Unit 4: Electrical phenomena in the biological system

Living cells maintain a voltage difference across their plasma membranes, and the controlled movement of ions through that membrane generates the signals that underlie nerve conduction, muscle contraction and sensation. This unit builds from the physical basis of the membrane potential to the electrical behaviour of excitable cells and the protein machinery that produces it.

I. Foundations: The Membrane as a Charged Capacitor

Every cell is bounded by a lipid bilayer that separates ionic solutions of differing composition, creating a stored charge and a resting voltage that all later phenomena modify.

• Resting membrane potential (Vm): the steady voltage across the membrane at rest, typically −70 mV in neurons, measured inside relative to outside.
• Ionic gradients: Na⁺ and Cl⁻ are concentrated outside the cell; K⁺ and organic anions (A⁻) are concentrated inside. Typical neuronal values: [K⁺]in ≈ 140 mM, [K⁺]out ≈ 5 mM; [Na⁺]in ≈ 15 mM, [Na⁺]out ≈ 145 mM.
• Selective permeability: at rest the membrane is far more permeable to K⁺ than to Na⁺ (roughly 25:1), so Vm sits near the K⁺ equilibrium.
• Capacitance and conductance: the bilayer behaves as a capacitor (Cm ≈ 1 µF/cm²) in parallel with ion-conducting resistors (channels). Charge Q = Cm·Vm.
• Sign convention: depolarisation = Vm becomes less negative (toward 0); hyperpolarisation = Vm becomes more negative.

II. Electrical Phenomena in Excitable Cells

How ionic gradients are converted into a measurable, stable voltage.

A. Origin of the resting potential

The resting potential arises because ions diffuse down concentration gradients until the electrical force opposing them balances the chemical force.

• Nernst equation: gives the equilibrium potential of a single ion.

E_ion = (RT / zF) · ln([ion]_out / [ion]_in)
     ≈ (61.5/z) · log10([ion]_out / [ion]_in)   mV at 37°C

• R: gas constant (8.314 J·mol⁻¹·K⁻¹); T: absolute temperature (K); z: ion valence; F: Faraday constant (96 485 C·mol⁻¹).
• Worked value: for K⁺, E_K = 61.5·log10(5/140) ≈ −89 mV.
  • Goldman–Hodgkin–Katz (GHK) equation: combines several ions weighted by permeability P.

Vm = 61.5 · log10( (P_K[K]o + P_Na[Na]o + P_Cl[Cl]i) /
                    (P_K[K]i + P_Na[Na]i + P_Cl[Cl]o) )

• Because P_K ≫ P_Na at rest, Vm lies close to E_K but is pulled slightly positive by Na⁺ leak.

B. Maintenance and the electrogenic pump

• Na⁺/K⁺-ATPase: exports 3 Na⁺ and imports 2 K⁺ per ATP, restoring gradients dissipated by leak channels.
• Electrogenic contribution: the 3:2 stoichiometry moves net positive charge outward, adding a few mV of hyperpolarisation directly.
• Steady state, not equilibrium: the resting potential is a dynamic balance requiring continuous ATP; blocking the pump (e.g. with ouabain) slowly collapses Vm.

III. Electrically Excitable Cells

Cells whose membranes can generate regenerative, all-or-none voltage responses.

A. Defining property of excitability

• Threshold behaviour: a stimulus that depolarises Vm past a critical value (≈ −55 mV) triggers a self-amplifying response; smaller stimuli decay passively.
• Voltage-gated channels: excitability depends on channels that open in response to depolarisation, producing positive feedback (depolarisation → Na⁺ entry → more depolarisation).
• Cell types: neurons, skeletal/cardiac/smooth muscle fibres, and some endocrine cells (e.g. pancreatic β-cells).

B. Passive electrical properties

• Length constant (λ): distance over which a passive signal decays to 1/e (≈ 37%) of its value.

λ = √(r_m / r_i)

• r_m: membrane resistance per unit length; r_i: axial (internal) resistance per unit length. Larger λ → signal spreads farther.
  • Time constant (τ): τ = R_m·C_m, the delay for the membrane voltage to reach 63% of its final value after a step of current.
  • Cable theory: treats the axon as a leaky cable; passive spread alone cannot carry signals far, which is why active regeneration is needed.

IV. Electrical Signals of Nerve Cells

The action potential and its propagation along the axon.

A. Phases of the action potential

The action potential is a rapid, stereotyped reversal of Vm produced by sequential opening of voltage-gated Na⁺ and K⁺ channels.

• 1. Depolarising (rising) phase: Na⁺ channels open, gNa rises sharply, Vm shoots toward E_Na (≈ +60 mV).
• 2. Repolarising (falling) phase: Na⁺ channels inactivate while delayed K⁺ channels open, driving Vm back toward E_K.
• After-hyperpolarisation: lingering K⁺ conductance briefly pushes Vm below rest before recovery.
• Refractory periods:
  • Absolute: Na⁺ channels inactivated — no second spike is possible.
  • Relative: recovering channels allow a spike only with a stronger stimulus.

B. Signal propagation

• Continuous conduction: local currents from an active region depolarise the adjacent membrane to threshold; the spike regenerates without decrement.
• Saltatory conduction: in myelinated axons the impulse jumps between nodes of Ranvier, raising velocity (up to ~120 m/s) and saving metabolic energy.
• All-or-none law: amplitude is fixed once threshold is crossed; stimulus intensity is coded by firing frequency, not spike size.
• Conduction velocity: increases with axon diameter and with myelination (higher λ, lower Cm at internodes).

V. The Ionic Hypothesis and Rules of Ionic Electricity

The Hodgkin–Huxley framework explaining the action potential as time- and voltage-dependent ionic conductances.

A. The ionic hypothesis

Formulated by Hodgkin and Huxley from squid giant axon experiments (1952), the hypothesis states that the action potential results from separate, independently gated Na⁺ and K⁺ conductances that change with voltage and time.

• Voltage clamp evidence: holding Vm fixed revealed an early inward (Na⁺) current followed by a sustained outward (K⁺) current.
• Ion substitution: removing external Na⁺ abolished the inward current, confirming its carrier.
• Conductance model: total current is the sum of ionic and capacitive terms.

I_m = C_m(dV/dt) + g_Na(V − E_Na) + g_K(V − E_K) + g_L(V − E_L)

• g_L, E_L: leak conductance and its reversal potential; g_Na, g_K: voltage- and time-dependent conductances.

B. Rules of ionic electricity

• Driving force: current for an ion is proportional to (Vm − E_ion); it is zero at the reversal potential and reverses sign across it.
• Ohm's law for channels: I_ion = g_ion·(Vm − E_ion), with g the conductance in siemens.
• Independence principle: each ion crosses the membrane independently of others, allowing conductances to be summed.
• Electroneutrality: bulk solutions stay electrically neutral; only a tiny excess charge near the membrane sets Vm.
• Space-charge minimality: the number of ions that must cross to change Vm is minute (order 10⁻¹² mol/cm²), so gradients are barely depleted per spike.

VI. Membrane Proteins

The molecular devices that create, gate and pump the ionic currents behind every electrical signal.

A. Ion channels

• Function: form aqueous pores allowing selective, high-throughput (10⁶–10⁸ ions/s) passive flux down electrochemical gradients.
• Selectivity filter: narrow region (e.g. the K⁺ channel's Gly-Tyr-Gly signature) that discriminates ions by size and dehydration energy.
• Gating modes:
  • Voltage-gated: open on membrane depolarisation (Nav, Kv, Cav).
  • Ligand-gated: open on binding of a neurotransmitter (nicotinic acetylcholine receptor).
  • Mechanically gated: respond to membrane stretch (touch/stretch receptors).

B. Transporters (carriers)

• Function: bind ions/solutes and change conformation to move them, far slower than channels (10²–10⁴ ions/s).
• Types: uniporters (one species), symporters (co-transport same direction, e.g. Na⁺-glucose SGLT), antiporters (exchange, e.g. Na⁺/Ca²⁺ exchanger).

C. Pumps (active transporters)

• Function: use ATP hydrolysis to move ions against gradients, building the potential energy channels later release.
• Na⁺/K⁺-ATPase: a P-type ATPase, the master generator of Na⁺ and K⁺ gradients (3 Na⁺ out / 2 K⁺ in).
• Ca²⁺-ATPase (PMCA/SERCA): keeps cytosolic Ca²⁺ low (~100 nM), essential for signalling reset.

D. Structural basis of function

• Transmembrane domains: hydrophobic α-helices anchor the protein; charged residues (the S4 helix's arginine ladder) act as the voltage sensor in Nav/Kv channels.
• Conformational states: channels cycle through closed, open and inactivated states; the inactivation "ball-and-chain" peptide plugs the open Na⁺ pore.
• Pharmacological relevance: tetrodotoxin blocks Nav channels; tetraethylammonium blocks Kv channels — the same tools that dissected the ionic hypothesis.