Membrane Potential Basics and Ion Gradients

SECTION 1

Historical Context & Motivation

The realization that living cells generate electrical signals is one of the most consequential discoveries in the history of biology and medicine. Long before scientists understood the molecular machinery responsible, early experimenters observed that biological tissue could produce and respond to electricity. The quest to explain membrane potential — the voltage difference across a cell's plasma membrane — ultimately unified concepts from physics, chemistry, and biology into a coherent framework for understanding how neurons fire, muscles contract, and cells regulate their internal environments.

1791

Galvani's "Animal Electricity"

Luigi Galvani demonstrated that frog leg muscles twitched in response to electrical stimulation, proposing an intrinsic "animal electricity" generated within living tissues themselves.

1902

Bernstein's Membrane Hypothesis

Julius Bernstein proposed that cell membranes are selectively permeable and that the resting potential arises from differential potassium ion diffusion, laying the theoretical groundwork for modern electrophysiology.

1949

Hodgkin & Katz Experiments

Using the giant squid axon, Alan Hodgkin and Bernard Katz confirmed that multiple ions — not just potassium — contribute to membrane potential, and that sodium permeability drives the action potential.

1952

Hodgkin–Huxley Model

Hodgkin and Huxley published their landmark mathematical model of the action potential, quantitatively describing how voltage-gated sodium and potassium conductances produce the nerve impulse — work that earned the 1963 Nobel Prize.

1997

Nobel Prize for Na⁺/K⁺-ATPase

Jens Christian Skou received the Nobel Prize in Chemistry for discovering sodium–potassium ATPase, the molecular pump that actively maintains the ion gradients essential for membrane potential.

The central question that drove centuries of research remains strikingly elegant: How does a thin lipid bilayer — only about 7 to 8 nanometers thick — sustain a measurable voltage difference, and why is this voltage essential for life? To answer this, we must understand the interplay between ion concentration gradients, selective membrane permeability, and the active transport mechanisms that keep the system far from thermodynamic equilibrium.

SECTION 2

Core Principles & Definitions

The membrane potential of a typical animal cell at rest hovers around −70 mV, meaning the interior of the cell is negatively charged relative to the extracellular fluid. This seemingly modest voltage is maintained by a sophisticated interplay of passive and active processes. To appreciate how and why the resting membrane potential exists, four foundational concepts must be understood in concert.

Electrochemical Gradient

Each ion experiences two simultaneous forces: a chemical gradient (concentration difference across the membrane) and an electrical gradient (voltage difference). Together, these form the electrochemical gradient that determines the net driving force on any ion.

Selective Permeability

The plasma membrane's lipid bilayer is largely impermeable to ions. Ion channels and transporters confer selective permeability — at rest the membrane is roughly 50–100 times more permeable to K⁺ than to Na⁺, which is critical for establishing the resting potential.

Equilibrium Potential

For any single ion species, there exists a specific voltage — the equilibrium potential (E_ion) — at which the electrical and chemical driving forces balance exactly, producing zero net flux of that ion across the membrane.

Na⁺/K⁺-ATPase Pump

This active transporter hydrolyzes ATP to move 3 Na⁺ out and 2 K⁺ in per cycle, maintaining the concentration gradients that passive diffusion would otherwise dissipate. The pump is electrogenic — it contributes approximately −3 to −5 mV directly to the resting potential.

✦ KEY TAKEAWAY

Think of the cell membrane as a dam separating two reservoirs at different heights. The concentration gradient is like the water-level difference (chemical potential energy), and the electrical gradient is like an electric field pushing charged water molecules. Ion channels are the sluice gates — when specific gates open, ions flow downhill along their electrochemical gradient. Meanwhile, the Na⁺/K⁺-ATPase is a pump continuously working to refill the upper reservoir, ensuring the dam never reaches equilibrium.

SECTION 3

Visualizing the Resting Membrane Potential

A clear spatial picture of ion distribution across the membrane is essential for understanding the resting potential. The diagram below shows the key ions, their approximate concentrations on each side of a typical mammalian neuron's plasma membrane, and the direction of their electrochemical driving forces at rest.

Ion distribution across a typical mammalian neuron at rest. Circle size represents relative concentration. Na⁺ (pink) is concentrated extracellularly at 145 mM versus 15 mM inside. K⁺ (cyan) is concentrated intracellularly at 150 mM versus 5 mM outside. Large intracellular organic anions (A⁻, orange) cannot cross the membrane. The Na⁺/K⁺-ATPase pump (gold box) actively maintains these gradients.

Several features of this diagram deserve emphasis. First, notice the dramatic asymmetry: potassium is roughly 30 times more concentrated inside the cell than outside, while sodium exhibits the reverse pattern at nearly a 10:1 extracellular-to-intracellular ratio. Second, note the large organic anions (A⁻) — proteins, nucleic acids, and organic phosphates — trapped inside the cell because they are too large to pass through ion channels. These impermeant anions contribute a fixed negative charge to the intracellular compartment. Third, the Na⁺/K⁺-ATPase runs continuously, consuming roughly 20–40% of a neuron's total ATP budget to maintain these gradients against the relentless tendency of ions to leak down their concentration differences.

SECTION 4

Mathematical Framework

Two equations form the quantitative backbone of membrane potential physiology. The Nernst equation predicts the equilibrium potential for a single ion species, while the Goldman–Hodgkin–Katz (GHK) equation integrates contributions from all permeant ions to predict the actual resting membrane potential. Understanding both is essential for interpreting electrophysiological data and predicting how changes in ion concentration or permeability shift membrane voltage.

The Nernst Equation

NERNST EQUATION

E_ion = (RT / zF) × ln([ion]_outside / [ion]_inside)

E_ion = equilibrium potential (V); R = gas constant (8.314 J·mol⁻¹·K⁻¹); T = temperature (K); z = ion valence (charge number, including sign); F = Faraday's constant (96,485 C·mol⁻¹); [ion]_outside and [ion]_inside = extracellular and intracellular concentrations.

At mammalian body temperature (37 °C = 310 K), the factor RT/F evaluates to approximately 26.7 mV. For monovalent cations (z = +1), converting from natural logarithm to base-10 logarithm introduces a factor of 2.303, yielding the simplified form frequently used in physiology:

NERNST EQUATION (SIMPLIFIED, 37 °C)

E_ion = (61.5 mV / z) × log₁₀([ion]_outside / [ion]_inside)

The constant 61.5 mV applies specifically at 37 °C. At 25 °C (298 K), the constant is approximately 58.2 mV. For negative ions (e.g., Cl⁻ with z = −1), the sign of z inverts the ratio.

The Goldman–Hodgkin–Katz (GHK) Equation

The Nernst equation applies to one ion at a time — but real membranes are permeable to multiple ions simultaneously. The GHK equation weights each ion's contribution by its relative membrane permeability (P), producing a single predicted membrane voltage. For the three most physiologically relevant ions:

GOLDMAN–HODGKIN–KATZ EQUATION

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) )

V_m = membrane potential; P_K, P_Na, P_Cl = relative permeabilities for each ion. Note that Cl⁻ (an anion) has its inside and outside concentrations swapped in the numerator and denominator to account for its negative charge.

📐 Why Cl⁻ Terms Are Flipped

Because chloride carries a negative charge (z = −1), the Nernst equation for Cl⁻ naturally produces a sign inversion. Rather than carrying a negative z through the GHK derivation, the convention is to place [Cl⁻]_i in the numerator and [Cl⁻]_o in the denominator — the reverse of the cation pattern. This is not a special case; it is a direct algebraic consequence of z = −1.

SECTION 5

Ion Channels, Pumps, and Gradient Maintenance

Ion gradients do not maintain themselves. If the membrane's leak channels allowed ions to diffuse freely for long enough without active compensation, concentration differences would eventually collapse and the membrane potential would dissipate. The cell employs distinct molecular machines for passive ion movement (channels) and active gradient restoration (pumps and transporters). The diagram below classifies the major pathways by which ions cross the plasma membrane.

Overview of passive and active membrane transport mechanisms. Passive channels (left, cyan/blue/green/pink borders) allow ions to flow down their electrochemical gradient without energy input. Active transporters (right, gold/orange/red borders) use ATP hydrolysis — directly or indirectly — to move ions against their gradient, maintaining the concentration asymmetries essential for the resting potential.

Approximate ion concentrations and equilibrium potentials for a typical mammalian neuron at 37 °C

Ion	[Extracellular] (mM)	[Intracellular] (mM)	E_ion (37 °C)	Driving Force at V_m = −70 mV
Na⁺	145	15	+60 mV	−130 mV (strong inward)
K⁺	5	150	−91 mV	+21 mV (mild outward)
Cl⁻	120	10	−66 mV	−4 mV (slight inward)
Ca²⁺	2	0.0001	+132 mV	−202 mV (very strong inward)

The driving force on each ion is calculated as V_m − E_ion. A negative driving force on a cation means it is driven inward; a positive driving force drives it outward. Notice that the resting membrane potential (−70 mV) sits much closer to E_K (−91 mV) than to E_Na (+60 mV), reflecting the membrane's far greater resting permeability to potassium. If sodium permeability were to suddenly increase — as happens during an action potential — the membrane potential would shift rapidly toward +60 mV.

SECTION 6

Worked Example: Calculating E_K and V_m

Let us apply the Nernst equation to calculate the equilibrium potential for potassium and then use the GHK equation to estimate the resting membrane potential of a neuron, given the following typical mammalian values at 37 °C.

Calculating E_K and Resting V_m

Step 1 — State the Given Values[K⁺]_o = 5 mM, [K⁺]_i = 150 mM, [Na⁺]_o = 145 mM, [Na⁺]_i = 15 mM, [Cl⁻]_o = 120 mM, [Cl⁻]_i = 10 mM. Temperature = 37 °C. Relative permeabilities: P_K : P_Na : P_Cl = 1 : 0.04 : 0.45.

Step 2 — Calculate E_K Using the Nernst EquationE_K = (61.5 mV / +1) × log₁₀(5 / 150) = 61.5 × log₁₀(0.0333) = 61.5 × (−1.477).

E_K ≈ −90.8 mV

Step 3 — Build the GHK NumeratorNumerator = P_K[K⁺]_o + P_Na[Na⁺]_o + P_Cl[Cl⁻]_i = (1)(5) + (0.04)(145) + (0.45)(10) = 5 + 5.8 + 4.5 = 15.3

Step 4 — Build the GHK DenominatorDenominator = P_K[K⁺]_i + P_Na[Na⁺]_i + P_Cl[Cl⁻]_o = (1)(150) + (0.04)(15) + (0.45)(120) = 150 + 0.6 + 54 = 204.6

Step 5 — Compute V_mV_m = 61.5 × log₁₀(15.3 / 204.6) = 61.5 × log₁₀(0.0748) = 61.5 × (−1.126).

V_m ≈ −69.3 mV, which is consistent with the typical resting membrane potential of approximately −70 mV.

💡 Interpreting the Result

The calculated V_m of −69.3 mV is very close to E_K (−90.8 mV) but far from E_Na (+60 mV). This confirms that the resting membrane is dominated by potassium permeability. The slight depolarization from E_K toward zero is caused by the small but non-negligible sodium and chloride conductances.

SECTION 7

Nernst vs. GHK: Strengths and Limitations

Both the Nernst and GHK equations are indispensable tools in electrophysiology, but they apply under different assumptions and conditions. Understanding when each is appropriate — and where each falls short — is critical for correctly interpreting experimental data and clinical measurements.

Comparison of the Nernst and Goldman–Hodgkin–Katz equations

Feature	Nernst Equation	GHK Equation
Number of ions	Single ion species at a time	Multiple ions simultaneously
Output	Equilibrium potential (E_ion)	Resting membrane potential (V_m)
Permeability data needed?	No — only concentrations and valence	Yes — requires relative permeability ratios
Accounts for active transport?	No	Indirectly (through maintained concentrations), but does not model pump current
Assumes steady state?	Assumes thermodynamic equilibrium for one ion	Assumes steady-state (constant field) for all ions
Clinical use	Quick prediction of one ion's reversal potential	Prediction of net membrane voltage under varying conditions

✦ KEY TAKEAWAY

Think of the Nernst equation as a specialized instrument — like a single-channel oscilloscope that shows you the voltage contribution of one ion in isolation. The GHK equation is more like a mixer board in a recording studio: it takes the separate signals (each ion's concentration and permeability) and blends them into a single composite output — the actual membrane potential. Neither equation accounts for the electrogenic pump current directly, which is typically small (3–5 mV) but becomes clinically relevant when pump activity is pharmacologically inhibited (e.g., by cardiac glycosides like digoxin).

SECTION 8

Connection to Action Potentials and Clinical Physiology

The resting membrane potential is not merely an academic abstraction — it is the loaded spring from which all electrical signaling originates. When a neuron receives sufficient synaptic input to depolarize the membrane from −70 mV past the threshold (approximately −55 mV), voltage-gated Na⁺ channels open explosively. The membrane potential surges toward E_Na (+60 mV), producing the upstroke of the action potential. The subsequent opening of delayed-rectifier K⁺ channels repolarizes the membrane back toward E_K. This entire cycle depends fundamentally on the ion gradients established at rest.

Resting potential concepts vs. action potential concepts

Concept	Resting Membrane Potential (This Lesson)	Action Potential (Advanced)
State	Steady-state; ion fluxes balanced	Transient; rapid sequential changes in permeability
Dominant permeability	K⁺ (leak channels)	Na⁺ (depolarization) then K⁺ (repolarization)
V_m range	≈ −70 mV (stable)	−70 mV → +30 mV → −80 mV (1–2 ms)
Key equation	Goldman–Hodgkin–Katz	Hodgkin–Huxley differential model
Clinical disruption	Hyperkalemia shifts V_m toward threshold	Local anesthetics block Na⁺ channels, preventing firing

Clinically, perturbations to ion gradients produce measurable — and often dangerous — consequences. Hyperkalemia (elevated extracellular K⁺) reduces the K⁺ concentration gradient, making E_K less negative and depolarizing the resting potential. In cardiac myocytes, this can impair repolarization and trigger lethal arrhythmias. Conversely, hypokalemia hyperpolarizes cells, making them harder to excite and potentially causing muscle weakness and cardiac conduction defects. A solid grasp of ion gradients and the Nernst equation is therefore essential for interpreting arterial blood gas panels and electrolyte lab values in clinical practice.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

The resting membrane potential of a typical neuron is approximately −70 mV, yet the equilibrium potential for potassium is about −90 mV. Explain, in terms of ion permeabilities, why V_m at rest does not equal E_K.

PROBLEM 2 — BASIC CALCULATION

Using the simplified Nernst equation at 37 °C, calculate the equilibrium potential for Na⁺ given [Na⁺]_o = 145 mM and [Na⁺]_i = 15 mM.

PROBLEM 3 — INTERMEDIATE

A patient's lab work shows serum K⁺ = 7.0 mM (normal ≈ 5 mM). Assuming intracellular [K⁺] remains 150 mM, calculate the new E_K at 37 °C and predict the qualitative effect on the resting membrane potential and cardiac excitability.

PROBLEM 4 — APPLIED

Ouabain is a cardiac glycoside that inhibits the Na⁺/K⁺-ATPase. Predict the short-term and long-term effects of ouabain on (a) intracellular Na⁺ concentration, (b) the Na⁺ concentration gradient, (c) E_Na, and (d) the resting membrane potential. Explain your reasoning using concepts from this lesson.

PROBLEM 5 — CRITICAL THINKING

The GHK equation assumes a constant electric field across the membrane (the "constant field assumption"). Under what physiological conditions might this assumption break down, and how would violations affect the accuracy of V_m predictions? Consider the presence of fixed charges within the membrane and the dynamic behavior of voltage-gated channels.

SUMMARY

Lesson Summary

The resting membrane potential of approximately −70 mV in a typical neuron arises from an unequal distribution of ions across the selectively permeable plasma membrane. Potassium is concentrated inside the cell (~150 mM intracellular vs. ~5 mM extracellular), while sodium is concentrated outside (~145 mM vs. ~15 mM). Because the resting membrane is far more permeable to K⁺ than to Na⁺, the resting V_m sits close to the potassium equilibrium potential (E_K ≈ −91 mV) but is partially depolarized by sodium leak. The Na⁺/K⁺-ATPase actively maintains these gradients by pumping 3 Na⁺ out and 2 K⁺ in per ATP consumed.

The Nernst equation calculates the equilibrium potential for a single ion, while the Goldman–Hodgkin–Katz equation integrates all permeant ions to predict the actual membrane voltage. The driving force on any ion is V_m − E_ion, and understanding these forces is essential for predicting how changes in ion concentration (e.g., hyperkalemia) or permeability (e.g., channel opening during an action potential) will shift the membrane potential. These foundational principles underpin all of neural, cardiac, and muscular electrophysiology.