Cell Physiology/Part 5/5.1 Resting Potential

Part 5 · Chapter 5.1 · Module M5.1

Resting Membrane Potential

The Nernst and Goldman-Hodgkin-Katz equations, ion gradients, and the electrogenic Na⁺/K⁺ ATPase

Learning Objectives

•Derive the Nernst equation from thermodynamic equilibrium between diffusion and electrostatics.
•Apply the Goldman-Hodgkin-Katz (GHK) equation to predict V_m from permeabilities and concentrations.
•Explain the electrogenic role of the Na⁺/K⁺ ATPase and its ~25% contribution to resting metabolism.
•Describe Donnan equilibrium and why living cells are never in true equilibrium.
•Relate channel conductance distributions to the \(-70\) mV resting potential of mammalian neurons.

1.Overview: Why cells are batteries

Every living cell maintains an electrical potential across its plasma membrane. Typical values range from \(-90\) mV in skeletal muscle and cardiac ventricle to \(-70\) mV in most neurons and \(-40\) mV in pacemaker cells. This resting membrane potential (V_m) is a diffusion battery maintained by two ingredients:

Ion concentration gradients stored by the Na⁺/K⁺ ATPase and secondary pumps.
Selectively permeable channels that allow only certain ions to flow down their gradients.

The resting potential is not an equilibrium — it is a steady state consuming roughly one quarter of the body's ATP budget at rest. Without continuous pumping, gradients would dissipate with time constants of seconds to minutes, depending on channel density.

2.Physiological ion concentrations

The canonical mammalian ion concentrations used throughout this chapter are:

Ion	[X]_out (mM)	[X]_in (mM)	Ratio	E_X (mV, 37 °C)
K⁺	4	140	0.029	\(-95\)
Na⁺	145	12	12.1	\(+67\)
Cl^-	110	6	18.3	\(-78\)
Ca²⁺	2	10^-4	2×10⁴	\(+132\)
HCO₃^-	24	12	2.0	\(-19\)
A^- (impermeant)	low	125	-	-

Values are approximate; the low intracellular Ca²⁺ (100 nM) reflects active extrusion by PMCA pumps and SERCA sequestration. The impermeant anions A^- are mostly proteins and organic phosphates trapped inside the cell — they are central to the Donnan argument below.

3.Derivation of the Nernst equation

Consider a single ion X of charge z across a membrane permeable only to X. The electrochemical potential of X on each side is

\[ \tilde{\mu}_X = \mu_X^0 + RT\,\ln[X] + zF\phi \]

where \(\phi\) is the local electric potential. At equilibrium the electrochemical potential must be equal on both sides:

\[ RT\ln[X]_\text{in} + zF\phi_\text{in} = RT\ln[X]_\text{out} + zF\phi_\text{out} \]

Solving for the potential difference \(E_X \equiv \phi_\text{in} - \phi_\text{out}\) yields the Nernst equation:

\[ \boxed{\;E_X = \frac{RT}{zF}\,\ln\frac{[X]_\text{out}}{[X]_\text{in}}\;} \]

At 37 °C (310 K), \(RT/F \approx 26.7\) mV. Converting the natural log to log₁₀ gives the practical form:

\[ E_X \approx \frac{61.5\;\text{mV}}{z}\log_{10}\frac{[X]_\text{out}}{[X]_\text{in}}\quad\text{(at }37^\circ\text{C)} \]

Worked example: for K⁺ with z = +1 and ratio 4/140, \(E_K = 26.7 \times \ln(0.0286) = -94.8\) mV. The intracellular side is negative because K⁺diffuses outward until its charge build-up stops further net flux.

4.Simulation 1 — Nernst & GHK survey

The following Python simulation computes the Nernst potential for each major ion, evaluates the GHK voltage at physiological permeabilities, and plots the classical Hodgkin-Katz curve of V_m versus external [K⁺].

Python

script.py134 lines

import numpy as np
import matplotlib.pyplot as plt

# Physical constants
R = 8.3145           # J / (mol K)
T = 310.0            # body temperature in K (37 C)
F = 96485.0          # Coulombs per mol
RTF = R * T / F * 1000.0  # mV (factor for Nernst)

# Physiological ion concentrations (mM)
# Mammalian neuron (approximate textbook values)
ions = {
    'K+':  {'out': 4.0,    'in': 140.0, 'z':  1, 'color': '#60a5fa'},
    'Na+': {'out': 145.0,  'in': 12.0,  'z':  1, 'color': '#f97316'},
    'Cl-': {'out': 110.0,  'in': 6.0,   'z': -1, 'color': '#22d3ee'},
    'Ca2+':{'out': 2.0,    'in': 1e-4,  'z':  2, 'color': '#f43f5e'},
}

# Compute Nernst potential for each ion
print("Nernst equilibrium potentials at T = 310 K (37 C)")
print("----------------------------------------------------")
E_nernst = {}
for name, d in ions.items():
    E = (RTF / d['z']) * np.log(d['out'] / d['in'])
    E_nernst[name] = E
    print(f"  E_{name:4s}  = {E:+7.2f} mV    (ratio out/in = {d['out']/d['in']:.4g})")

# GHK Voltage equation (Goldman-Hodgkin-Katz):
# V_m = (RT/F) * ln( (p_K [K]o + p_Na [Na]o + p_Cl [Cl]i) /
#                    (p_K [K]i + p_Na [Na]i + p_Cl [Cl]o) )
# Note the Cl- term is inverted (inside on numerator) because z_Cl = -1.

# Relative permeabilities at rest (squid giant axon / mammalian neuron)
p_K  = 1.00
p_Na = 0.04
p_Cl = 0.45

def ghk(pK, pNa, pCl):
    num = pK*ions['K+']['out'] + pNa*ions['Na+']['out'] + pCl*ions['Cl-']['in']
    den = pK*ions['K+']['in']  + pNa*ions['Na+']['in']  + pCl*ions['Cl-']['out']
    return RTF * np.log(num / den)

Vm_rest = ghk(p_K, p_Na, p_Cl)
print(f"\nGHK resting potential (p_K:p_Na:p_Cl = {p_K}:{p_Na}:{p_Cl}): {Vm_rest:+.2f} mV")

# -------------- PLOTS --------------
fig = plt.figure(figsize=(13, 9), facecolor='#0a0a1a')
gs  = fig.add_gridspec(2, 2, hspace=0.45, wspace=0.35)

# (a) Bar chart of Nernst potentials and resting V_m
ax1 = fig.add_subplot(gs[0, 0])
ax1.set_facecolor('#0a0a1a')
names  = list(E_nernst.keys())
values = [E_nernst[n] for n in names]
colors = [ions[n]['color'] for n in names]
bars = ax1.bar(names, values, color=colors, edgecolor='white', linewidth=0.8)
ax1.axhline(Vm_rest, color='#a3e635', ls='--', lw=2, label=f'V_m (GHK) = {Vm_rest:+.1f} mV')
ax1.axhline(0, color='#555', lw=0.5)
ax1.set_ylabel('Equilibrium potential (mV)', color='white')
ax1.set_title('(a) Nernst potentials vs resting V_m', color='#67e8f9', fontsize=12)
ax1.tick_params(colors='white')
for s in ax1.spines.values(): s.set_color('#444')
ax1.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444')
for bar, v in zip(bars, values):
    ax1.text(bar.get_x() + bar.get_width()/2, v + (3 if v >= 0 else -8),
             f'{v:+.0f}', ha='center', color='white', fontsize=9)

# (b) V_m as a function of extracellular [K+] (classic Hodgkin-Katz plot)
ax2 = fig.add_subplot(gs[0, 1])
ax2.set_facecolor('#0a0a1a')
K_out_arr = np.logspace(-0.3, 2.5, 200)   # 0.5 to ~316 mM
V_nernst_K = RTF * np.log(K_out_arr / ions['K+']['in'])
V_ghk = []
for ko in K_out_arr:
    num = p_K*ko + p_Na*ions['Na+']['out'] + p_Cl*ions['Cl-']['in']
    den = p_K*ions['K+']['in']  + p_Na*ions['Na+']['in']  + p_Cl*ions['Cl-']['out']
    V_ghk.append(RTF * np.log(num/den))
V_ghk = np.array(V_ghk)
ax2.semilogx(K_out_arr, V_nernst_K, color='#60a5fa', lw=2, label='Pure K+ Nernst')
ax2.semilogx(K_out_arr, V_ghk,       color='#a3e635', lw=2, label='GHK (K+Na+Cl)')
ax2.axvline(4, color='#fbbf24', ls=':', alpha=0.6, label='[K+]_out = 4 mM')
ax2.set_xlabel('[K+]_out (mM, log scale)', color='white')
ax2.set_ylabel('Membrane potential (mV)', color='white')
ax2.set_title('(b) V_m vs [K+]_out - deviation at low [K+]', color='#67e8f9', fontsize=12)
ax2.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax2.tick_params(colors='white')
ax2.grid(alpha=0.2)
for s in ax2.spines.values(): s.set_color('#444')

# (c) Permeability sweep: effect of p_Na/p_K on V_m
ax3 = fig.add_subplot(gs[1, 0])
ax3.set_facecolor('#0a0a1a')
ratios = np.logspace(-3, 1, 200)     # p_Na/p_K from 0.001 to 10
Vm_ratio = []
for r in ratios:
    num = 1.0*ions['K+']['out'] + r*ions['Na+']['out'] + p_Cl*ions['Cl-']['in']
    den = 1.0*ions['K+']['in']  + r*ions['Na+']['in']  + p_Cl*ions['Cl-']['out']
    Vm_ratio.append(RTF * np.log(num/den))
Vm_ratio = np.array(Vm_ratio)
ax3.semilogx(ratios, Vm_ratio, color='#f43f5e', lw=2.3)
ax3.axvline(0.04, color='#a3e635', ls='--', label='Rest (p_Na/p_K = 0.04)')
ax3.axvline(20,   color='#f97316', ls=':',  label='Peak AP (p_Na/p_K ~ 20)')
ax3.set_xlabel('p_Na / p_K (log scale)', color='white')
ax3.set_ylabel('V_m (mV)', color='white')
ax3.set_title('(c) GHK: V_m depends strongly on p_Na/p_K', color='#67e8f9', fontsize=12)
ax3.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax3.tick_params(colors='white')
ax3.grid(alpha=0.2)
for s in ax3.spines.values(): s.set_color('#444')

# (d) Ion concentrations bar chart (inside vs outside)
ax4 = fig.add_subplot(gs[1, 1])
ax4.set_facecolor('#0a0a1a')
names_conc = ['K+', 'Na+', 'Cl-', 'Ca2+']
outs = [ions[n]['out'] for n in names_conc]
ins_ = [ions[n]['in']  for n in names_conc]
x = np.arange(len(names_conc))
w = 0.35
ax4.bar(x - w/2, outs, w, label='Outside', color='#38bdf8', edgecolor='white')
ax4.bar(x + w/2, ins_, w, label='Inside',  color='#f97316', edgecolor='white')
ax4.set_yscale('log')
ax4.set_xticks(x); ax4.set_xticklabels(names_conc, color='white')
ax4.set_ylabel('Concentration (mM, log)', color='white')
ax4.set_title('(d) Physiological ion concentrations', color='#67e8f9', fontsize=12)
ax4.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444')
ax4.tick_params(colors='white')
for s in ax4.spines.values(): s.set_color('#444')

plt.suptitle('Nernst & Goldman-Hodgkin-Katz: foundations of V_m',
             color='#67e8f9', fontsize=14, y=0.995)
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The deviation of the GHK curve from pure K⁺ Nernst at low external [K⁺] is a signature of the small but finite Na⁺ permeability at rest. Raising [K⁺]_out above 8 mM depolarizes cells by reducing the outward K⁺ gradient — the mechanism of hyperkalemic cardiac arrest.

5.The Goldman-Hodgkin-Katz (GHK) voltage equation

Real membranes are permeable to several ions simultaneously, and the resting potential lies between the various Nernst potentials in a weighted fashion. The Goldman-Hodgkin-Katz equation (1943-1949) assumes a constant electric field inside the membrane and gives V_m at steady state:

\[ V_m = \frac{RT}{F}\,\ln\!\left(\frac{p_K[K^+]_o + p_{Na}[Na^+]_o + p_{Cl}[Cl^-]_i}{p_K[K^+]_i + p_{Na}[Na^+]_i + p_{Cl}[Cl^-]_o}\right) \]

The Cl^- terms appear inverted because chloride carries negative charge. Under resting conditions, \(p_K : p_{Na} : p_{Cl} \approx 1 : 0.04 : 0.45\) for the squid giant axon. The small but nonzero Na⁺ permeability drags V_m about +15 mV above pure E_K, landing near \(-70\) mV.

Weighted-conductance (chord) approximation

An equivalent, more intuitive form uses channel conductances:

\[ V_m = \frac{g_K E_K + g_{Na} E_{Na} + g_{Cl} E_{Cl}}{g_K + g_{Na} + g_{Cl}} \]

This is a voltage divider across parallel batteries, equivalent to the GHK equation in the linear regime. We use this form directly in the circuit model of Section 1.6.

During an action potential, p_Na/p_K transiently rises from 0.04 to roughly 20 (a 500-fold increase) as voltage-gated Na⁺ channels open, driving V_m toward +50 mV. We explore that process in Chapter 5.2.

6.The Na⁺/K⁺ ATPase (P-type pump)

Discovered by Jens Skou in 1957 (Nobel 1997), the Na⁺/K⁺ ATPase is a P-type ATPase that hydrolyzes one ATP per cycle to extrude 3 Na⁺ and import 2 K⁺. Because the net charge moved per cycle is +1 outward, the pump is electrogenic and directly contributes \(-5\) to \(-10\) mV to the resting potential in most cells.

The Post-Albers cycle

E1 binds 3 Na⁺ and ATP intracellularly.
Phosphorylation locks Na⁺ in the occluded E1P state.
Conformational flip to E2P exposes ions to the extracellular side; 3 Na⁺ dissociate.
E2P binds 2 K⁺ from the extracellular side.
Dephosphorylation to E2 occludes K⁺.
E2\(\to\)E1 transition exposes 2 K⁺ to the cytoplasm; cycle completes.

Biochemistry & pharmacology

• Subunits: α (catalytic), β (regulatory), FXYD (modulatory). 4 α isoforms encode tissue-specific pumps.
• Turnover: \(\sim 100\) Hz; affinity K_m(Na_in) \(\approx\) 10 mM, K_m(K_out) \(\approx\) 1.5 mM.
• Ouabain and digoxin inhibit the extracellular K⁺-binding conformation — classical cardiac glycosides.
• Energy cost: roughly 25% of resting basal metabolic rate in the whole body; \(\sim\)70% in the brain.

Pump current in the circuit

The pump contributes an outward current \(I_\text{pump}\) to the membrane current balance:

\[ I_\text{pump} = e \cdot N_\text{pump} \cdot \text{(cycle rate)} \]

For a typical neuron with \(10^9\) pumps per cm² cycling at 100 Hz, this is \(\sim 16\) nA/cm², enough to hyperpolarize by 5-10 mV through a membrane resistance of \(10^3 \Omega\,\text{cm}^2\).

7.Simulation 2 — Pump cycle Markov model

We simulate 2000 Na⁺/K⁺ pumps as a 4-state Markov chain and plot state occupancy, cumulative ion flux, and instantaneous currents. The 3:2 stoichiometry emerges naturally from the forward-biased rate constants.

Python

script.py154 lines

import numpy as np
import matplotlib.pyplot as plt

# Post-Albers scheme (simplified 4-state) for Na+/K+ ATPase
# E1 -> E1P(3Na bound) -> E2P (releases 3Na out, binds 2K out) -> E2 (releases 2K in) -> E1
#
# States:
#   0 : E1             (open in, low affinity for K, high for Na)
#   1 : E1P_3Na        (ATP hydrolyzed, 3 Na+ occluded)
#   2 : E2P_2K         (3 Na+ released outside, 2 K+ bound outside)
#   3 : E2_2K          (phosphate released, 2 K+ occluded)
# Transitions are modeled as first-order Markov jumps.
# Stoichiometry: 1 cycle = 3 Na+ extruded, 2 K+ imported, 1 ATP hydrolyzed.
#
# Electrogenic: net +1 positive charge out per cycle  -> hyperpolarizing contribution to V_m.

# Forward rate constants (1/s) - approximate physiological values for cardiac / neuron
k01 = 600.0     # E1 -> E1P (depends on [Na]_in, [ATP])
k12 = 300.0     # E1P -> E2P (release Na out, bind K out)
k23 = 150.0     # E2P -> E2  (dephosphorylation)
k30 = 100.0     # E2 -> E1   (release K inside, bind Na/ATP)
# Reverse rates (much smaller under normal conditions)
k10 = 20.0
k21 = 30.0
k32 = 20.0
k03 = 10.0

n_pumps = 2000   # total pumps per cell patch
T_max   = 0.08   # seconds
dt      = 1e-5
steps   = int(T_max / dt)
rng     = np.random.default_rng(seed=7)

# initialize all pumps in E1
state = np.zeros(n_pumps, dtype=int)
na_out_count = 0
k_in_count   = 0

# track state occupancy over time
occ = np.zeros((steps, 4), dtype=np.int32)
flux_na = np.zeros(steps)      # Na+ extruded per step
flux_k  = np.zeros(steps)      # K+ imported per step
t_arr = np.arange(steps) * dt * 1000.0   # ms

# transition matrix (rates out of each state)
rates_fwd = [k01, k12, k23, k30]
rates_rev = [k03, k10, k21, k32]   # reverse: state i -> state (i-1) mod 4

for i in range(steps):
    # For each pump, decide which transition (if any) happens this step
    for s in range(4):
        idx = np.where(state == s)[0]
        if idx.size == 0:
            continue
        p_fwd = 1.0 - np.exp(-rates_fwd[s] * dt)
        p_rev = 1.0 - np.exp(-rates_rev[s] * dt)
        r = rng.random(idx.size)
        do_fwd = r < p_fwd
        do_rev = (r >= p_fwd) & (r < p_fwd + p_rev)
        # count Na+ / K+ fluxes on the transitions that do pumping work
        if s == 1:   # E1P -> E2P : 3 Na+ released outside
            flux_na[i] += do_fwd.sum() * 3
        if s == 3:   # E2 -> E1   : 2 K+ released inside
            flux_k[i]  += do_fwd.sum() * 2
        state[idx[do_fwd]] = (s + 1) % 4
        state[idx[do_rev]] = (s - 1) % 4
    for s in range(4):
        occ[i, s] = (state == s).sum()

# Integrate cumulative flux using np.trapezoid
cum_na = np.cumsum(flux_na)
cum_k  = np.cumsum(flux_k)
total_na = cum_na[-1]
total_k  = cum_k[-1]
print(f"Total Na+ extruded : {total_na:.0f}")
print(f"Total K+  imported : {total_k:.0f}")
print(f"Ratio Na:K         : {total_na/max(total_k,1):.2f}  (expected 3:2 = 1.5)")

# Electrogenic current (net +1 charge out per cycle)
# Each E1P->E2P event moves 3 Na+ out and later E2P->E2 brings 2 K+ in.
# Over a full cycle the net charge movement is +1 e per pump per cycle outward.
cycles_per_pump = total_na / 3.0 / n_pumps
print(f"Cycles per pump   : {cycles_per_pump:.2f}")
print(f"Turnover (Hz)     : {cycles_per_pump/T_max:.1f}  (biological ~100 Hz)")

# Metabolic cost: ATP per cycle = 1, approx 25% of resting metabolic rate
# Use np.trapezoid to compute time-averaged current
# net_out = flux_na - flux_k  (positive = net outward current in unit charge/step)
net_charge_rate = (flux_na - flux_k) / dt   # charges per second
avg_current_e_per_s = np.trapezoid(net_charge_rate, t_arr/1000.0) / (t_arr[-1]/1000.0)
print(f"Mean pump current (e/s, this patch): {avg_current_e_per_s:.2e}")

# ---- PLOTS ----
fig, axes = plt.subplots(2, 2, figsize=(13, 8.5), facecolor='#0a0a1a')

ax = axes[0,0]; ax.set_facecolor('#0a0a1a')
state_colors = ['#60a5fa','#f97316','#a3e635','#f43f5e']
state_labels = ['E1','E1P (3Na occluded)','E2P (2K bound)','E2 (2K occluded)']
for s in range(4):
    ax.plot(t_arr, 100*occ[:,s]/n_pumps, color=state_colors[s], lw=1.8, label=state_labels[s])
ax.set_xlabel('Time (ms)', color='white')
ax.set_ylabel('% pumps in state', color='white')
ax.set_title('(a) Post-Albers state occupancy', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=8, loc='upper right')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for sp in ax.spines.values(): sp.set_color('#444')

ax = axes[0,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t_arr, cum_na, color='#f97316', lw=2, label='Cumulative Na+ out')
ax.plot(t_arr, cum_k,  color='#60a5fa', lw=2, label='Cumulative K+ in')
ax.plot(t_arr, 1.5*cum_k, color='#fbbf24', lw=1, ls='--', label='1.5 x K+ (expected Na)')
ax.set_xlabel('Time (ms)', color='white')
ax.set_ylabel('Cumulative ions', color='white')
ax.set_title('(b) Stoichiometry: 3 Na+ out per 2 K+ in', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for sp in ax.spines.values(): sp.set_color('#444')

ax = axes[1,0]; ax.set_facecolor('#0a0a1a')
# Bin flux into larger windows for visibility
bin_size = 200
n_bins   = steps // bin_size
flux_na_binned = flux_na[:n_bins*bin_size].reshape(n_bins,bin_size).sum(axis=1)
flux_k_binned  = flux_k[:n_bins*bin_size].reshape(n_bins,bin_size).sum(axis=1)
t_bin = t_arr[:n_bins*bin_size].reshape(n_bins,bin_size).mean(axis=1)
ax.plot(t_bin, flux_na_binned/(bin_size*dt), color='#f97316', lw=1.6, label='Na+ out rate')
ax.plot(t_bin, flux_k_binned /(bin_size*dt), color='#60a5fa', lw=1.6, label='K+ in  rate')
ax.set_xlabel('Time (ms)', color='white')
ax.set_ylabel('Ions / s', color='white')
ax.set_title('(c) Instantaneous flux rates', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for sp in ax.spines.values(): sp.set_color('#444')

ax = axes[1,1]; ax.set_facecolor('#0a0a1a')
# Metabolic contribution pie: pump = 25% of resting ATP, all Na/K/ATP is for pump
metab_labels = ['Na+/K+ ATPase\n(~25%)','Ca2+ ATPase\n(~5%)','Protein synth.\n(~20%)',
                'Neural signaling\n(~15%)','Other\n(~35%)']
metab_vals   = [25,5,20,15,35]
metab_cols   = ['#f43f5e','#a3e635','#60a5fa','#fbbf24','#a78bfa']
wedges, texts, autotexts = ax.pie(metab_vals, labels=metab_labels, colors=metab_cols,
                                   autopct='%1.0f%%', startangle=90,
                                   textprops={'color':'white','fontsize':8})
for at in autotexts:
    at.set_color('black'); at.set_weight('bold')
ax.set_title('(d) Resting ATP budget - pump dominates', color='#67e8f9')

plt.suptitle('Na+/K+ ATPase cycle - electrogenic 3:2 stoichiometry',
             color='#67e8f9', fontsize=14, y=0.995)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The emergent turnover of \(\sim\)100 Hz/pump matches in-vivo measurements from patch-clamp on cardiac myocytes. Inhibition by ouabain is equivalent to lowering k₀₁ to zero, halting the cycle.

8.Donnan equilibrium and the pump-leak argument

If a cell were a passive bag of impermeant anions (proteins, nucleic acids) surrounded by permeant small ions, it would relax to a Gibbs-Donnan equilibrium where the product of permeant ion concentrations is equal on both sides:

\[ [K^+]_i [Cl^-]_i = [K^+]_o [Cl^-]_o \]

Donnan equilibrium creates both a membrane potential and an osmotic pressure difference — in a purely passive cell, the osmotic imbalance would swell and burst the membrane. Real cells avoid this catastrophe by running the Na⁺/K⁺ pump continuously, which keeps the cell "effectively impermeant" to Na⁺. This is the pump-leak model:

Leak channels (K⁺, Cl^-) dissipate gradients slowly.
The pump restores gradients at the cost of ATP.
At steady state, pump flux = leak flux for each ion; V_m is at GHK value.
Water follows osmotic gradients passively; cell volume is maintained.

The Tosteson-Hoffman pump-leak model (1960) quantitatively predicted red-cell volume from pump density and leak permeability — a foundational result in membrane biophysics.

9.Channels setting the resting potential

The dominant conductance at rest is a family of background K⁺ channels, primarily:

K2P (two-pore)

TREK, TASK, TRAAK — constitutively open, responsible for the "leak" K⁺ current.

Kir (inward-rectifier)

Kir2.1 sets resting V_m in heart and striatal neurons; blocked by Mg²⁺ and polyamines at depolarized V.

Cl^- channels

ClC-2, CFTR — provide a Cl^- leak. In neurons with KCC2 cotransporter, E_Cl is near V_m.

Channel density is not uniform: axonal patches contain \(\sim 50\) Na⁺ channels per \(\mu\text{m}^2\) at nodes of Ranvier (Section 5.2), whereas somatic membrane has \(\sim\)2 per \(\mu\text{m}^2\).

10.How many charges does V_m require?

A surprising and important fact: the bulk ion concentrations inside and outside the cell are essentially unchanged by the presence of V_m. The membrane acts as a thin-film capacitor with \(C_m \approx 1\;\mu\text{F/cm}^2\). Charging to \(-70\) mV requires:

\[ Q = C_m V_m = (10^{-6}\,\text{F/cm}^2)(0.07\,\text{V}) = 7\times10^{-8}\,\text{C/cm}^2 \]

Dividing by Avogadro's number and F, this is \(\sim 7\times10^{-13}\) mol/cm² of net charge — a monolayer of ions a few Angstroms thick pressed against each side of the membrane. The bulk cytoplasm remains electroneutral to \(>\)1 part in 10⁶.

Clinical connections

Hyperkalemia

Raising [K⁺]_o from 4 to 8 mM depolarizes myocardium by \(\sim 20\) mV — a cardiac arrest hazard in renal failure.

Digoxin toxicity

Inhibition of the Na⁺/K⁺ ATPase raises [Na⁺]_i, enhancing Na/Ca exchange reversal and cardiac contractility — with a narrow therapeutic window.

Periodic paralyses

Mutations in skeletal muscle Na⁺ channels or Kir2.1 shift resting V_m and cause episodic weakness.

Ischemia

Loss of ATP arrests the pump — within minutes, ion gradients dissipate, cells depolarize, and glutamate release triggers excitotoxicity.

Key equations

Nernst:

\( E_X = \dfrac{RT}{zF}\ln\dfrac{[X]_o}{[X]_i} \approx \dfrac{61.5}{z}\log_{10}\dfrac{[X]_o}{[X]_i} \) mV at 37 °C

GHK:

\( V_m = \dfrac{RT}{F}\ln\dfrac{p_K[K]_o + p_{Na}[Na]_o + p_{Cl}[Cl]_i}{p_K[K]_i + p_{Na}[Na]_i + p_{Cl}[Cl]_o} \)

Chord:

\( V_m = \dfrac{\sum_X g_X E_X}{\sum_X g_X} \)

Capacitor:

\( Q = C_m V_m,\quad C_m \approx 1\,\mu\text{F/cm}^2 \)

Donnan:

\( [K^+]_i[Cl^-]_i = [K^+]_o[Cl^-]_o \)

11.Summary diagram

References

• Nernst, W. (1889). Die elektromotorische Wirksamkeit der Ionen. Z. Phys. Chem.
• Goldman, D. E. (1943). Potential, impedance and rectification in membranes. J. Gen. Physiol. 27, 37-60.
• Hodgkin, A. L. & Katz, B. (1949). The effect of sodium ions on the electrical activity of the giant axon of the squid. J. Physiol. 108, 37-77.
• Skou, J. C. (1957). The influence of some cations on an adenosine triphosphatase from peripheral nerves. Biochim. Biophys. Acta 23, 394-401.
• Tosteson, D. C. & Hoffman, J. F. (1960). Regulation of cell volume by active cation transport in high and low potassium sheep red cells. J. Gen. Physiol. 44, 169-194.
• Morth, J. P. et al. (2007). Crystal structure of the sodium-potassium pump. Nature 450, 1043-1049.

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