At rest, a neuron has P_K/P_Na ≈ 40:1. The K⁺ equilibrium potential is −90 mV and the Na⁺ equilibrium potential is +60 mV. Which statement best predicts the resting membrane potential according to the Goldman equation?
AIt equals −90 mV, since K⁺ completely dominates and the membrane potential converges exactly to E_K
BIt equals the arithmetic average of −90 and +60 mV (approximately −15 mV), since both ions contribute equally
CIt is slightly less negative than −90 mV — close to E_K but pulled a few millivolts toward E_Na by the small Na⁺ permeability
DIt equals 0 mV, since the opposing K⁺ and Na⁺ gradients exactly cancel each other out
The GHK equation weights each ion's equilibrium potential by its permeability. The large K⁺ permeability pulls V_m strongly toward E_K (−90 mV), but the small residual Na⁺ permeability through leak channels exerts a constant depolarizing pull toward E_Na (+60 mV). The result is approximately −70 mV — dominated by K⁺ but not equal to E_K. Option A is the most tempting misconception: students often equate 'K⁺ dominates' with 'V_m = E_K,' but any nonzero Na⁺ permeability prevents the membrane from reaching the K⁺ equilibrium potential. Options B and D both misunderstand the weighting — the average is not arithmetic, and the contributions do not cancel.
Question 2 Multiple Choice
During the rising phase of an action potential, voltage-gated Na⁺ channels open, increasing P_Na roughly 500-fold. What does the Goldman equation predict for the membrane potential at this moment?
AThe potential becomes more negative, since Na⁺ influx adds positive charge inside and repels the existing negative interior potential
BThe potential swings toward E_Na (+60 mV), since the massive increase in Na⁺ permeability now dominates the weighting
CThe potential stays near −70 mV, since the K⁺ concentration gradient is larger and resists displacement
DThe potential reaches exactly 0 mV, since equal inward Na⁺ and outward K⁺ currents temporarily balance
When P_Na increases 500-fold, it completely overwhelms the previously dominant K⁺ permeability. In the GHK equation, Na⁺ now carries almost all the weight, and the predicted V_m approaches E_Na (+60 mV). This is precisely the depolarizing upstroke of the action potential. The membrane does not reach +60 mV exactly because K⁺ permeability doesn't drop to zero, but the potential swings dramatically positive before Na⁺ channels inactivate. Option A has the direction wrong — Na⁺ ions carry positive charge INTO the cell, depolarizing (making more positive) the interior, not more negative.
Question 3 True / False
If the concentrations of K⁺ and Na⁺ were suddenly equalized across the membrane (same inside and outside), while most permeabilities remained unchanged, the resting membrane potential would be unaffected.
TTrue
FFalse
Answer: False
The GHK equation depends on the ratio of outside-to-inside ion concentrations inside the logarithm. If all concentrations were equalized, every logarithmic concentration ratio becomes 1, and ln(1) = 0, so the entire equation evaluates to 0 mV regardless of the permeability values. The permeabilities weight the relative influence of each ion, but there must be concentration gradients to generate a potential in the first place. Permeability without concentration difference produces no driving force — the two inputs (concentration gradients and permeabilities) are multiplicative, not additive.
Question 4 True / False
According to the Goldman equation, an ion with zero membrane permeability contributes nothing to the resting membrane potential, even if it has a steep concentration gradient across the membrane.
TTrue
FFalse
Answer: True
Permeability in the GHK equation acts as a multiplicative weight. If P_ion = 0, the ion's term drops out of both numerator and denominator entirely — its concentration ratio has no effect on V_m. This is physically sensible: if the membrane is completely impermeable to an ion, no current can flow through it, and it exerts no electrochemical driving force on the membrane voltage. Large impermeant anions like proteins and nucleic acids inside neurons contribute to the overall charge balance but not directly to the membrane potential through the GHK mechanism.
Question 5 Short Answer
Why does the Goldman equation predict a resting membrane potential of approximately −70 mV rather than the K⁺ equilibrium potential of −90 mV, even though K⁺ dominates membrane permeability at rest?
Think about your answer, then reveal below.
Model answer: Because the membrane is not exclusively permeable to K⁺ — there is a small but non-zero Na⁺ permeability through leak channels. In the GHK equation, each ion's equilibrium potential is weighted by its permeability. The high K⁺ permeability pulls V_m strongly toward E_K (−90 mV), but the small Na⁺ permeability exerts a constant depolarizing tug toward E_Na (+60 mV). The resting potential (~−70 mV) is the equilibrium where these opposing influences balance, landing closer to E_K than E_Na because K⁺ conductance is about 40 times larger, but not equal to E_K because Na⁺ conductance is not zero.
This is the key conceptual difference between the Nernst and Goldman equations. The Nernst equation gives the potential for a membrane permeable to only one ion; the Goldman equation gives the potential for a membrane with multiple competing permeabilities. At rest, K⁺ 'wins' the competition but does not win completely. This partial victory (~20 mV short of E_K) has important consequences: it means the Na-K ATPase must continuously pump to maintain the concentration gradients, since the small persistent Na⁺ leak keeps driving Na⁺ in, and the K⁺ leak drives K⁺ out.