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Membrane Potential
Ion Gradients

Understanding how electrical charge differences across the cell membrane drive every nerve impulse, muscle contraction, and heartbeat in your body.

Anatomy & Physiology

What Is Membrane Potential?

Every living cell has a voltage difference across its plasma membrane. The inside of the cell is slightly negative relative to the outside. This difference is called the membrane potential, measured in millivolts (mV).

Think of it like a tiny battery: the membrane stores electrical energy that the cell can use to send signals, transport molecules, and perform work.

The Key Ions Involved

Na⁺ Sodium
High concentration outside the cell. Wants to rush inward.
K⁺ Potassium
High concentration inside the cell. Tends to leak outward.
Cl⁻ Chloride
High concentration outside. Moves inward through specific channels.
A⁻ Anions
Large proteins trapped inside the cell. Cannot cross the membrane.

Ion Distribution Across the Membrane

Plasma Membrane Extracellular (Outside) Relatively Positive Intracellular (Inside) Relatively Negative (\( -70 \text{ mV} \)) Na⁺ Na⁺ Na⁺ Cl⁻ Cl⁻ K⁺ K⁺ K⁺ A⁻ A⁻ + -

The Electrochemical Gradient

Two forces act on every ion near the membrane. Together, they form the electrochemical gradient:

Chemical Gradient
Ions diffuse from high to low concentration (down their concentration gradient).
Electrical Gradient
Ions are attracted toward the opposite charge and repelled by the same charge.

The Nernst Equation

The Nernst equation calculates the equilibrium potential for a single ion: the voltage at which the electrical force exactly balances the concentration gradient, so there is no net movement of that ion.

\( R \)
Gas constant: \( 8.314 \text{ J/(mol·K)} \)
\( T \)
Temperature in Kelvin (\( 310 \text{ K} \) at body temp)
\( z \)
Ion charge (e.g., \( +1 \) for Na⁺, \( -1 \) for Cl⁻)
\( F \)
Faraday's constant: \( 96{,}485 \text{ C/mol} \)

The Goldman Equation

Real cells are permeable to multiple ions at once. The Goldman-Hodgkin-Katz (GHK) equation accounts for all permeable ions and their relative permeabilities:

Nernst vs. Goldman

Nernst Equation Goldman Equation
Ions Considered One ion at a time Multiple ions simultaneously
Output Equilibrium potential (\( E_{\text{ion}} \)) Resting membrane potential (\( V_m \))
Permeability Not considered Weighted by \( P \) values
Use Case Predict one ion's reversal voltage Predict actual cell voltage at rest
Clinical Value Understanding single-channel drugs Predicting overall membrane behavior

The Na⁺/K⁺ ATPase Pump

Membrane Na⁺/K⁺ ATPase Na⁺ Na⁺ Na⁺ K⁺ K⁺ ATP → ADP + P Outside Inside 3 Na⁺ out 2 K⁺ in

Why the Inside Is Negative

1
K⁺ leak channels are abundant at rest. K⁺ flows out of the cell down its concentration gradient, carrying positive charge with it.
2
Large protein anions (A⁻) are trapped inside the cell. They cannot follow K⁺ out, leaving a net negative charge behind.
3
Na⁺ channels are mostly closed at rest. Only a small Na⁺ leak partially offsets the negativity.
4
The Na⁺/K⁺ pump expels 3 positive charges for every 2 it brings in, adding a small extra negativity.

How to Calculate \( E_{\text{ion}} \)

1
Identify the ion and its charge (\( z \)). For K⁺, \( z = +1 \). For Cl⁻, \( z = -1 \).
2
Find concentrations: \([\text{ion}]_{\text{outside}}\) and \([\text{ion}]_{\text{inside}}\) in mM.
3
Plug into the simplified Nernst equation: \( E = \frac{61}{z} \log \frac{[\text{ion}]_o}{[\text{ion}]_i} \).
4
Evaluate the logarithm and multiply. Watch the sign: a negative result means the inside is more negative at equilibrium.
5
Interpret: this voltage is where the electrical force exactly cancels the concentration gradient for that ion.

Example: K⁺ Equilibrium Potential

Given typical mammalian neuron values:
\([\text{K}^+]_o = 5 \text{ mM}\), \([\text{K}^+]_i = 140 \text{ mM}\), \( z = +1 \).

Example: Na⁺ Equilibrium Potential

Given: \([\text{Na}^+]_o = 145 \text{ mM}\), \([\text{Na}^+]_i = 12 \text{ mM}\), \( z = +1 \).

Example: Cl⁻ Equilibrium Potential

Given: \([\text{Cl}^-]_o = 120 \text{ mM}\), \([\text{Cl}^-]_i = 4 \text{ mM}\), \( z = -1 \).

Understanding Driving Force

The driving force on an ion tells you which direction (and how strongly) that ion will move when a channel opens:

-90 E(K⁺) -70 V(m) E(Cl⁻) 0 +66 E(Na⁺) Na⁺ driving force: -136 mV K⁺: +18 mV

Common Mistakes to Avoid

✕ Confusing Equilibrium with Rest
The resting potential (\( V_m \)) is NOT the equilibrium potential of any single ion. It is a weighted average of all permeable ions.
✕ Forgetting the Sign of \( z \)
For anions like Cl⁻, \( z = -1 \). Forgetting the negative sign gives you an equilibrium potential with the wrong sign.
✕ Swapping Inside/Outside
The Nernst equation uses outside over inside (\( [\text{ion}]_o / [\text{ion}]_i \)). Inverting the ratio flips the sign of your answer.

Ion Summary Table

Ion Outside (mM) Inside (mM) \( E_{\text{ion}} \) (mV) Driving Force (mV)
Na⁺ 145 12 +66 -136 (inward)
K⁺ 5 140 -88 +18 (outward)
Cl⁻ 120 4 -90 +20 (inward)
Ca²⁺ 2.5 0.0001 +134 -204 (inward)

Question 1 of 5

The resting membrane potential of a typical neuron is approximately \( -70 \text{ mV} \). Which ion's equilibrium potential is closest to this value?

Question 2 of 5

The Na⁺/K⁺ ATPase pump moves ions in a 3:2 ratio. What does this mean for the membrane potential?

Question 3 of 5

Using the simplified Nernst equation at \( 37°\text{C} \), if \( [\text{K}^+]_o = 10 \text{ mM} \) and \( [\text{K}^+]_i = 100 \text{ mM} \), what is \( E_K \)?

Question 4 of 5

If the membrane suddenly became equally permeable to Na⁺ and K⁺, the membrane potential would move closest to:

Question 5 of 5

A student calculates \( E_{\text{Cl}} \) and gets \( +90 \text{ mV} \). What is the most likely error?