17.09.01 · mol-cell-bio / cellular-neuroscience

Resting membrane potential and ion channels

shipped3 tiersLean: none

Anchor (Master): Hille, *Ion Channels of Excitable Membranes* (3rd ed., Sinauer 2001), Ch. 1-4; Hodgkin & Huxley 1952; Nernst 1888; Goldman 1943

Intuition [Beginner]

Every cell has an electrical charge difference across its membrane — the membrane potential. The inside of the cell is negative relative to the outside. In neurons, this resting potential is about -70 millivolts (mV).

This voltage exists because of three things. First, ion concentrations are unequal inside and out. Potassium (K+) is high inside; sodium (Na+) is high outside. Second, the membrane is selectively permeable — at rest, it lets K+ pass much more than Na+. Third, the sodium-potassium pump constantly exports 3 Na+ and imports 2 K+, using ATP, to maintain the concentration gradients.

Because the membrane at rest is mostly permeable to K+, potassium leaks out through potassium leak channels, driven by its concentration gradient. Each K+ that leaves carries one positive charge, making the inside more negative. This builds up an electrical force that pulls K+ back in. At the Nernst equilibrium, the concentration gradient pushing K+ out exactly balances the electrical gradient pulling it in. For K+, this equilibrium is about -89 mV.

The actual resting potential (-70 mV) is less negative because the membrane is not perfectly selective — some Na+ leaks in, partially offsetting the K+ outward current. The Goldman equation accounts for all permeable ions.

Visual [Beginner]

Picture the cell membrane as a wall. On the outside: high Na+ (145 mM), low K+ (5 mM). On the inside: low Na+ (12 mM), high K+ (140 mM). The wall has small doors (ion channels). At rest, the K+ doors are open (leak channels), the Na+ doors are mostly closed.

K+ ions stream out through the open doors, carrying positive charge. The inside becomes negative. The negativity pulls K+ back. When outward chemical push equals inward electrical pull, equilibrium is reached.

Worked example [Beginner]

Calculate the Nernst potential for K+ at body temperature (37 C = 310 K):

$E_{K} = \frac{R T}{z F} ln \frac{[ K ^{+} ] _{out}}{[ K ^{+} ] _{in}}$

With $R = 8.314$ J/(mol K), $T = 310$ K, $z = + 1$ for K+, $F = 96, 485$ C/mol:

$E_{K} = \frac{8.314 \times 310}{1 \times 96485} ln \frac{5}{140} = 0.0267 \times ln (0.0357)$

$= 0.0267 \times (- 3.33) = - 0.089$ V = -89 mV.

At -89 mV, the electrical force pulling K+ in exactly balances the concentration force pushing K+ out. The resting potential is -70 mV (less negative), meaning the membrane is not at K+ equilibrium — there is a small net K+ efflux at rest, balanced by the Na+/K+ pump.

What this tells us: the resting potential is close to the K+ equilibrium because the membrane is most permeable to K+ at rest, but small contributions from Na+ and Cl- pull it away from the pure K+ value.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The resting membrane potential ( $V_{m}$ ) is the electrical potential difference across the plasma membrane of a cell at steady state, resulting from the unequal distribution of ions and the selective permeability of the membrane.

The Nernst equation

For a single ion species X with charge z at electrochemical equilibrium ^{[Nernst 1888]}:

$E_{X} = \frac{R T}{z F} ln \frac{[ X ] _{out}}{[ X ] _{in}}$

At 37 C, $R T / F \approx 26.7$ mV. For monovalent ions (z = 1), converting to log base 10:

$E_{X} = 61.5 lo g_{10} \frac{[ X ] _{out}}{[ X ] _{in}} mV$

Typical equilibrium potentials (mammalian neuron, 37 C):

Ion	[out] (mM)	[in] (mM)	$E_{X}$ (mV)
K+	5	140	-89
Na+	145	12	+67
Cl-	110	4	-89
Ca2+	2	0.0001	+126

The Goldman-Hodgkin-Katz (GHK) equation

When the membrane is permeable to multiple ions, the resting potential is a weighted average of their equilibrium potentials ^{[Goldman 1943]}:

$V_{m} = \frac{R T}{F} ln \frac{P _{K} [ K ^{+} ] _{out} + P _{N a} [ N a ^{+} ] _{out} + P _{C l} [ C l ^{-} ] _{in}}{P _{K} [ K ^{+} ] _{in} + P _{N a} [ N a ^{+} ] _{in} + P _{C l} [ C l ^{-} ] _{out}}$

where $P_{K}$ , $P_{N a}$ , $P_{C l}$ are the membrane permeability coefficients for each ion. At rest, $P_{K} ≫ P_{N a} ≫ P_{C l}$ , with the ratio $P_{K} : P_{N a} : P_{C l} \approx 100 : 1 : 10$ in a typical neuron. This is why $V_{m}$ is close to $E_{K}$ .

For the typical resting neuron: $V_{m} \approx - 70$ mV, between $E_{K} = - 89$ mV and $E_{N a} = + 67$ mV, pulled toward $E_{K}$ by the high K+ permeability.

Ion channels

Ion channels are transmembrane proteins that form aqueous pores through which specific ions pass. They are classified by:

Gating mechanism:

Leak channels: Constitutively open (or weakly voltage-sensitive). K+ leak channels (KCNK/TASK/TREK families) set the resting potential.
Voltage-gated channels: Open or close in response to changes in membrane potential. Key families: NaV (voltage-gated Na+), KV (voltage-gated K+), CaV (voltage-gated Ca2+). Contain voltage-sensor domains (S4 helix with positively charged arginine residues).
Ligand-gated channels: Open when a specific molecule binds. Nicotinic acetylcholine receptor (nAChR), GABA-A receptor, glutamate receptors (AMPA, NMDA).
Mechanically-gated channels: Respond to membrane stretch or tension.

Selectivity:

K+ channels are ~10,000-fold selective for K+ over Na+. The selectivity filter contains a signature sequence (TVGYG) with backbone carbonyl oxygens that coordinate dehydrated K+ ions. Na+ is too small to interact optimally with these coordination sites.
Na+ channels select for Na+ over K+ using a narrow selectivity filter with conserved aspartate, glutamate, lysine, and alanine (DEKA motif).
Ca2+ channels use a ring of glutamate residues (EEEE motif) that preferentially binds Ca2+ over Na+ by its higher charge density.

Counterexamples to common slips

The Na+/K+ pump directly generates the membrane potential. The pump creates the concentration gradients, but the potential is generated by K+ diffusion through leak channels. Pump inhibition (ouabain) depolarizes slowly over minutes as gradients run down, not instantly.
Ions flow freely through any open channel. Ion channels are selective and gated. K+ channels exclude Na+ by 10,000-fold despite Na+ being smaller. Channels open only when their specific gating stimulus is present (voltage change, ligand binding, mechanical force).
The cell interior is negative because of negatively charged proteins. The charge separation is a surface phenomenon involving only ~~$1 0^{- 7}$ M of uncompensated ions in a thin boundary layer (~~1 nm) at each membrane face. The bulk cytoplasm is electrically neutral. The potential arises from the distribution of a tiny number of charges across the membrane capacitor, not from a bulk charge imbalance.

Key theorem with proof [Intermediate+]

Theorem (Goldman-Hodgkin-Katz voltage equation). Consider a membrane of thickness $δ$ at temperature $T$ , permeable to the monovalent ions K+, Na+, and Cl- with permeability coefficients $P_{K}$ , $P_{N a}$ , $P_{C l}$ . Under the constant-field assumption (the electric field $V_{m} / δ$ is uniform across the membrane), the steady-state membrane potential at zero net current is

$V_{m} = \frac{R T}{F} ln \frac{P _{K} [ K ^{+} ] _{out} + P _{N a} [ N a ^{+} ] _{out} + P _{C l} [ C l ^{-} ] _{in}}{P _{K} [ K ^{+} ] _{in} + P _{N a} [ N a ^{+} ] _{in} + P _{C l} [ C l ^{-} ] _{out}} .$

Proof. The electrodiffusive flux of ion X through the membrane obeys the Nernst-Planck equation:

$J_{X} = - P_{X} (\frac{d [ X ]}{d x} + \frac{z _{X} F}{R T} [X] \frac{d V}{d x})$

Under the constant-field assumption, $d V / d x = - V_{m} / δ$ (constant). Writing $u := F V_{m} / (R T)$ , the equation becomes a first-order linear ODE in $[X] (x)$ :

$\frac{d [ X ]}{d x} - \frac{z _{X} u}{δ} [X] = - \frac{J _{X}}{P _{X}}$

Multiplying by the integrating factor $e^{- z_{X} ux / δ}$ and integrating from $x = 0$ (outside) to $x = δ$ (inside):

$[X]_{in} e^{- z_{X} u} - [X]_{out} = - \frac{J _{X} δ}{P _{X} z _{X} u} (1 - e^{- z_{X} u})$

Solving for $J_{X}$ and computing the ionic current $I_{X} = z_{X} F J_{X}$ :

$I_{X} = \frac{P _{X} z _{X}^{2} F ^{2} V _{m}}{R T} \cdot \frac{[ X ] _{out} - [ X ] _{in} e ^{- z_{X} u}}{1 - e ^{- z_{X} u}}$

At steady state with zero net current, $I_{K} + I_{N a} + I_{C l} = 0$ . For the monovalent ions ( $z_{K} = z_{N a} = + 1$ , $z_{C l} = - 1$ ), substituting and collecting terms with $e^{- u}$ :

$P_{K} [K^{+}]_{out} + P_{N a} [N a^{+}]_{out} + P_{C l} [C l^{-}]_{in} = (P_{K} [K^{+}]_{in} + P_{N a} [N a^{+}]_{in} + P_{C l} [C l^{-}]_{out}) e^{- u}$

Taking logarithms and substituting $u = F V_{m} / (R T)$ :

Note the reversal of Cl- concentrations (inside in numerator, outside in denominator) because Cl- carries negative charge. $□$

Bridge. The GHK equation builds toward 17.09.02 pending, where transient increases in $P_{Na}$ during voltage-gated sodium channel opening shift $V_{m}$ from its resting value near $E_{K}$ toward $E_{Na}$ — the upstroke of the action potential. The foundational reason the resting potential sits near $E_{K}$ is the hundred-fold dominance of $P_{K}$ at rest, and this is exactly the content of the GHK weighted average: the permeability ratio determines which equilibrium potential dominates. The equation generalises to any number of permeant ion species and appears again in 11.04.01 pending as the biological application of the canonical-ensemble equilibrium calculation applied to a multi-ion electrochemical system.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Ouabain is a drug that inhibits the Na+/K+ pump. Predict what happens to the resting membrane potential over time after ouabain is applied. Consider both the immediate and long-term effects.

Hint

The pump maintains concentration gradients. Without it, what happens to those gradients?

Answer

Immediate effect: Very little change in membrane potential. The resting potential is primarily set by K+ leak channels and the K+ gradient, and the gradient changes slowly. The -4 mV electrogenic contribution of the pump is lost, causing a slight depolarization (~4 mV).

Long-term effect (minutes to hours): Without the pump, the Na+ and K+ gradients gradually run down. Na+ leaks in, K+ leaks out. Intracellular K+ decreases, extracellular K+ increases, and $E_{K}$ becomes less negative. The resting potential slowly depolarizes toward 0 mV as the gradients collapse, eventually leading to cell death.

Exercise 5 (medium, numeric).

A patch of neuronal membrane has specific capacitance $C_{m} = 1$ $μ$ F/cm $^{2}$ and a resting K+ conductance of $g_{K} = 0.5$ mS/cm $^{2}$ . Calculate the membrane time constant $τ_{m}$ . If a small current step is injected, after how many time constants has the voltage reached 95% of its final value?

Hint

$τ_{m} = C_{m} / g_{m}$ where $g_{m}$ is the total membrane conductance. For the exponential charging curve, $1 - e^{- t / τ} = 0.95$ gives $t / τ$ .

Answer

$τ_{m} = C_{m} / g_{K} = 1 μ F/cm^{2} /0.5 mS/cm^{2} = 2$ ms.

For exponential charging, $V (t) = V_{\infty} (1 - e^{- t / τ})$ . Setting $V (t) = 0.95 V_{\infty}$ : $1 - e^{- t / τ} = 0.95$ , so $e^{- t / τ} = 0.05$ , giving $t / τ = - ln (0.05) = 3.0$ .

The voltage reaches 95% of its final value after 3 time constants (~6 ms for this membrane). The membrane time constant determines how rapidly the cell integrates synaptic inputs.

Exercise 6 (hard, symbolic).

K+ channels are 10,000-fold selective for K+ over Na+, despite Na+ being smaller than K+. Explain the structural basis for this selectivity.

Hint

Both ions must shed their hydration shell to pass through. What coordinates the ion in the filter?

Answer

K+ and Na+ both travel through ion channels as dehydrated ions — they must shed their water shell to enter the narrow selectivity filter. In K+ channels, the selectivity filter contains four binding sites, each formed by a ring of backbone carbonyl oxygen atoms from the signature TVGYG sequence. These oxygens are positioned at exactly the right distance (~2.8 A) to coordinate a dehydrated K+ ion (radius 1.33 A) with optimal geometry, replacing its water shell.

Na+ (radius 0.95 A) is too small to contact all the carbonyl oxygens simultaneously. The "snug fit" that makes K+ coordination energetically favorable does not work for Na+: Na+ would sit loosely in the K+-sized binding site, with fewer favorable interactions than it had with water. The energetic cost of dehydration exceeds the energetic gain from partial coordination, so Na+ does not enter the filter.

Roderick MacKinnon determined the K+ channel structure in 1998 (Nobel Prize 2003).

Exercise 7 (hard, symbolic).

Tetrodotoxin (TTX, from puffer fish) blocks voltage-gated Na+ channels with very high affinity (Kd ~ 1-10 nM). Explain why TTX causes paralysis and why puffer fish are immune to their own toxin.

Hint

Na+ channels are essential for action potentials. What happens if they are blocked? And why do puffer fish neurons still fire?

Answer

Paralysis mechanism: Voltage-gated Na+ channels carry the inward Na+ current that drives the depolarizing upstroke of the action potential. TTX binds to the outer mouth of the Na+ channel pore with high affinity, physically occluding it. Without functional Na+ channels, neurons cannot fire action potentials. Motor neurons cannot signal muscles, causing flaccid paralysis. Diaphragm paralysis leads to respiratory failure.

Puffer fish immunity: Puffer fish have Na+ channels with specific amino acid substitutions in the TTX-binding site that reduce TTX affinity by ~1,000-fold. A single mutation (e.g., a phenylalanine to cysteine change in the pore loop of domain I) is sufficient to confer TTX resistance while preserving normal Na+ channel function. Garter snakes that eat TTX-bearing newts have also evolved TTX-resistant Na+ channels through convergent amino acid substitutions.

Exercise 8 (hard, symbolic).

Derive the Nernst equation from thermodynamic first principles. Start from the condition of electrochemical equilibrium and show that the electrical work of moving an ion across the membrane must equal the chemical work against the concentration gradient.

Hint

At equilibrium, the electrochemical potential $μ = μ^{0} + R T ln [X] + z F V$ is equal on both sides of the membrane.

Answer

The electrochemical potential of ion X on each side of the membrane is:

Inside: $μ_{in} = μ^{0} + R T ln [X]_{in} + z F V_{in}$

Outside: $μ_{out} = μ^{0} + R T ln [X]_{out} + z F V_{out}$

At equilibrium, $μ_{in} = μ_{out}$ :

$R T ln [X]_{in} + z F V_{in} = R T ln [X]_{out} + z F V_{out}$

$z F (V_{in} - V_{out}) = R T ln \frac{[ X ] _{out}}{[ X ] _{in}}$

$V_{m} = V_{in} - V_{out} = \frac{R T}{z F} ln \frac{[ X ] _{out}}{[ X ] _{in}}$

The physical meaning: at equilibrium, the electrical energy cost of moving a charge across the potential ( $z F V_{m}$ ) exactly equals the chemical free energy gained by moving down the concentration gradient ( $R T ln ([X]_{out} / [X]_{in})$ ).

Exercise 9 (hard, symbolic).

A cell has a Cl- equilibrium potential of -89 mV and a resting potential of -70 mV. Is Cl- at equilibrium? If not, in which direction is Cl- driven, and what mechanism maintains the Cl- gradient?

Hint

$E_{C l} = - 89$ mV, $V_{m} = - 70$ mV. The driving force is $V_{m} - E_{C l}$ .

Answer

Cl- is not at equilibrium. The driving force on Cl- is: $V_{m} - E_{C l} = - 70 - (- 89) = + 19$ mV.

For a negative ion, a positive driving force means Cl- is driven inward. At $E_{C l} = - 89$ mV, the concentration and electrical gradients balance. Since $V_{m} = - 70$ mV is above $E_{C l}$ (less negative), the membrane is not negative enough to hold Cl- inside — Cl- has a net tendency to enter the cell.

In most neurons, Cl- is kept near equilibrium by KCC2 (a K+-Cl- cotransporter that extrudes Cl-) and NKCC1 (a Na+-K+-2Cl- cotransporter that imports Cl-). In mature neurons, KCC2 dominates, keeping $[C l^{-}]_{in}$ low. In immature neurons, NKCC1 dominates, raising $[C l^{-}]_{in}$ and making GABA-A receptor activation depolarizing rather than hyperpolarizing in developing neurons.

Advanced results [Master]

Theorem 1 (Nernst equation). For an ion X at electrochemical equilibrium across a selectively permeable membrane, the equilibrium potential is $E_{X} = (R T / z_{X} F) ln ([X]_{out} / [X]_{in})$ .

The Nernst equation derives from setting the electrochemical potential equal on both sides of the membrane (see Exercise 8). It applies exactly when the membrane is permeable to only one ion species, or when the system has reached Donnan equilibrium ^{[Nernst 1888]}.

Theorem 2 (GHK voltage equation). Under the constant-field assumption for a membrane permeable to K+, Na+, and Cl-, the steady-state resting potential is given by the GHK equation (proved above) ^{[Goldman 1943]}.

Hodgkin and Katz (1949) confirmed this quantitatively on the squid giant axon by varying external ion concentrations and measuring the shift in resting potential ^{[Hodgkin & Katz 1949]}. The measured resting potential of -65 mV matched the GHK prediction within a few millivolts when permeability ratios were estimated from flux measurements.

Theorem 3 (Chord conductance equation). At steady state, the membrane potential can be expressed as a conductance-weighted average of the equilibrium potentials:

$V_{m} = \frac{g _{K} E _{K} + g _{N a} E _{N a} + g _{C l} E _{C l}}{g _{K} + g _{N a} + g _{C l}}$

This is equivalent to the GHK equation in the limit of small driving forces (linearised current-voltage relation). It is the form used by Hodgkin and Huxley in their 1952 analysis ^{[Hodgkin & Huxley 1952]}.

Theorem 4 (Membrane time constant). A passive membrane patch with specific capacitance $C_{m}$ and total conductance $g_{m}$ charges exponentially with time constant $τ_{m} = C_{m} / g_{m}$ . For typical neurons, $τ_{m} \approx 1$ - $20$ ms.

The time constant determines how rapidly the membrane potential responds to current injection and sets the temporal resolution of synaptic integration. Excitatory postsynaptic potentials (EPSPs) arriving within $τ_{m}$ of each other summate; those arriving further apart do not.

Theorem 5 (Cable equation). Subthreshold voltage spread along a cylindrical neurite of radius $a$ with intracellular resistivity $r_{i}$ and specific membrane resistance $R_{m}$ obeys:

$τ_{m} \frac{\partial V}{\partial t} = λ^{2} \frac{\partial ^{2} V}{\partial x ^{2}} - V$

where $λ = a R_{m} / (2 r_{i})$ is the length constant and $τ_{m} = R_{m} C_{m}$ is the membrane time constant. Steady-state voltage decays as $V (x) = V_{0} e^{- x / λ}$ .

The cable equation is a reaction-diffusion PDE. The diffusion term ( $λ^{2} \partial_{xx} V$ ) spreads voltage passively; the reaction term ( $- V$ ) represents the leak through the membrane. Adding active HH currents to the right-hand side gives the active cable equation that supports propagating action potentials in 17.09.02 pending.

Theorem 6 (Ussing flux ratio). For passive ion flux through a membrane, the ratio of unidirectional influx to efflux satisfies:

$\frac{J _{in}}{J _{out}} = \frac{[ X ] _{out}}{[ X ] _{in}} exp (\frac{z _{X} F V _{m}}{R T})$

This provides a test for whether ion transport is passive (obeying the flux ratio theorem) or involves active coupling to metabolic energy. Deviations from the Ussing ratio indicate either active transport, co-transport, or single-file diffusion through narrow pores.

Theorem 7 (Eisenman selectivity sequences). The selectivity of an ion-binding site for cations follows one of eleven sequences (Eisenman sequences I-XI), determined by the field strength of the coordinating anionic groups. High-field-strength sites (high charge density) select small cations (Li+ > Na+ > K+ > Rb+ > Cs+, sequence XI). Low-field-strength sites select large cations (Cs+ > Rb+ > K+ > Na+ > Li+, sequence I). Biological K+ channels achieve K+ selectivity through high-field-strength carbonyl coordination sites that match K+ ionic radius.

Theorem 8 (Donnan equilibrium). When a membrane separates two compartments containing a permeant ion and an impermeant charged species (e.g., intracellular proteins), the equilibrium distribution of the permeant ions is constrained by electroneutrality and the Gibbs-Donnan condition: the product of diffusible cation and anion concentrations is equal on both sides. The resulting Donnan potential contributes to the resting potential but cannot by itself maintain a stable steady state — without active pumping, the osmotic imbalance would cause the cell to swell. The Na+/K+ pump breaks the Donnan equilibrium, maintaining cell volume and the resting potential simultaneously by consuming ATP to oppose the passive redistribution of ions.

Synthesis. The resting membrane potential is the foundational reason that electrical signaling exists in biology: selective permeability to K+ at rest stores ~70 mV of electrochemical energy across a 5 nm lipid bilayer, producing an electric field of ~140,000 V/cm. The central insight is that ion channels are molecular machines combining selectivity with precise gating control, and putting these together — the Nernst equilibrium for each ion, the GHK permeability-weighted average, the chord conductance formulation, and the cable equation for spatial spread — identifies the membrane with an equivalent electrical circuit whose behavior is quantitatively predictable from thermodynamic first principles. This is exactly the biophysical substrate that 17.09.02 pending exploits to generate the action potential: voltage-gated channels change the permeability ratios and the GHK equation predicts the resulting voltage trajectory. The bridge is between equilibrium thermodynamics (Nernst, GHK) and dynamical excitability (Hodgkin-Huxley), and the pattern generalises from the squid giant axon to every excitable cell in every animal.

Ion channel structure and selectivity [Master]

The atomic-resolution understanding of ion selectivity began with the crystal structure of the KcsA potassium channel from Streptomyces lividans, solved by Doyle, Cabral, MacKinnon and collaborators at 3.2 A resolution in 1998 ^{[Doyle et al. 1998]}. The structure revealed a homotetramer with each subunit contributing two transmembrane helices (inner and outer) and a pore loop containing the selectivity filter. The filter itself is formed by the conserved signature sequence TVGYG, with backbone carbonyl oxygens from the valine, glycine, and tyrosine residues creating four sequential K+ binding sites (S1-S4).

Zhou, Morais-Cabral, and MacKinnon (2001) determined the KcsA structure at 2.0 A resolution with K+ ions bound, revealing the precise coordination geometry ^{[Zhou et al. 2001]}. Each K+ ion sits at the centre of a cage formed by eight carbonyl oxygen atoms (four from each of two adjacent subunits), at a K+-oxygen distance of 2.8 A that precisely matches the K+-water oxygen distance in the hydrated ion. The energetic cost of stripping the hydration shell is exactly compensated by the favorable interactions with the filter oxygens: the filter is a "surrogate water" sized for K+ but not for Na+.

The filter conducts by a "knock-on" mechanism in which multiple K+ ions occupy the filter simultaneously, separated by water molecules. Electrostatic repulsion between adjacent K+ ions in the single-file column provides the driving force for rapid conduction: a new K+ ion entering from one side pushes the column forward, ejecting the ion at the opposite end. This mechanism achieves near-diffusion-limited conduction rates of ~ $1 0^{8}$ ions per second per channel while maintaining 10,000-fold selectivity.

Voltage-gated sodium channels (NaV) use a different selectivity mechanism. The NaV pore is formed by four homologous domains of a single polypeptide. The selectivity filter contains the DEKA motif (aspartate from domain I, glutamate from domain II, lysine from domain III, alanine from domain IV). The crystal structure of NavAb, a bacterial voltage-gated sodium channel, at 2.7 A resolution ^{[Payandeh et al. 2011]} showed a selectivity filter with two ion binding sites formed by glutamate side chains, creating a high-field-strength site that selects Na+ over K+ based on size: the filter cavity is sized for Na+ (radius 0.95 A), and K+ (radius 1.33 A) is too large to fit optimally.

CLC chloride channels select for Cl- over other anions using a fundamentally different architecture. Rather than a pore formed by symmetric subunits, the CLC channel is a homodimer in which each subunit forms its own independent pore ("double-barrelled" channel). The selectivity filter contains a conserved motif with a bound Cl- ion coordinated by backbone amide nitrogens and serine hydroxyl groups. The selectivity arises from the precise geometry of these hydrogen bonds, which matches the Cl- radius (1.81 A) but excludes larger anions like phosphate.

Two-pore-domain potassium channels (K2P, including TASK and TREK subfamilies) are the molecular identity of the potassium "leak" conductance that sets the resting potential. Each K2P subunit contains four transmembrane helices and two pore domains; the functional channel is a dimer with two pores in parallel. K2P channels are constitutively open (or very weakly gated), providing the background K+ conductance that dominates at rest. They are regulated by pH (TASK channels), mechanical stretch (TREK channels), lipids, and volatile anaesthetics, providing a mechanistic link between cellular metabolism, mechanical state, and resting excitability. The TREK-1 channel (KCNK2) is activated by polyunsaturated fatty acids, lysophospholipids, and membrane stretch, and inhibited by neurotransmitters acting through Gq-coupled receptors via protein kinase C phosphorylation. This makes TREK-1 a convergence point for neuromodulatory control of resting potential: Gq signalling depolarises the neuron by closing K2P channels, reducing $P_{K}$ and shifting $V_{m}$ away from $E_{K}$ toward $E_{Na}$ . The TASK-1 and TASK-3 channels (KCNK3, KCNK9) are sensitive to extracellular pH and are highly expressed in brainstem respiratory centres, where they couple CO2/pH sensing to ventilatory drive — loss of TASK channel function causes central hypoventilation syndrome.

Gating mechanisms [Master]

Ion channels are gated: they transition between open and closed conformations in response to specific stimuli. The gating mechanism determines what signals the channel transduces into electrical activity.

Voltage-dependent gating is the mechanism underlying the action potential. Voltage-gated channels contain a voltage-sensor domain formed by the S4 transmembrane helix, which carries regularly spaced positively charged residues (arginine or lysine) at every third position. In the resting state (negative $V_{m}$ ), the S4 helix is pulled inward by the transmembrane electric field. Depolarisation reduces this electrical force, allowing S4 to move outward through the membrane. This movement is coupled mechanically to the pore-lining S6 helices, opening the activation gate at the intracellular end of the pore.

The gating charge — the total charge moved across the membrane during the voltage-sensor transition — has been measured directly. For the squid NaV channel, the total gating charge is approximately 13 elementary charges ( $e_{0}$ ) per channel, distributed across four voltage sensors (one in each domain). This means each S4 helix moves approximately 3-4 positive charges across the membrane electric field. The resulting gating current ( $I_{g} = d Q_{g} / d t$ ) is a measurable capacitive current that precedes the ionic current during depolarisation, and its time course provides direct information about voltage-sensor kinetics independent of pore opening.

Ligand-gated channels open when a specific molecule binds to a site on the channel protein. The nicotinic acetylcholine receptor (nAChR) at the neuromuscular junction is the prototypical ligand-gated channel. It is a pentamer of subunits (typically $α_{2} β γ δ$ in embryonic muscle) arranged around a central pore. Acetylcholine binding at the two $α$ -subunit interfaces induces a conformational change in the extracellular domain that propagates through the transmembrane helices, rotating the pore-lining M2 helices to widen the channel. The open nAChR is permeable to Na+, K+, and Ca2+, producing a depolarising inward current. The 5-HT3 (serotonin) receptor has a similar pentameric architecture.

Mechanosensitive channels open in response to membrane tension. The Piezo family (Piezo1 and Piezo2) are large trimeric channels (~900 kDa) with a distinctive blade-like architecture: each subunit contributes an extended blade of transmembrane helices that curves through the membrane like a nanoscale dome. Membrane tension flattens the dome, mechanically pulling the central pore open. Piezo1 mediates red blood cell volume regulation and vascular development; Piezo2 mediates touch sensation in dorsal root ganglion neurons.

Inactivation is the process by which an open channel closes despite the continued presence of the activating stimulus. Two mechanistically distinct forms exist. N-type ("ball-and-chain") inactivation involves a tethered peptide "ball" at the N-terminus of the channel that swings into the intracellular mouth of the pore, physically blocking ion flow. This was demonstrated for the Shaker K+ channel by Armstrong and Bezanilla (1977) and confirmed by deletion experiments: removing the N-terminal ball abolishes fast inactivation, and applying the free peptide restores it. C-type inactivation involves a constriction of the selectivity filter itself — the filter collapses from its conductive conformation into a non-conductive state. C-type inactivation is slower than N-type and depends on the occupancy of the filter by permeant ions.

Patch-clamp recording by Neher and Sakmann (1976) ^{[Neher & Sakmann 1976]} resolved single-channel gating as discrete stochastic transitions between conducting and non-conducting states. At the single-channel level, gating is a continuous-time Markov process: the channel occupies a finite set of states (closed, open, inactivated), transitioning between them with voltage-dependent rate constants. The dwell time in each state is exponentially distributed, with the mean dwell time equal to the reciprocal of the sum of the exit rates from that state.

The Hodgkin-Huxley $m^{3} h$ formulation approximates this Markov process under the assumption of independent gating particles. High-resolution patch-clamp data reveal systematic deviations from independence that require full Markov-chain models with multiple closed and inactivated states. For example, the sodium channel requires a minimum of five states (three closed, one open, one inactivated) to reproduce single-channel data, and some models include additional slow-inactivated states that correspond to clinically relevant phenomena such as use-dependent drug block. The macroscopic current in the Hodgkin-Huxley formalism is recovered in the limit where the number of channels is large and the stochastic transitions are averaged — the law of large numbers converts the discrete Markov chain into a deterministic system of ODEs for the state occupancies.

Channelopathies and pharmacology [Master]

Mutations in ion channel genes cause a class of inherited diseases called channelopathies, which provide direct evidence for the physiological roles of specific channel types.

Epilepsy provides a striking example. Benign familial neonatal seizures (BFNS) are caused by mutations in KCNQ2 or KCNQ3, the subunits of the M-type potassium current ( $I_{M}$ ). This current is a voltage-gated K+ conductance that activates at subthreshold voltages and limits neuronal excitability. Loss-of-function mutations reduce $I_{M}$ , lowering the threshold for action potential firing and producing hyperexcitability manifesting as neonatal seizures. Retigabine (ezogabine), a KCNQ channel opener that shifts the voltage-dependence of activation to more negative potentials, was developed as an antiepileptic drug that specifically enhances the impaired current.

Periodic paralysis results from mutations in voltage-gated sodium or calcium channels in skeletal muscle. Hyperkalemic periodic paralysis (HyperPP) is caused by missense mutations in SCN4A, encoding the Nav1.4 skeletal muscle sodium channel. The most common mutations (T704M, M1592V) cause incomplete channel inactivation, producing a persistent Na+ "leak" current that depolarises the muscle fibre. When extracellular K+ rises (after exercise or a potassium-rich meal), the additional depolarisation inactivates the remaining healthy Na+ channels, rendering the muscle inexcitable — the paralysis episode. Hypokalemic periodic paralysis (HypoPP) is most often caused by mutations in CACNA1S (Cav1.1, the L-type calcium channel) or SCN4A, producing "gating pore currents" in which the mutated voltage sensor leaks protons or sodium ions through the voltage-sensor domain itself rather than through the main pore.

Long QT syndrome (LQTS) is a cardiac channelopathy that causes delayed ventricular repolarisation (prolonged QT interval on the ECG) and predisposes to torsades de pointes ventricular tachycardia and sudden cardiac death. LQTS type 1 (KCNQ1 mutations) impairs the slow delayed-rectifier K+ current ( $I_{K s}$ ); LQTS type 2 (KCNH2/HERG mutations) impairs the rapid delayed-rectifier ( $I_{K r}$ ); LQTS type 3 (SCN5A mutations) produces a persistent late Na+ current that delays repolarisation. Each mutation type points to a different repolarising current in the cardiac action potential.

Local anaesthetics block voltage-gated Na+ channels in a state-dependent manner. Lidocaine, bupivacaine, and related compounds bind to a site in the inner mouth of the pore (accessible from the intracellular side) with higher affinity for the open and inactivated states than for the resting state. This produces "use-dependent block": the drug preferentially blocks channels that are repeatedly opening (as in firing neurons), sparing channels at rest. The molecular basis involves binding to conserved aromatic residues in the S6 helix of domain IV.

Toxin tools have been indispensable for dissecting ion channel function. Tetrodotoxin (TTX) from puffer fish blocks NaV channels from the extracellular side by plugging the outer pore mouth (Kd ~1-10 nM for most NaV isoforms). Saxitoxin (STX), produced by marine dinoflagellates and responsible for paralytic shellfish poisoning, blocks NaV channels through a similar mechanism at the external pore vestibule. Tetraethylammonium (TEA) blocks K+ channels from either side: externally it binds near the selectivity filter entrance, internally it enters the pore from the cytoplasmic side. Apamin (from bee venom) blocks small-conductance calcium-activated K+ channels (SK channels). Charybdotoxin (from scorpion venom) blocks large-conductance BK channels and some KV channels by binding to the external vestibule. Dendrotoxins (from mamba snake venom) block certain KV channels, while 4-aminopyridine (4-AP) blocks KV channels from the intracellular side and is used clinically to improve conduction in demyelinated axons (multiple sclerosis). Each toxin's specificity has been exploited to isolate and characterise individual current components in voltage-clamp experiments, enabling the Hodgkin-Huxley decomposition of membrane current into distinct ionic conductances.

Full proof set [Master]

Proposition 1 (Chord conductance equation). At steady state with zero external current, the membrane potential equals the conductance-weighted average of the equilibrium potentials: $V_{m} = (g_{K} E_{K} + g_{N a} E_{N a} + g_{C l} E_{C l}) / (g_{K} + g_{N a} + g_{C l})$ .

Proof. Kirchhoff's current law at the membrane requires $I_{K} + I_{N a} + I_{C l} = 0$ at steady state. Using the linear current-voltage relation $I_{X} = g_{X} (V_{m} - E_{X})$ for each ion:

$g_{K} (V_{m} - E_{K}) + g_{N a} (V_{m} - E_{N a}) + g_{C l} (V_{m} - E_{C l}) = 0$

$V_{m} (g_{K} + g_{N a} + g_{C l}) = g_{K} E_{K} + g_{N a} E_{N a} + g_{C l} E_{C l}$

$V_{m} = \frac{g _{K} E _{K} + g _{N a} E _{N a} + g _{C l} E _{C l}}{g _{K} + g _{N a} + g _{C l}}$

The membrane potential is the conductance-weighted mean of the equilibrium potentials. When $g_{K} ≫ g_{N a}, g_{C l}$ , this reduces to $V_{m} \approx E_{K}$ . $□$

Proposition 2 (Membrane time constant and exponential charging). A membrane patch with capacitance $C_{m}$ and total leak conductance $g_{L}$ responds to a current step $I_{0}$ starting at $t = 0$ with $V_{m} (t) = V_{\infty} + (V_{0} - V_{\infty}) e^{- t / τ_{m}}$ , where $τ_{m} = C_{m} / g_{L}$ and $V_{\infty} = V_{0} + I_{0} / g_{L}$ .

Proof. The circuit equation is $C_{m} d V_{m} / d t = - g_{L} (V_{m} - E_{L}) + I_{0}$ . Writing $V_{\infty} = E_{L} + I_{0} / g_{L}$ (the steady state where $d V / d t = 0$ ) and substituting $w = V_{m} - V_{\infty}$ :

$C_{m} \frac{d w}{d t} = - g_{L} w$

$w (t) = w (0) e^{- g_{L} t / C_{m}} = (V_{0} - V_{\infty}) e^{- t / τ_{m}}$

Therefore $V_{m} (t) = V_{\infty} + (V_{0} - V_{\infty}) e^{- t / τ_{m}}$ with $τ_{m} = C_{m} / g_{L}$ . For typical neuronal parameters ( $C_{m} = 1$ $μ$ F/cm $^{2}$ , $g_{L} = 0.3$ mS/cm $^{2}$ ), $τ_{m} \approx 3.3$ ms. $□$

Proposition 3 (Cable equation: steady-state exponential decay). For a semi-infinite cable with sealed end at $x = 0$ and voltage $V (0) = V_{0}$ applied at $x = 0$ , the steady-state voltage is $V (x) = V_{0} e^{- x / λ}$ where $λ = R_{m} a / (2 r_{i})$ is the length constant, $R_{m}$ is the specific membrane resistance, $a$ is the cable radius, and $r_{i}$ is the intracellular resistivity.

Proof. Kirchhoff's current law on an infinitesimal cable element of length $d x$ gives:

$- \frac{1}{r _{i}} \frac{\partial ^{2} V}{\partial x ^{2}} d x = \frac{V}{R _{m}} \cdot 2 π a d x + C_{m} \cdot 2 π a d x \cdot \frac{\partial V}{\partial t}$

At steady state ( $\partial V / \partial t = 0$ ), dividing by $d x$ and rearranging:

$\frac{1}{r _{i}} \frac{d ^{2} V}{d x ^{2}} = \frac{2 π aV}{R _{m} \cdot 2 π a \cdot 1} \cdot \frac{2 π a}{1}$

More directly, using per-unit-length parameters $r_{m} = R_{m} / (2 π a)$ (membrane resistance times length) and $r_{a} = r_{i} / (π a^{2})$ (axial resistance per length):

$\frac{d ^{2} V}{d x ^{2}} = \frac{V}{λ ^{2}}, λ = r_{m} / r_{a} = R_{m} a / (2 r_{i})$

The general solution is $V (x) = A e^{- x / λ} + B e^{x / λ}$ . For a semi-infinite cable, $B = 0$ (voltage must remain bounded). The boundary condition $V (0) = V_{0}$ gives $A = V_{0}$ .

Therefore $V (x) = V_{0} e^{- x / λ}$ . For a squid giant axon ( $a = 250$ $μ$ m, $R_{m} = 2000$ $Ω \cdot cm^{2}$ , $r_{i} = 30$ $Ω \cdot cm$ ), $λ = 2000 \times 0.025/60 \approx 0.65$ cm = 6.5 mm. $□$

Connections [Master]

The action potential — ionic basis 17.09.02 pending. Builds directly on the resting potential established here: voltage-gated Na+ channels transiently increase $P_{Na}$ by orders of magnitude, shifting the GHK weighted average from $E_{K}$ toward $E_{Na}$ and generating the spike upstroke. The cable equation derived here becomes the active cable equation when HH currents are added, supporting propagating action potentials.
Canonical ensemble and partition function 11.04.01 pending. The Nernst equation derives directly from the Boltzmann factor of the canonical ensemble applied to an ion at electrochemical equilibrium. The GHK equation is the multi-ion generalisation. This is the densest biological application of equilibrium stat-mech in the curriculum.
Membrane transport 17.02.02. The Na+/K+ pump and secondary active transporters (KCC2, NKCC1) maintain the ion concentration gradients that drive the resting potential. Without active transport, the gradients run down and $V_{m}$ collapses to 0 mV.
Electrochemistry — Nernst equation and cells 14.11.01. The Nernst equation of cellular biophysics is the same thermodynamic relation derived for electrochemical half-cells. The membrane potential is a biological galvanic cell in which the "electrodes" are the ion-selective channels and the "electrolyte" is the salt solution on either side of the membrane.
Cell signalling — receptors and GPCRs 17.07.01 pending. Ligand-gated ion channels (nAChR, GABA-A, NMDA) are both ion channels and signal-transducing receptors. The resting potential provides the electrochemical driving force that makes ligand-gated channel opening electrically consequential: at $V_{m} = - 70$ mV, opening a Na+-permeable channel produces a large inward current precisely because the resting potential stores the energy that the channel releases.
Cell cycle and mitosis 17.08.01. Cell cycle progression in electrically excitable cells (neurons, cardiomyocytes) is modulated by membrane potential: depolarisation promotes proliferation in neural progenitor cells, while hyperpolarisation is associated with cell cycle exit. The ion-channel biophysics described here intersects the CDK-cyclin machinery of the cell cycle unit in stem-cell biology and cancer electrophysiology.

Historical & philosophical context [Master]

Julius Bernstein proposed the "membrane theory" of bioelectricity in 1902 ^{[Bernstein 1902]}, arguing that the resting potential arises from selective K+ permeability of the cell membrane. Bernstein measured the resting potential of muscle at ~-70 mV using capillary electrometers and showed that it approximately equalled the K+ Nernst potential. His theory predicted that cell death or membrane damage would cause the potential to collapse to zero (the "injury potential"), which was experimentally confirmed. Bernstein's theory could not, however, explain the overshoot of the action potential to positive values — that required the sodium hypothesis developed decades later.

David Goldman derived the constant-field voltage equation in 1943 ^{[Goldman 1943]}, providing the mathematical framework for multi-ion membrane potentials. Hodgkin and Katz (1949) ^{[Hodgkin & Katz 1949]} applied the Goldman equation to the squid giant axon, demonstrating that replacing external Na+ with choline (an impermeant cation) abolished the action potential upstroke while leaving the resting potential largely unchanged. This established that the resting potential is primarily a K+ diffusion potential while the action potential upstroke requires Na+ entry.

The ion channel concept solidified with two technical breakthroughs. The voltage clamp, developed by Hodgkin, Huxley, and Katz ^{[Hodgkin & Huxley 1952]}, allowed measurement of membrane current at controlled voltages, separating the individual ionic conductances. Patch-clamp recording by Neher and Sakmann (1976) ^{[Neher & Sakmann 1976]} resolved single-channel currents, demonstrating that channels open and close as discrete stochastic events with unitary conductances of 1-300 pS. The KcsA crystal structure by Doyle, Cabral, MacKinnon and colleagues (1998) ^{[Doyle et al. 1998]} revealed the atomic basis of K+ selectivity at 3.2 A resolution, earning MacKinnon the 2003 Nobel Prize in Chemistry.

Bibliography [Master]

Primary literature.

Nernst, W., "Zur Kinetik der in Losung befindlichen Korper", Z. Phys. Chem. 2 (1888), 613-637.

Bernstein, J., "Untersuchungen zur Thermodynamik der bioelektrischen Strome", Pflugers Arch. 92 (1902), 521-562.

Goldman, D. E., "Potential, impedance, and rectification in membranes", J. Gen. Physiol. 27 (1943), 37-60.

Hodgkin, A. L. & Katz, B., "The effect of sodium ions on the electrical activity of the giant axon of the squid", J. Physiol. 108 (1949), 37-77.

Hodgkin, A. L. & Huxley, A. F., "A quantitative description of membrane current and its application to conduction and excitation in nerve", J. Physiol. 117 (1952), 500-544.

Neher, E. & Sakmann, B., "Single-channel currents recorded from membrane of denervated frog muscle fibres", Nature 260 (1976), 799-802.

Doyle, D. A. et al., "The structure of the potassium channel: molecular basis of K+ conduction and selectivity", Science 280 (1998), 69-77.

Zhou, Y., Morais-Cabral, J. H., Kaufman, A. & MacKinnon, R., "Chemistry of ion coordination and hydration revealed by a K+ channel-Fab complex at 2.0 A resolution", Nature 414 (2001), 43-48.

Payandeh, J., Scheuer, T., Zheng, N. & Catterall, W. A., "The crystal structure of a voltage-gated sodium channel", Nature 475 (2011), 353-358.

Textbook and monograph.

Hille, B., Ion Channels of Excitable Membranes, 3rd ed. (Sinauer, 2001).

Kandel, E. R. et al., Principles of Neural Science, 5th ed. (Mcraw-Hill, 2013).

Alberts, B. et al., Molecular Biology of the Cell, 6th ed. (Garland, 2014).

Hodgkin, A. L., The Conduction of the Nervous Impulse (Liverpool University Press, 1964).

Prerequisites

17.02.02

Used in

17.09.02
18.05.01

Tier anchors

beginner: Khan Academy (neuronal synapses); Crash Course — Nervous System
intermediate: Alberts et al., *Molecular Biology of the Cell* (6th ed., Garland 2014), Ch. 11; Kandel et al., *Principles of Neural Science* (5th ed., McGraw-Hill 2013), Ch. 6-7
master: Hille, *Ion Channels of Excitable Membranes* (3rd ed., Sinauer 2001), Ch. 1-4; Hodgkin & Huxley 1952; Nernst 1888; Goldman 1943

References

TODO_REF
Hille — Ion Channels of Excitable Membranes (3rd ed., Sinauer 2001) · Chs. 1-4: classical biophysics, selectivity, voltage-gated channels
TODO_REF pending
Alberts et al. — Molecular Biology of the Cell (6th ed., Garland 2014) · Ch. 11 — Membrane Transport of Small Molecules and the Electrical Properties of Membranes · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-mboc-ch11
TODO_REF pending
Kandel et al. — Principles of Neural Science (5th ed., McGraw-Hill 2013) · Ch. 6-7: membrane potential and passive electrical properties · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-kandel-pns
TODO_REF pending
Nernst 1888 — Zur Kinetik der in Losung befindlichen Korper · Z. Phys. Chem. 2 (1888) 613-637; the Nernst equation originator · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-nernst1888
TODO_REF pending
Goldman 1943 — Potential, impedance, and rectification in membranes · J. Gen. Physiol. 27 (1943) 37-60; the constant-field equation · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-goldman1943
TODO_REF pending
Hodgkin & Katz 1949 — The effect of sodium ions on the electrical activity of the giant axon of the squid · J. Physiol. 108 (1949) 37-77 · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-hk1949
TODO_REF pending
Hodgkin & Huxley 1952 — A quantitative description of membrane current and its application to conduction and excitation in nerve · J. Physiol. 117 (1952) 500-544 · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-hh1952
TODO_REF pending
Bernstein 1902 — Untersuchungen zur Thermodynamik der bioelektrischen Strome · Pflugers Arch. 92 (1902) 521-562; the membrane theory of the resting potential · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-bernstein1902
TODO_REF pending
Doyle et al. 1998 — The structure of the potassium channel: molecular basis of K+ conduction and selectivity · Science 280 (1998) 69-77; the KcsA crystal structure · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-doyle1998
TODO_REF pending
Zhou et al. 2001 — Chemistry of ion coordination and hydration revealed by a K+ channel-Fab complex at 2.0 A resolution · Nature 414 (2001) 43-48; four K+ binding sites in the selectivity filter · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-zhou2001
TODO_REF pending
Neher & Sakmann 1976 — Single-channel currents recorded from membrane of denervated frog muscle fibres · Nature 260 (1976) 799-802 · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-ns1976
TODO_REF pending
Payandeh et al. 2011 — The crystal structure of a voltage-gated sodium channel · Nature 475 (2011) 353-358; NavAb structure revealing the voltage-sensor mechanism · see docs/catalogs/NEED_TO_SOURCE.md#bio-wave1-payandeh2011

Reviewer

Tyler (pending external biology reviewer per BIOLOGY_PLAN §6)

Estimated time

beginner: 15m
intermediate: 35m
master: 70m