Na,i and κ_K,i are the electrolytic conductivities contributed by the sodium and potassium ions in the intracellular fluid, respectively.

The electrolytic conductivity of a solution contributed by a kind of ion is as follows:¹¹ $κ (i) = Z (i) F U (i) C (i),$ (12)

where κ(i) is the electrolytic conductivity contributed by the ion i in the solution and has the units of S/m;

U(i) is the ionic mobility of the ion i and has the units of m²/(s·V);

C(i) is the concentration of the ion i and has the units of mol/m³;

F = 96,500 J/(V·mol) is the Faraday's constant;

Z(i) is the valence charge of the ion i.

So, according to Equation (12), $κ_{N a, o} = Z_{N a} F U_{N a} C_{N a, o}, κ_{K, o} = Z_{K} F U_{K} C_{K, o},$

κ_{N a, i} = Z_{N a} F U_{N a} C_{N a, i}, κ_{K, i} = Z_{K} F U_{K} C_{K, i}

(12′)

where U_Na and U_K are the ionic mobilities of sodium and potassium ions, respectively, and Z_Na = Z_K = 1. Substituting Equation (12′) into (11), (11′), we have ${I_{c}}_{, 1} = (n_{n a} ∕ 3) 4 π ψ 1 0^{- 10} F (U_{N a} C_{N a, o} + U_{K} C_{K, o})$ (13)

{I_{c}}_{, 2} = (n_{k} ∕ 3) 4 π ψ 1 0^{- 10} F (U_{N a} C_{N a, i} + U_{K} C_{K, i})

(13′)

We know that I_c, I_c_,1, and I_c_,2 all indicate the part of the charging current flow that goes on to the membrane. The sodium component of I_c_,1 and the potassium component of I_c_,2 will affect the membrane electric double layers for the resting potential, so they should be equal to each other, that is, $(n_{n a} ∕ 3) 4 π ψ 1 0^{- 10} F U_{N a} C_{N a, o} = (n_{k} ∕ 3) 4 π ψ 1 0^{- 10} F U_{K} C_{K, i}, o r n_{n a} ∕ n_{k} = (U_{K} ∕ U_{N a}) (C_{K, i} ∕ C_{N a, o})$ (14)

The potassium component of I_c_,1 and the sodium component of I_c_,2 can be used by the Na⁺/K⁺ pump, so they should be compatible with the pumping ratio, that is, $(n_{n a} ∕ 3) 4 π ψ {1 0^{-}}^{10} F U_{K} C_{K, o} = (n_{k} ∕ n_{n a}) (n_{k} ∕ 3) 4 π ψ {1 0^{-}}^{10} F U_{N a} C_{N a, i}, o r {(n_{n a} ∕ n_{k})}^{2} = (U_{N a} ∕ U_{K}) (C_{N a, i} ∕ C_{K, o})$ (15)

Equations (14) and (15) relate the pumping ratio, the ionic mobilities of Na⁺ and K⁺ ions, and the intracellular and the extracellular concentrations of sodium and potassium ions. By using these two equations, we can calculate C_K,o/C_Na,i and C_K,i/C_Na,o according to different pumping ratios of n_na/n_k.

Results

Table 1 shows the calculated C_K,o/C_Na,i and C_K,i/C_Na,o according to the pumping ratio n_na/n_k, where n_na and n_k are assumed to take 1 to 3.

From Table 1 we can see that when n_na/n_k = 3/2, C_K,i/C_Na,o ≈ 1. We know that sodium ions are the most abundant extracellular cations and potassium ions are the most abundant intracellular cations, and they are the major sources of the extracellular and the intracellular osmotic pressures, respectively¹² (also see Table 2 in Discussion section). So the condition

Table 2.

Osmolar Substances in Extracellular and Intracellular Fluids

	Plasma, mOsmol/L H₂O	Interstitial, mOsmol/L H₂O	Intracellular, mOsmol/L H₂O
Na⁺	144	137	10
K⁺	5	4.7	141
Ca⁺⁺	2.5	2.4	0
Mg⁺⁺	1.5	1.4	31
Cl⁻	107	112.7	4
HCO₃⁻	27	28.3	10
HPO₄⁻⁻, H₂PO₄⁻	2	2	11
SO₄⁻⁻	0.5	0.5	1
Phosphocreatine			45
Carnosine			14
Amino acids	2	2	8
Creatine	0.2	0.2	9
Lactate	1.2	1.2	1.5
Adenosine triphosphate			5
Hexose monophosphate			3.7
Glucose	5.6	5.6
Protein	1.2	0.2	4
Urea	4	4	4
Total, mOsmol/L	303.7	302.2	302.2

Adapted partly from Table 33_2 in Ref.¹⁴

C_{K, i} ∕ C_{N a, o} = 1

(16)

can actually ensure that the cell would maintain osmotic pressure balance. I call this relationship osmotic balance condition. This condition can be used for selection of n_na/n_k from Equation (14). Substituting Equation (16) into Equation (14), together with U_K = 7.62 × 10⁻⁸ m²/(s·V) and U_Na = 5.19 × 10⁻⁸ m²/(s·V), we have $n_{n a} ∕ n_{k} = U_{K} ∕ U_{N a} = 1.47 \approx 3 ∕ 2$ (17)

Equation (17) shows that the pumping ratio of 3 Na⁺ to 2 K⁺ is determined by the ratio of the ionic mobilities of K⁺ to Na⁺ ions. This relation results from the dipole-charging model together with the osmotic balance condition.

Discussion

The dipole-charging model and the osmotic balance condition

Table 1 is calculated from Equations (14) and (15). We can see from Table 1 that C_K,i/C_Na,o = 1 only when n_na/n_k = 3/2. We know that the body fluid contains cations and anions (or negatively charged molecules), and they each contribute to about one half of the total osmolarity, respectively (Table 2). As stated above, sodium ions outside the cell and potassium ions inside the cell are the most abundant cations. From Table 2 we can see that, for interstitial fluid, sodium ions contribute to 137/302.2 ≈ 45.33% of the total osmolarity; for intracellular fluid, potassium ions contribute to 141/302.2 ≈ 46.66% of the total osmolarity. The interstitial fluid also contains a large number of chloride ions, and the intracellular fluid contains a high concentration of negatively charged molecules,^13,14 but all their contributions to osmolarity belong to the anion (or negatively charged molecules) part (Table 2). So the concentration equivalence C_K,i/C_Na,o = 1 is enough to ensure the osmotic pressure balance between the interstitial and the intracellular fluids. In this way, only the pumping ratio n_na/n_k = 3/2 has been selected during evolution.

Stability of the dipole-charging model

For other pumping ratios, the corresponding values of C_K,i/C_Na,o in Table 1 are far away from 1, so they cannot ensure the osmotic balance condition (16). This is attributed to the stability of the dipole-charging model. We can see that Equations (5) and (6) as well as Equations (14) and (15) are outputs of the group of Equations (1) to (4). We consider Equation (14) here, the inputs of which are n_na, n_k and C_K,i, C_Na,o. We can see that Equation (14) is more sensitive to n_na and n_k than to C_K,i and C_Na,o. This is because, for each step, n_na or n_k changes discretely, while C_K,i or C_Na,o changes continuously. Therefore, the left-hand side of Equation (14) changes significantly with change of n_k or n_na, but the right-hand side of Equation (14) changes gradually with change of C_K,i or C_Na,o. This results in two consequences. First, n_na/n_k can only be 3/2. If n_na/n_k takes other values, according to Equation (14), C_K,i/C_Na,o will be far away from 1 and cannot satisfy the osmotic balance condition (16). Second, in the case of n_na/n_k = 3/2, Equation (14) can allow small variations of C_K,i and C_Na,o without changing n_na/n_k to other ratios. These variations are necessary for different cells or animals where C_K,i and C_Na,o can be adjusted to a small extent in accordance with different concentration combinations of other osmolar substances (Table 2). So here we can see the robustness of the dipole-charging model, which ensures both the pumping ratio n_na/n_k = 3/2 and the osmotic balance of the cell.

Stability of the dipole-charging process and mechanism for the electrogenic contribution of the Na⁺/K⁺ pump

We know that, at rest, there is a higher concentration of sodium ions outside the cell than inside and a higher concentration of potassium ions inside the cell than outside, and there is an electric potential difference ϕ across the cell membrane being negative inside and positive outside. This electric potential difference ϕ is called the membrane resting potential. Therefore, sodium ions tend to enter the cell, which are driven by their concentration gradient and ϕ; potassium ions tend to leave the cell, which are driven by their concentration gradient but are prevented by ϕ. The membrane potential that can balance the concentration gradient of an ion is called the equilibrium potential of the ion. The equilibrium potential can be calculated by the Nernst equation as follows: $E = (R T ∕ Z F) l n (C_{o} ∕ C_{i})$ (18)

where C_o and C_i are concentrations of the ion outside and inside the cell, respectively; E is the ion's equilibrium potential; R = 8.31 J/(mol·K) is the gas constant; T is the absolute temperature (K), T = 300K; F = 96,500 J/(V·mol) or C/mol is the Faraday's constant; and Z is the valence charge of the ion.

Let i_Na denote the sodium current driven by its concentration gradient and ϕ, and let i_K denote the potassium current driven by its concentration gradient and prevented by ϕ. So we have the following: $i_{N a} = g_{N a} (φ - E_{N a})$ (19) $i_{K} = g_{K} (φ - E_{K})$ (19′)

where g_Na is the electric conductance of the membrane ion channels to sodium ions and has the unit of /Ω or S;

g_K is the electric conductance of the membrane ion channels to potassium ions and has the unit of /Ω or S;

ϕ is the membrane resting potential defined as ϕ = ϕ(in) − ϕ(out) and has the unit of V;

E_K is the potassium equilibrium potential and has the unit of V;

E_Na is the sodium equilibrium potential and has the unit of V.

The function of the Na⁺/K⁺ pump is to continuously pump sodium ions toward outside the cell and pump potassium ions toward inside the cell to compensate for the consumption of ions used by i_Na and i_K. Let j_Na denote the pump current of sodium ions, and let j_K denote the pump current of potassium ions. So it should be that $j_{N a} = - i_{N a}, j_{K} = - i_{K}$ (20)

For the pumping ratio n_na/n_k = 3/2, j_Na = −(3/2)j_K.

So i_Na = −(3/2)i_K, or as in Ref.:¹⁵

g_Na(ϕ − E_Na) = −(3/2) g_K(ϕ − E_K).

This equation governs the membrane potential ϕ.

However, as we noted above, the pumping rate of the Na⁺/K⁺ pump can be 100 times/s, the density of the Na⁺/K⁺ pump can be several thousand pumps/μm², and the amount of charge required for producing a membrane potential of −70 mV such as in neurons is only about 4350 monovalent ions/μm². So any accumulation of charges on the membrane by the dipole charging process may cause significant change in the membrane potential because the density and the pumping rate of the Na⁺/K⁺ pump are high.

For example, if we assume that the pumping rate is 100 times/s and the density of the Na⁺/K⁺ pump is 2000 pumps/μm², then because the pumping ratio is 3 Na⁺ to 2 K⁺, for every second there will be (3 − 2) × 100 × 2000 = 2 × 10⁵ net monovalent ions/μm² being transported. This will create (2 × 10⁵/4350) × (−70 mV) = −3.2 × 10³ mV/s, which is much larger (in absolute value) than −70 mV. So to avoid accumulation of such a large number of charges on the membrane, there should be some mechanisms to regulate the pumping process of the Na⁺/K⁺ pump. This includes two aspects.

The pumping process of the Na⁺/K⁺ pump can be regarded as two charging processes by two dipoles: the sodium dipole and the potassium dipole. Because the two dipoles have opposite directions, the sodium charging current and the potassium charging current will cancel each other out, but that process should be steady so as to avoid any variation in membrane potential. This is ensured by both Equations (14) and (15). In other words, the sodium component of the sodium charging current and the potassium component of the potassium charging current are equal to each other so that they can cancel each other out steadily, and the potassium component of the sodium charging current and the sodium component of the potassium charging current are compatible with the pumping ratio n_k/n_na so that they can be used by the Na⁺/K⁺ pump again.

After the two dipole charging currents cancel each other out, there will still be some charges left on the membrane. This is because the pumping ratio may not be equal (such as 3 Na⁺ to 2K⁺), and not all pumped ions will charge the membrane but some of them will go through ion channels [as i₂ in Equation (3)]. So these remaining charges will create a change in the membrane resting potential, and this is known as the electrogenic contribution of the Na⁺/K⁺ pump. This electrogenic contribution can be shown by a small depolarization of the membrane potential occurring immediately after the Na⁺/K⁺ pump is blocked.¹⁶ It is reported that this electrogenic contribution of the Na⁺/K⁺ pump is less than 11 mV,¹⁵ or about 2 to 16 mV, depending on cell type.¹⁶ We can do some calculations to explain this phenomenon.

Let D denote the density of the Na⁺/K⁺ pump, and let H denote the pumping rate of the Na⁺/K⁺ pump. So the pump current of sodium ions j_Na = 3DHF/L, where F = 96,500 C/mol is the Faraday's constant, L = 6.02 × 10²³/mol is the Avogadro's number, and 3 indicates three sodium ions being pumped out per pumping cycle. According to Equation (20), i_Na = −j_Na = −3DHF/L. Substituting i_Na into Equation (19), we get r_Na = g_Na⁻¹ = (ϕ − E_Na)/i_Na = −(ϕ − E_Na)L/(3DHF), where r_Na is the electric resistance of the membrane ion channels to sodium ions. The accumulated charges on the membrane caused by the electrogenic contribution of the Na⁺/K⁺ pump will discharge through the membrane ion channels. The capacitive time constant for the discharging process of sodium ions can be calculated as follows: $τ_{N a} = r_{N a} C_{m} = - (φ - E_{N a}) L C_{m} ∕ (3 D H F)$ (21)

where C_m is the electric capacitance per unit area of the cell membrane as defined in Equation (4).

Similarly, the pump current of potassium ions j_K = −2DHF/L, where 2 indicates two potassium ions being pumped in per pumping cycle, and the minus sign means the direction of j_K is opposite to that of j_Na. According to Equation (20), i_K = −j_K = 2DHF/L. Substituting i_K into Equation (19′), we get r_K = g_K⁻¹ = (ϕ − E_K)/i_K = (ϕ − E_K)L/(2DHF), where r_K is the electric resistance of the membrane ion channels to potassium ions. The capacitive time constant for the discharging process of potassium ions can be calculated as follows: $τ_{K} = r_{K} C_{m} = (φ - E_{K}) L C_{m} ∕ (2 D H F)$ (22)

According to Equation (18), the sodium equilibrium potential is as follows: $E_{N a} = (R T ∕ Z_{N a} F) l n (C_{N a, o} ∕ C_{N a, i}) = (R T ∕ F) l n (C_{N a, o} ∕ C_{N a, i}),$

and the potassium equilibrium potential is as follows: $E_{K} = (R T ∕ Z_{K} F) l n (C_{K, o} ∕ C_{K, i}) = (R T ∕ F) l n (C_{K, o} ∕ C_{K, i}) .$

Substituting E_Na, E_K into Equations (21) and (22), we have the following: $τ_{N a} = - [φ - (R T ∕ F) l n (C_{N a, o} ∕ C_{N a, i})] L C_{m} ∕ (3 D H F)$ (23) $τ_{K} = [φ - (R T ∕ F) l n (C_{K, o} ∕ C_{K, i})] L C_{m} ∕ (2 D H F)$ (24)

We know that for humans,¹⁷ C_Na,o = 142 mM, C_Na,i = 12 mM, C_K,o = 4.3 mM, and C_K,i = 139 mM. Assume that the pump density D = 2000 pumps/μm² = 2.0 × 10¹⁵ pumps/m², the pumping rate H = 100 times/s, ϕ = −70 mV = −7.0 × 10⁻² V, and the electric capacitance C_m = 1 μF/cm² = 1.0 × 10⁻² F/m². Substituting these quantities with R = 8.31 J/(mol·K), T = 300K, F = 96,500 J/(V·mol), L = 6.02 × 10²³/mol into Equations (23) and (24), we get τ_Na = 0.0139 s ≈ 14 ms, and τ_K = 0.0031 s ≈ 3.1 ms.

We can see that both τ_Na = 14 ms and τ_K = 3.1 ms are comparable with the pumping cycle 10 ms (corresponding to pumping rate 100 times/s). Considering here that this is a rough estimation, in a real process it can be regarded that the accumulated charges would be able to discharge within one pumping cycle. Thus the averaged accumulated charges staying on the membrane over one pumping cycle can be estimated as half of the overall accumulated charges in one cycle, which is (1/2) × [(3Na⁺ − 2K⁺) × 1 cycle × 2000 pumps/μm²] = 1000 net monovalent ions/μm². As we have noted, 4350 monovalent ions/μm² corresponds to −70 mV, so this will create (1000/4350) × (−70 mV) ≈ −16 mV. This value is just what we would expect for the electrogenic contribution of the Na⁺/K⁺ pump.¹⁶

We can also see that, in Equations (21) and (22), τ_Na and τ_K are functions of variables E_Na, E_K, ϕ (and therefore r_Na, r_K), D, H, and C_m; in Equations (23) and (24), τ_Na and τ_K are even functions of variables C_Na,o, C_Na,i, C_K,o, C_K,i, ϕ, D, H, and C_m. The cell can adjust all these variables so that the accumulated charges would discharge in time and the direct electrogenic contribution of the Na⁺/K⁺ pump to the membrane resting potential is small.

Effectiveness and prospect of the dipole-charging model

Our current knowledge about the Na⁺/K⁺ pump remains at the stage where much attention is paid to the structural and functional study of the Na⁺/K⁺ pump itself, but less attention is paid to the relationship between the function of the Na⁺/K⁺ pump and other cellular functions. The main reason for this is that we are unaware of what the pumping process is and why the Na⁺/K⁺ pump takes 3 Na⁺ to 2 K⁺ as the pumping ratio.

The proposal of the dipole-charging model, which takes the pumping process as a dipole-charging process and makes a complete description accordingly, solves these questions. It not only shows that the 3 Na⁺ to 2 K⁺ pumping ratio is due to the ratio of the ionic mobilities of K⁺ to Na⁺ ions, but also links the pumping ratio, the pumping rate, the pump density, and other cellular functional parameters, such as the intracellular and extracellular concentrations of Na⁺ and K⁺ ions and the membrane capacitance, therefore revealing the relationship between the function of the Na⁺/K⁺ pump and other cellular functions, such as the membrane potential and the osmotic balance condition. The effectiveness of the dipole-charging model can be shown by the predictions it makes in the following three aspects: 1.

On the condition that the pumping ratio n_na/n_k = 3/2, the dipole-charging model predicts for the first time that the concentration ratios C_K,i/C_Na,o ≈ 1.0 and C_K,o/C_Na,i ≈ 0.30.⁶ The observation values of C_K,i/C_Na,o and C_K,o/C_Na,i for different cell types are shown to be consistent with these two predicted values (see Table 1 in Ref.⁶).

In this article, I further propose that the concentration equivalence C_K,i/C_Na,o = 1 can ensure the osmotic pressure balance of the cell and can be regarded as osmotic balance condition [Equation (16)]. The dipole-charging model gives the result that C_K,i/C_Na,o = 1 when and only when n_na/n_k = U_K/U_Na ≈ 3/2 [Equation (17)]. For other pumping ratios, the corresponding values of C_K,i/C_Na,o are far away from 1 (Table 1), and they cannot ensure the osmotic balance condition. So the pumping ratio of 3 Na⁺ to 2 K⁺ is shown to be determined by the ratio of the ionic mobilities of K⁺ to Na⁺ ions.

According to the dipole-charging model, the pumping process of the Na⁺/K⁺ pump can be regarded as two charging processes by the sodium dipole and the potassium dipole. After the cancellation of the two dipole charging currents, there will still be some charges left on the membrane. This will create a small change in the membrane resting potential, which is known as the electrogenic contribution of the Na⁺/K⁺ pump. These accumulated charges on the membrane should discharge through the membrane ion channels in time to ensure that the change created by these charges in the membrane potential is small. This is the mechanism I propose in this article to explain the electrogenic contribution of the Na⁺/K⁺ pump. The dipole-charging model can be used to explain this mechanism.

As we have calculated, when the pump density D = 2000 pumps/μm² and the pumping rate H = 100 times/s, the capacitive time constant for the discharging process of sodium ions is τ_Na = 14 ms and the capacitive time constant for the discharging process of potassium ions is τ_K = 3.1 ms. Both of them are comparable with the pumping cycle 10 ms (corresponding to the pumping rate 100 times/s). So the accumulated charges on the membrane should be able to discharge through the membrane ion channels in the time of one pumping cycle to ensure that this direct electrogenic contribution of the Na⁺/K⁺ pump is small. So this mechanism is supported by the dipole-charging model.

The dipole-charging model provides a complete description of the pumping process of the Na⁺/K⁺ pump. Any phenomenon related to the pumping process, such as the electrogenic contribution of the Na⁺/K⁺ pump, can be explained by this model. So the dipole-charging model would be applicable in medicinal practice. For example, under pathological conditions such as transient ischemia or hypoxia, the cell will undergo depolarization, and the electrogenic contribution of the Na⁺/K⁺ pump may affect this process.^16,17 This electrogenic contribution may also affect automaticity of the nodal cells of the heart, as well as of other cells that exhibit automaticity.¹⁶ In these cases, we may need to know what factors would influence the electrogenic contribution of the Na⁺/K⁺ pump, and what kind of influences these factors may have. Equations (21) to (24) provide information for us.

As already noted, τ_Na and τ_K are functions of variables E_Na, E_K, or C_Na,o, C_Na,i, C_K,o, C_K,i, and ϕ (and therefore r_Na, r_K), D, H, and C_m. For example, the membrane capacitance C_m is due to the lipid bilayer matrix. Under pathological conditions, the structural and chemical composition of the cell membrane may be changed, and the capacitance C_m will change accordingly. This has influence on τ_Na and therefore on the electrogenic contribution of the Na⁺/K⁺ pump. In addition, some drugs and toxins exert primary or secondary effects on the electrical properties of the cell membrane. For an understanding of the mode of action of these therapeutic drugs, toxic agents, neurotransmitters, hormones, and plasma electrolytes, it is necessary to understand the electrical properties and behavior of the cell membrane.¹⁶ The dipole-charging model would be useful in these areas. So the dipole-charging model has a broad prospect in medicinal and biological applications.

The pumping ratio of the Na⁺/K⁺ pump is not due to the energy requirement

There is an opinion that the pumping ratio of the Na⁺/K⁺ pump is due to the energy requirement. In Ref.,⁶ I have discussed the free energy change of the coupled overall reaction for the Na⁺/K⁺ pump [Reaction (26) in Ref.⁶]: $3 N a^{+} (i n) + 2 K^{+} (o u t) + A T P \to 3 N a^{+} (o u t) + 2 K^{+} (i n) + A D P + P i$ (25)

where “in” indicates intracellular and “out” indicates extracellular, and ATP, adenosine diphosphate (ADP), and Pi are in the intracellular solution.

The free energy change of Reaction (25) is [Equation (39) in Ref.⁶]: $Δ G_{p u m p} = - R T l n K_{A T P} + R T l n \{[{(C_{N a, o})}^{3} {(C_{K, i})}^{2}] ∕ [{(C_{N a, i})}^{3} {(C_{K, o})}^{2}]\} + R T l n [(C_{A D P}) (C_{P i}) ∕ (C_{A T P})] - (3 Z_{N a} - 2 Z_{K}) φ F$ (26)

where K_ATP is the equilibrium constant for ATP hydrolysis, ϕ is the membrane potential as defined before, and C_ATP, C_ADP, C_Pi denote the concentrations of ATP, ADP, and Pi, respectively.

The free energy used for pumping 1 mol Na⁺ ions, ΔG_Na, and for pumping 1 mol K⁺ ions, ΔG_K [Equation (42) in Ref.⁶], can be obviously obtained from Equation (26): $Δ G_{N a} = R T l n [(C_{N a, o}) ∕ (C_{N a, i})] - Z_{N a} φ F = F (E_{N a} - φ), Δ G_{K} = R T l n [(C_{K, i}) ∕ (C_{K, o})] + Z_{K} φ F = F (- E_{K} + φ)$ (26′)

where E_Na = (RT/F)ln[(C_Na,o)/(C_Na,i)] >0 and E_K = (RT/F)ln[(C_K,o)/(C_K,i)] <0 are equilibrium potentials for Na⁺ and K⁺ ions, respectively; ϕ = ϕ(in) − ϕ(out) <0, and Z_Na = Z_K = 1. Two cases have been calculated in Ref.,⁶ for mammalian skeletal muscle and for the squid axon. For mammalian skeletal muscle, C_Na,i = 12 mM, C_Na,o = 145 mM, C_K,i = 155 mM, C_K,o = 4 mM, and ϕ = −80 mV, and we have ΔG_Na = 13.9 kJ/mol and ΔG_K = 1.40 kJ/mol. For the squid axon, C_Na,i = 50 mM, C_Na,o = 440 mM, C_K,i = 400 mM, C_K,o = 20 mM, and ϕ = −70 mV, and we have ΔG_Na = 12.2 kJ/mol and ΔG_K = 0.713 kJ/mol.

These results show that the free energy used for pumping 1 mol Na⁺ ions is about 10-fold larger than that for pumping 1 mol K⁺ ions. This is because the K⁺ equilibrium potential E_K is close to the resting potential ϕ, and the resting potential favors the pumping of K⁺ ions from outside to inside the cell [Equation (26′)]. So if only energy requirement is considered, the number of K⁺ ions pumped in each cycle by the Na⁺/K⁺ pump can actually be arbitrary.

In this study, I make a further discussion on the use of the free energy in Equation (26). Let $Δ G_{i o n} = R T l n \{[{(C_{N a, o})}^{3} {(C_{K, i})}^{2}] ∕ [{(C_{N a, i})}^{3} {(C_{K, o})}^{2}]\} - (3 Z_{N a} - 2 Z_{K}) φ F$ (27)

ΔG_ion is the free energy used for pumping 3 mol Na⁺ ions from inside to outside the cell and 2 mol K⁺ ions from outside to inside the cell, corresponding to the pumping ratio of 3 Na⁺ to 2 K⁺. Apparently, ΔG_ion depends not only on the pumping ratio but also on the concentrations of Na⁺ and K⁺ ions in the intracellular and the extracellular solutions. For example, using the data above for mammalian skeletal muscle and for the squid axon, we can calculate that the ΔG_ion for the mammalian skeletal muscle is 44.6 kJ/mol and the ΔG_ion for the squid axon is 38.0 kJ/mol. The difference between these two values of ΔG_ion is 6.6 kJ/mol, which is caused mainly by the difference in the intracellular and extracellular concentrations of Na⁺ and K⁺ ions between the mammalian skeletal muscle and the squid axon (the difference in the membrane potential ϕ makes a little contribution). This 6.6 kJ/mol difference is apparently large enough to allow more K⁺ ions to be pumped in each cycle by the Na⁺/K⁺ pump in case of the squid axon (ΔG_K = 0.713 kJ/mol as above), but actually it does not cause any change in the pumping ratio in the squid axon. This shows that the pumping ratio of the Na⁺/K⁺ pump is not due to the energy requirement.

Moreover, as we can see from Tables 4–6 in Ref.,⁶ the extracellular ion concentrations decrease dramatically from marine animals to freshwater and terrestrial animals, with those of marine animals being similar to those of seawater. As discussed above, a change in ion concentrations would cause a change in ΔG_ion. So if the pumping ratio were dependent on the energy requirement, such a change in ion concentrations or in ΔG_ion should have given rise to mutation of pumping ratio during evolution. In another way we can say that the ion concentrations of the body fluid could adjust themselves to any pumping ratio within certain energy available from the hydrolysis of 1 mole ATP. In such cases, the number of Na⁺ ions pumped in each cycle by the Na⁺/K⁺ pump could also be varied, just like that the number of K⁺ ions could be varied in the above cases of the mammalian skeletal muscle and the squid axon. However, as we know, the Na⁺/K⁺ pump only takes 3 Na⁺ to 2 K⁺ as the pumping ratio. So we can say that the energy requirement is not the factor that determines the pumping ratio of the Na⁺/K⁺ pump during evolution.

Some reports on different pumping ratios of the Na⁺/K⁺ pump

There have been some reports on different pumping ratios of the Na⁺/K⁺ pump. For example, Ref.¹⁸ reported a conversion of the normal 3 Na⁺/2 K⁺/1 ATP stoichiometry of the Na⁺/K⁺-ATPase to electroneutral 2 Na⁺/2 K⁺/1 ATP stoichiometry. By sequence comparing of the electrogenic Na⁺/K⁺-ATPase and the nonelectrogenic H⁺/K⁺-ATPase, the researchers found that a cysteine conserved in the α-subunit transmembrane helix M8 of all Na⁺/K⁺-ATPases was replaced by arginine in H⁺/K⁺-ATPases at the corresponding position. Replacement of this cysteine C932 (rat α₁ Na⁺/K⁺-ATPase numbering) in the transmembrane helix M8 of the Na⁺/K⁺-ATPase with arginine converted the 3 Na⁺/2 K⁺/1 ATP stoichiometry of the Na⁺, K⁺-ATPase to 2 Na⁺/2 K⁺/1 ATP stoichiometry, similar to the electroneutral transport mode of the H⁺/K⁺-ATPase. The mechanism is that the guanidium group of the M8 arginine, which works as an internal cation, prevents Na⁺ binding at the third Na⁺-specific site, while allowing Na⁺ to bind at the two other sites and become transported. For our purposes, this work provides the structural basis for 2 Na⁺ to 2 K⁺ pumping ratio of the Na⁺/K⁺ pump, which means that during evolution, the occurrence of a 3 Na⁺ to 2 K⁺ pumping ratio, rather than a 2 Na⁺ to 2 K⁺ pumping ratio, is not due to the structure restriction of the Na⁺/K⁺ pump.

Furthermore, Ref.¹⁸ found that the substitution of this M8 cysteine of the Na⁺/K⁺-ATPase with phenylalanine led to spectacular reductions of apparent Na⁺ affinity, which could explain the function of its presence in a patient with alternating hemiplegia of childhood.¹⁹ These facts indicate that the mutation of this M8 cysteine of the Na⁺/K⁺-ATPase to arginine may also have appeared in the course of evolution but has not been conserved because it might be a more severe mutation than that of the M8 cysteine to phenylalanine.

Another exception is found in the brine shrimp Artemia, which can thrive at extreme salt concentrations in hypersaline lakes.²⁰ The shrimp drinks seawater, excreting salt from salt glands and maintaining low salt concentrations (155–185 mM) in the extracellular hemolymph fluid.^21,22 There are two isoforms of the Na⁺/K⁺-ATPase in Artemia franciscana, the α₁-NN-subunit²³ and the lysine-substituted α₂-KK-subunit.²⁴ In the α₂-KK-subunit, two lysines are substituted for two asparagines in the transmembrane segments M4 and M5, introducing two positively charged ɛ-amino groups of lysine near the center of the binding cavities for cations,²⁵ which interfere with the binding of Na⁺ and K⁺ ions.²⁰ The lysine substitutions of α₂-KK-Na⁺/K⁺-ATPase act as fixed internal cations to reduce the number of ions transported per ATP hydrolyzed and enable the Na⁺/K⁺ pump to extrude Na⁺ ions against steeper gradients.

The α₂-KK-subunit is almost exclusively located in the salt gland of Artemia larvae, while the α₁-NN-subunit is ubiquitous.²⁶ An increase of the α₂ mRNA levels was noticed when salinity increased, whereas expression of α₁ remained unchanged.²⁰ Refs.^27,28 further determined the pumping ratio of the Artemia α₂-KK-Na⁺/K⁺-ATPase to be 2 Na⁺ to 1 K⁺ by studying the functional effects of the equivalent asparagine-to-lysine substitutions in the Xenopus. However, the α₂-KK-Na⁺/K⁺-ATPase with this pumping ratio only exists in the salt gland of Artemia, and its function is to extrude Na⁺ ions against steeper gradients in the salt gland, which contributes to the salt adaptation of Artemia. In most cases in Artemia, the α₁-NN-Na⁺/K⁺-ATPase with the normal 3 Na⁺ to 2 K⁺ pumping ratio plays a central role in cellular function.

Conclusion

In this article, the dipole-charging model is extended to obtain a universal group of Equations (14) and (15) that relate the pumping ratio of the Na⁺/K⁺ pump, the ionic mobilities of K⁺ and Na⁺ ions, and the intracellular and the extracellular concentrations of sodium and potassium ions. By combining with the osmotic balance condition C_K,i/C_Na,o = 1, the pumping ratio of 3 Na⁺ to 2 K⁺ can be determined as the ratio of the ionic mobilities of K⁺ to Na⁺ ions. The dipole-charging model is shown to be useful in medicinal and biological applications. Further analysis of the free energy change of the coupled overall reaction for the Na⁺/K⁺ pump shows that the pumping ratio of the Na⁺/K⁺ pump is not due to the energy requirement. Some reports on different pumping ratios of the Na⁺/K⁺ pump are also explained.

Footnotes

Author's Contribution

N.X. designed the research, performed the research, and wrote the article.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

No funding was received for this article.

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