Explain the resting membrane potential

Overview

The resting membrane potential is the electrical voltage difference across a neuron's plasma membrane when the cell is not actively transmitting a signal, typically around -70 mV (inside negative relative to outside). This voltage exists because of unequal distribution of ions and selective membrane permeability.

[!intuition] Core Intuition

Think of the neuron's membrane like a dam holding back water at different heights on each side. The "water" here is charged ions (Na⁺, K⁺, Cl⁻, A⁻), and they naturally want to flow to balance out. But the membrane is a selective dam – it has specific gates (channels) that let some ions through more easily than others.

WHY does voltage exist? Two competing forces:

1. Chemical gradient: Ions want to diffuse from high concentration → low concentration (entropy)
2. Electrical gradient: Opposite charges attract, like charges repel

When these two forces balance for a given ion, it reaches equilibrium potential. The resting membrane potential is a weighted average of all ion equilibrium potentials, weighted by how permeable the membrane is to each ion.

[!definition] Key Definitions

Resting Membrane Potential (RMP): The steady-state voltage across a neuron's membrane at rest, inside relative to outside. Typically -60 to -80 mV depending on cell type.

Equilibrium Potential (Eᵢₒₙ): The voltage at which the electrical force on an ion exactly balances its chemical (concentration) gradient, so there's no net movement. Calculated by the Nernst equation.

Selective Permeability: The membrane's property of allowing some ions to cross more easily than others, determined by the number and state of ion channels.

[!formula] The Nernst Equation — Deriving from First Principles

GOAL: Find the voltage (measured as $V_{\text{inside}} - V_{\text{outside}}$) where chemical and electrical forces on an ion balance.

Step 1: Chemical Force (Diffusion)

Ions diffuse from high → low concentration. Consider moving one ion from inside to outside. The chemical free-energy change is:

$W_{\text{chem}} = RT \ln\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right)$

WHY this form? Thermodynamics tells us the free energy change for moving a particle from concentration C₁ to C₂ is $RT \ln(C_2/C_1)$. Moving from inside (C₁) to outside (C₂) gives $RT\ln([\text{ion}]_{\text{out}}/[\text{ion}]_{\text{in}})$. Here R is the gas constant (8.314 J/(mol·K)), T is absolute temperature.

Step 2: Electrical Force

Moving a charged ion from inside to outside across the membrane voltage $V = V_{\text{in}} - V_{\text{out}}$ requires electrical work:

$W_{\text{elec}} = -zFV$

where:

• z = charge number (Na⁺ has z=+1, Ca²⁺ has z=+2, Cl⁻ has z=-1)
• F = Faraday constant (96,485 C/mol) — charge per mole of electrons
• V = membrane voltage ($V_{\text{in}} - V_{\text{out}}$)

WHY the minus sign? Work = charge × (potential change). Moving a positive charge from inside (potential $V_{\text{in}}$) to outside (potential $V_{\text{out}}$) means the potential drops by $(V_{\text{out}}-V_{\text{in}}) = -V$, so the work done is $zF(V_{\text{out}}-V_{\text{in}}) = -zFV$.

Step 3: Equilibrium Condition

At equilibrium, no NET work is done moving the ion across:

$W_{\text{chem}} + W_{\text{elec}} = 0$

$RT \ln\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right) - zFV = 0$

Step 4: Solve for V (Equilibrium Potential)

$E_{\text{ion}} = \frac{RT}{zF} \ln\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right)$

This is the correct sign convention for $V = V_{\text{in}} - V_{\text{out}}$: the ratio is outside over inside.

Converting to base-10 log (ln x = 2.303 log₁₀ x) and plugging in constants at 37°C (310 K):

$E_{\text{ion}} = \frac{2.303 \times 8.314 \times 310}{z \times 96485} \log_{10}\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right) = \frac{61.5 \text{ mV}}{z} \log_{10}\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right)$

This is the Nernst equation for 37°C. Note the clear division by z.

[!example] Worked Example 1: Potassium Equilibrium

Given typical concentrations:

• [K⁺]ᵢₙ = 140 mM
• [K⁺]ₒᵤₜ = 5 mM
• z = +1

Calculate E_K (using the correct out/in ratio):

$E_K = \frac{61.5}{+1} \log_{10}\left(\frac{[K^+]_{out}}{[K^+]_{in}}\right) = 61.5 \log_{10}\left(\frac{5}{140}\right) = 61.5 \log_{10}(0.0357) = 61.5 \times (-1.447) = -89 \text{ mV}$

Negative — inside negative! This makes sense: K⁺ is highly concentrated inside, so it tends to leak out. As positive charge leaves, the inside becomes negative. E_K is the voltage (about -89 mV, inside negative) at which the inward electrical pull exactly balances the outward diffusion, so net K⁺ flow stops.

Key insight: If K⁺ were the only permeant ion, the RMP would be -89 mV inside. Real RMP (-70 mV) sits near E_K but not exactly at it because other ions (especially Na⁺) also contribute.

[!example] Worked Example 2: Sodium Equilibrium

Given:

• [Na⁺]ᵢₙ = 12 mM
• [Na⁺]ₒᵤₜ = 145 mM
• z = +1

Apply the SAME formula (out/in ratio — no switching!):

$E_{Na} = \frac{61.5}{+1} \log_{10}\left(\frac{[Na^+]_{out}}{[Na^+]_{in}}\right) = 61.5 \log_{10}\left(\frac{145}{12}\right) = 61.5 \log_{10}(12.08) = 61.5 \times 1.082 = +66.5 \text{ mV}$

Interpretation: E_Na is positive (~+66.5 mV, inside positive). Na⁺ is concentrated outside, so it tends to rush in. To stop that inward flow you'd need the inside to be strongly positive (+66.5 mV) to electrically repel the incoming Na⁺.

Driving force at rest: At rest the inside is at -70 mV, but E_Na is +66.5 mV. The difference is a huge driving force (~136.5 mV) pushing Na⁺ to enter. Why doesn't Na⁺ flood in? Because Na⁺ channels are mostly CLOSED at rest – low permeability.

[!formula] The Goldman-Hodgkin-Katz (GHK) Equation

WHY do we need this? The Nernst equation gives equilibrium for one ion. But the real membrane is permeable to multiple ions simultaneously. The GHK equation gives the actual membrane potential as a weighted average:

$V_m = \frac{RT}{F} \ln\left(\frac{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}}\right)$

Derivation sketch (full derivation requires solving differential equations):

1. Goldman assumed constant electric field across the membrane
2. For each ion, the flux depends on both concentration gradient and voltage gradient
3. At steady state (resting), there's no net current (though individual ions move)
4. Setting total current = 0 and solving gives the GHK equation

Simplified form at 37°C:

$V_m = 61.5 \log_{10}\left(\frac{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}}\right)$

Note: Cl⁻ terms are "flipped" (in on top, out on bottom) because it's negatively charged (z = -1).

[!example] Worked Example 3: Calculating Resting Potential

Typical values:

• [K⁺]ᵢₙ = 140 mM, [K⁺]ₒᵤₜ = 5 mM
• [Na⁺]ᵢₙ = 12 mM, [Na⁺]ₒᵤₜ = 145 mM
• [Cl⁻]ᵢₙ = 4 mM, [Cl⁻]ₒᵤₜ = 110 mM
• Relative permeabilities: P_K : P_Na : P_Cl = 1 : 0.04 : 0.45

WHY these permeabilities? At rest, there are many "leak" K⁺ channels open, few Na⁺ channels, and moderate Cl⁻ permeability.

Calculate:

Numerator: $1 \times 5 + 0.04 \times 145 + 0.45 \times 4 = 5 + 5.8 + 1.8 = 12.6$

Denominator: $1 \times 140 + 0.04 \times 12 + 0.45 \times 110 = 140 + 0.48 + 49.5 = 189.98$

$V_m = 61.5 \log_{10}\left(\frac{12.6}{189.98}\right) = 61.5 \log_{10}(0.0663) = 61.5 \times (-1.178) = -72.5 \text{ mV}$

This matches typical measured resting potentials of -70 mV!

What this tells us:

• RMP is dominated by K⁺ (because P_K is highest), so it sits close to E_K (-89 mV)
• But it's pulled slightly positive of E_K by Na⁺ influx and Cl⁻ contributions
• The membrane is a "weighted average" machine

[!intuition] The Role of the Na⁺/K⁺-ATPase Pump

WAIT: If K⁺ leaks out and Na⁺ leaks in, won't the gradients eventually collapse?

YES – and that's where the sodium-potassium pump comes in. This is an active transporter (uses ATP) that pumps:

• 3 Na⁺ out for every 2 K⁺ in

WHY does this matter?

1. Maintains concentration gradients: Restores the ions that leaked
2. Contributes directly to voltage: Because it's 3:2 (not 1:1), it's electrogenic – it makes the inside slightly more negative (contributes about -5 to -10 mV to RMP)

Analogy: The pump is like a sump pump in a leaky basement. Water (ions) naturally seeps in, but the pump continuously removes it to maintain the gradient.

[!mistake] Common Misconception: "The Pump Creates the Voltage"

Wrong idea: "The Na⁺/K⁺ pump directly generates the -70 mV."

Why it feels right: The pump does move charge and uses energy, so it seems like it's the "battery."

The truth: The concentration gradients created by the pump are the battery. The leak channels (especially K⁺ leak channels) allow ions to flow down those gradients, which separates charge and creates voltage.

Analogy: The pump is like lifting water to create a height difference. The voltage is like the pressure at the bottom when you let some water flow down through a pipe.

Evidence: If you:

1. Block the pump with ouabain → gradients slowly dissipate, voltage decays over minutes
2. Block K⁺ channels with TEA → voltage immediately shifts toward E_Na (near 0 mV)

The immediate voltage is from leak channels; the pump maintains gradients for the long term.

[!mistake] Common Misconception: "Sign of the Nernst Equation Doesn't Matter"

Wrong idea: "You can write the ratio as in/out or out/in, whatever's convenient."

Why it feels right: Both give the same magnitude, and $\ln(a/b) = -\ln(b/a)$, so it seems interchangeable.

The truth: The sign must match your voltage convention. Since RMP is defined as $V_{\text{inside}} - V_{\text{outside}}$, the Nernst equation must use the out/in ratio: $E = \frac{61.5}{z}\log_{10}([\text{ion}]_{out}/[\text{ion}]_{in})$. Flip the ratio and you get the wrong sign — e.g. E_K would come out +89 mV instead of the correct -89 mV, which is physically backwards (K⁺ leaves and makes inside negative).

Fix: Always use out over in, and keep the $1/z$ factor so Cl⁻ (z = -1) flips correctly.

[!recall]- Explain It to a 12-Year-Old

Imagine your cell is like a special club with a bouncer (the membrane) at the door. Inside the club, there's way more potassium (K⁺) than outside, but way more sodium (Na⁺) outside than inside.

Now, the bouncer lets K⁺ people walk out easily (leak channels), but keeps the door mostly shut for Na⁺ people. When K⁺ people leave, they take their positive charge with them – so the inside becomes negative (like when your friends leave a party, it feels emptier).

But wait – if everyone leaves, won't the inside and outside become the same? That's where the "pump" comes in (the Na⁺/K⁺ pump). It's like a doorman who constantly kicks Na⁺ people out and brings K⁺ people back in, using energy (ATP) to do it. This keeps the "crowd difference" going forever.

The voltage (-70 mV) is like the "emptiness feeling" inside the club compared to outside. It's negative because more positive charges left than came in. This voltage is super important – it's what lets your neurons send messages (action potentials)!

[!mnemonic] Memory Aid: "K⁺ Leaks Out, Na⁺ Wants In"

"K-Leaks-Negative":

• K⁺ channels are open at rest (leak channels)
• K⁺ leaks out (high inside → low outside)
• Takes positive charge with it → inside becomes negative (E_K ≈ -89 mV)

"Pump's a 3-2-Outer":

• Pump moves 3 Na⁺ OUT, 2 K⁺ IN
• Net: 1 positive charge OUT per cycle
• Helps maintain negative inside

"OUT-over-IN, keep the sign":

• Nernst ratio is [out] / [in]
• Keeps equilibrium potentials with the correct sign (E_K negative, E_Na positive)

Key Mechanisms Summary

1. Ion gradients (high K⁺ inside, high Na⁺ outside) are maintained by Na⁺/K⁺-ATPase
2. Selective permeability (more K⁺ leak channels open than Na⁺) allows K⁺ to leave
3. Charge separation (positive K⁺ exits) makes inside negative
4. Equilibrium is reached when electrical pull-back on K⁺ balances diffusion outward (E_K ≈ -89 mV)
5. Result: Resting membrane potential of approximately -70 mV

Connections

• Nernst Equation — equilibrium potential for single ions
• Goldman-Hodgkin-Katz Equation — multi-ion membrane potential
• Sodium-Potassium Pump — maintains concentration gradients
• Action Potential — uses RMP as baseline for rapid depolarization
• Ion Channels — leak channels determine resting permeability
• Membrane Transport — active vs passive transport mechanisms
• Electrochemical Gradient — combined chemical + electrical driving forces

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