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Membrane Potential &
Electrochemical Gradients

Master the forces that drive ion movement across cell membranes: concentration gradients, electrical gradients, and the Nernst equation.

MCAT Biological Foundations

What Is Membrane Potential?

Every living cell maintains a voltage difference across its plasma membrane. This membrane potential (\( V_m \)) arises because the distribution of ions is unequal on either side of the lipid bilayer.

Intracellular
High \( \text{K}^+ \), low \( \text{Na}^+ \), many organic anions (\( \text{A}^- \))
Extracellular
High \( \text{Na}^+ \), high \( \text{Cl}^- \), low \( \text{K}^+ \)

Two Forces Drive Ion Movement

Chemical Gradient
Ions diffuse from high to low concentration, just like any solute following Fick's law.
Electrical Gradient
Cations are attracted toward negative regions; anions are attracted toward positive regions.

Visualizing the Electrochemical Gradient

Membrane INTRACELLULAR EXTRACELLULAR
\( \text{K}^+ \)
\( \text{K}^+ \)
\( \text{K}^+ \)
\( \text{K}^+ \)
\( \text{Na}^+ \)
\( \text{Na}^+ \)
\( \text{Na}^+ \)
\( \text{Na}^+ \)
\( \text{A}^- \)
\( \text{A}^- \)
\( \text{Cl}^- \)
 Chemical Chemical Net Charge: − Net Charge: +

The Nernst Equation

The Nernst equation calculates the equilibrium potential (\( E_{\text{ion}} \)) for a single ion species: the voltage at which the electrical and chemical forces on that ion are perfectly balanced.

\( R \)
Gas constant: \( 8.314 \; \text{J/(mol·K)} \)
\( T \)
Temperature in Kelvin (\( 310 \; \text{K} \) at body temp)
\( z \)
Valence of ion (\( +1 \) for \(\text{Na}^+\), \( -1 \) for \(\text{Cl}^-\))
\( F \)
Faraday's constant: \( 96{,}485 \; \text{C/mol} \)

The Goldman Equation

Cells are permeable to multiple ions simultaneously. The Goldman-Hodgkin-Katz (GHK) equation accounts for the relative permeability (\( P \)) of each ion to compute the overall resting membrane potential:

Nernst vs Goldman

Nernst Goldman (GHK)
Ions considered One ion at a time Multiple ions simultaneously
Output Equilibrium potential \( E_{\text{ion}} \) Resting membrane potential \( V_m \)
Permeability Not factored in Weighted by \( P_{\text{ion}} \)
When to use Predict where one ion "wants" to be Predict actual \( V_m \) of the cell
MCAT focus Calculate \( E_{\text{K}} \), \( E_{\text{Na}} \) Explain why \( V_m \approx -70 \text{ mV} \)

The \(\text{Na}^+/\text{K}^+\) ATPase

MEMBRANE
\( \text{Na}^+/\text{K}^+ \)
ATPase
\( 3 \text{ Na}^+ \) out
\( 2 \text{ K}^+ \) in
\( 1 \text{ ATP} \to \text{ADP} + \text{P}_i \)
 INSIDE (cytoplasm) OUTSIDE (ECF)

Equilibrium Potentials of Key Ions

Ion \([\text{ion}]_{\text{in}}\) (mM) \([\text{ion}]_{\text{out}}\) (mM) \( E_{\text{ion}} \) (mV)
\(\text{K}^+\) 140 5 −90
\(\text{Na}^+\) 12 145 +67
\(\text{Cl}^-\) 4 120 −91
\(\text{Ca}^{2+}\) 0.0001 2 +130

Solving Nernst Problems Step by Step

1
Identify the ion and its valence \( z \). Is it a cation (\( z = +1, +2 \)) or anion (\( z = -1 \))?
2
Find concentrations. Determine \([\text{ion}]_{\text{out}}\) and \([\text{ion}]_{\text{in}}\) from the problem or a reference table.
3
Apply the simplified Nernst equation: \( E = \frac{61.5}{z} \log_{10} \frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}} \)
4
Check the sign. For cations: higher outside → positive \( E \). For anions: the negative \( z \) flips the sign.
5
Compare to \( V_m \). If \( V_m > E_{\text{ion}} \), the driving force pushes the cation out. If \( V_m < E_{\text{ion}} \), it pushes the cation in.

Example 1: \(\text{K}^+\) Equilibrium Potential

Given: \([\text{K}^+]_{\text{in}} = 140 \; \text{mM}\), \([\text{K}^+]_{\text{out}} = 5 \; \text{mM}\), \( z = +1 \), body temperature.

Example 2: \(\text{Na}^+\) Equilibrium Potential

Given: \([\text{Na}^+]_{\text{in}} = 12 \; \text{mM}\), \([\text{Na}^+]_{\text{out}} = 145 \; \text{mM}\), \( z = +1 \).

Example 3: \(\text{Cl}^-\) with Negative \( z \)

Given: \([\text{Cl}^-]_{\text{in}} = 4 \; \text{mM}\), \([\text{Cl}^-]_{\text{out}} = 120 \; \text{mM}\), \( z = -1 \).

Electrochemical Driving Force

The driving force on any ion is the difference between the cell's actual membrane potential and that ion's equilibrium potential:

\( V_m - E < 0 \)
Net force pushes cations inward (or anions outward)
\( V_m - E > 0 \)
Net force pushes cations outward (or anions inward)
\( V_m - E = 0 \)
Ion is at equilibrium: no net movement

Common Mistakes to Avoid

✕ Flipping the Ratio
The Nernst equation uses \[[\text{out}]/[\text{in}]\]. Putting \([\text{in}]\] on top inverts the sign of your answer.
✕ Ignoring \( z \)
For \(\text{Cl}^-\) (\( z = -1 \)) or \(\text{Ca}^{2+}\) (\( z = +2 \)), forgetting the valence gives the wrong magnitude or sign.
✕ Nernst for Whole Cell
Nernst gives the equilibrium for one ion. To find \( V_m \), you need the Goldman equation (or be told to assume one ion dominates).

Summary: Ion Channels & Membrane Effects

Channel Opens \( V_m \) Moves Toward Effect
\(\text{K}^+\) leak −90 mV Hyperpolarization
\(\text{Na}^+\) voltage-gated +67 mV Depolarization
\(\text{Cl}^-\) ligand-gated −91 mV Hyperpolarization / stabilization
\(\text{Ca}^{2+}\) voltage-gated +130 mV Strong depolarization

Question 1 of 5

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

Question 2 of 5

Using the Nernst equation at \( 37^\circ \text{C} \), what is \( E_{\text{Na}} \) if \([\text{Na}^+]_{\text{out}} = 145 \text{ mM}\) and \([\text{Na}^+]_{\text{in}} = 12 \text{ mM}\)?

Question 3 of 5

The \(\text{Na}^+/\text{K}^+\) ATPase is described as electrogenic because it:

Question 4 of 5

If the extracellular \(\text{K}^+\) concentration is increased from \( 5 \text{ mM} \) to \( 20 \text{ mM} \), what happens to \( E_{\text{K}} \) and the resting \( V_m \)?

Question 5 of 5

In the Goldman equation, anion concentrations (e.g., \(\text{Cl}^-\)) appear in the opposite position compared to cations. Why?