VV

Ion Equilibrium, Currents, and Resting Membrane Potential Notes

Nernst Equation and Equilibrium Potential

  • Equilibrium potential (Eion) is the voltage at which there is no net flux of that ion across the membrane when the membrane is permeable only to that ion. It is the balancing point where electric forces and diffusion forces on the ion are equal and opposite.
  • Key idea: When the membrane is permeable to only one ion, the membrane potential that results is the equilibrium potential for that ion.
  • General form (thermodynamic basis): E{ion}= rac{RT}{zF}\, ext{ln}igg( rac{[ion]{out}}{[ion]_{in}}igg)
    • At 37°C, this can be written in base-10 form as:
      E{ion} \, ext{(at 37°C)} \,=\ \frac{61.54}{z}\,\log{10}igg(\frac{[ion]{out}}{[ion]{in}}\bigg)
  • For monovalent cations (z = +1), this simplifies to:
    E{K} = 61.54\log{10}\left(\frac{[K^+]{out}}{[K^+]{in}}\right)
    E{Na} = 61.54\log{10}\left(\frac{[Na^+]{out}}{[Na^+]{in}}\right)
  • Example (K+): If [K+]out = 4 mM and [K+]in = 140 mM, then
    E{K} \approx 61.54\log{10}\left(\frac{4}{140}\right) \approx -9.4 \text{ to } -95\ \text{mV}
    (depending on the precise constant used and rounding; commonly cited values are around -90 to -94 mV)
  • Important notes from the transcript examples:
    • EK is negative when [K+]out < [K+]in, reflecting that K+ tends to move inward if the inside is more negative; the actual resting potential is not equal to EK because the membrane is permeable to multiple ions.
    • Temperature affects the numerical constant (61.54 at 37°C). Higher temperature makes ions respond more quickly to gradients.
    • The Nernst potential for an ion would be the membrane potential if only that ion could permeate and if its concentration gradient were the only driving force.

Diffusion vs Electrostatic Forces and Driving Forces

  • Equilibrium occurs when electric force on ions balances the diffusion (concentration) gradient.
  • Driving force for an ion is defined as the difference between the membrane potential and the ion’s equilibrium potential:
    ext{Driving force}{ion}=Vm - E_{ion}
  • Current through an ion is the product of its conductance and the driving force:
    I{ion}=g{ion}ig(Vm - E{ion}ig)
  • Conditions for current to flow (as per the transcript):
    1) There must be ions to carry charge.
    2) There must be a passageway (ion channels open → nonzero conductance).
    3) There must be a driving force (concentration gradient and/or voltage).
  • If any condition is missing (no ions, channels closed, or Vm = Eion), current is zero.

Resting Membrane Potential and the Goldman Equation

  • The resting membrane potential (Vm) is not simply the sum of individual Nernst potentials for each ion; it results from the weighted influence of multiple ions, each with its own permeability (conductance):
    • PK, PNa, PCl, etc. contribute according to their permeabilities and concentrations.
  • Goldman-Hodgkin-Katz (GHK) equation generalizes the resting potential by accounting for multiple permeant ions and their relative permeabilities. Its common form for monovalent cations at 37°C:
    Vm = 61.54\log\left(\frac{PK[K^+]{out} + P{Na}[Na^+]{out}}{PK[K^+]{in} + P{Na}[Na^+]_{in}}\right)
  • More complete form including anions (e.g., Cl−) can be written as:
    Vm = 61.54\log\left(\frac{PK[K^+]{out} + P{Na}[Na^+]{out} + P{Cl}[Cl^-]{in}}{PK[K^+]{in} + P{Na}[Na^+]{in} + P{Cl}[Cl^-]_{out}}\right)
  • Key implications from the transcript:
    • Vm is drawn toward EK by potassium permeability and toward ENa by sodium permeability; the actual Vm is a balance of these competing forces.
    • Changes in membrane permeability for any ion cause Vm to move toward the corresponding ion’s equilibrium potential.
    • At rest, not all current is carried by K+; Na+ and other ions, though permeable, contribute to the resting current to a lesser extent depending on their conductances.

How Changing Permeabilities and Ion Concentrations Affect Vm

  • When extracellular K+ increases:
    • EK becomes less negative (less driving force for K+ to leave), causing Vm to depolarize toward 0 mV.
    • The Goldman equation predicts that an increased [K+]out shifts Vm in the depolarizing direction; the transcript notes a slope of about 58–61 mV per tenfold change in the K+ gradient (depicting the sensitivity of Vm to extracellular K+ changes).
  • The Nernst prediction (black line in the provided figure) often does not perfectly match the actual Vm (orange line) because not all current at rest is carried by K+; other ions contribute to resting conductance.
  • There are practical ranges in which EK would be zero if [K+]out equals [K+]in, in which case the driving force for K+ would vanish.

Sodium and Potassium Equilibrium Potentials at Rest

  • Sodium (Na+) equilibrium potential:
    E{Na} = 61.54 \log\left(\frac{[Na^+]{out}}{[Na^+]_{in}}\right)
  • Typical (example) scenarios from the transcript:
    • If membrane is permeable only to Na+ and external/internal concentrations are such that ENa is positive, opening Na+ channels would drive inward Na+ current when V_m is below ENa.
    • The resting Vm is usually far below ENa (often around -60 to -70 mV), so a large driving force exists for Na+ to enter if Na+ channels open, contributing to depolarizing currents during action potentials.
  • The Na+ equilibrium potential tends to be positive; as Vm becomes more positive (toward ENa) the driving force for Na+ decreases and the current decreases accordingly.
  • In neurons at rest, Vm is typically closer to EK than ENa because of higher permeability to K+ (PK > PNa) at rest.

When an Ion Isn’t at Its Happy Place

  • Resting current arises when the membrane is not at ion-specific equilibrium potentials for all permeant ions; current is generated only if there is conductance and a driving force.
  • If Vm equals Eion for a particular ion or if conductance for that ion is zero, that ion contributes no current.
  • Summary: resting Vm is a weighted result of multiple ions’ equilibrium potentials and their respective permeabilities. An ion’s current depends on whether there is both a driving force and a channel pathway to carry it.

The Goldman Equation in Practice: Predicting Resting Membrane Potential (RMP)

  • The Goldman equation shows Vm as a function of ion concentrations and permeabilities, not just a single ion’s Eion.
  • Practical observation from the transcript:
    • Increasing extracellular K+ depolarizes the cell (Vm moves toward 0 mV).
    • The Nernst prediction for Vm as [K+]out is varied (the black line) does not perfectly match the measured Vm (the orange line) because not all current at rest is carried by K+. Other ions (e.g., Na+) contribute to resting conductance.
  • The gradient and permeability interplay determine the actual RMP; small changes in permeability can significantly shift Vm.

Astrocytes and Extracellular Potassium Regulation (Potassium Spatial Buffering)

  • The brain employs mechanisms to regulate extracellular potassium concentration:
    • Blood-brain barrier reduces the passage of K+ between blood and brain tissue.
    • Astrocytes participate in potassium spatial buffering:
    • They express pumps to concentrate K+ intracellularly.
    • Their expansive branched networks help dissipate excess K+ by distributing it through their processes and into surrounding tissue, keeping extracellular [K+] in check.
  • This regulatory system helps maintain stable Vm and prevents excessive depolarization that could disrupt neuronal signaling.

Practical and Ethical Contexts Mentioned in the Transcript

  • A brief example included in the transcript describes lethal injection as a sequence of drugs and their systemic effects: sodium thiopental (puts the person to sleep), pancuronium bromide (muscle relaxant, paralyzes the diaphragm), and potassium chloride (stops the heart). This is provided as a real-world context for potassium-related physiology but is not a subject for electrophysiology calculations. It serves as a reminder of the ethical considerations surrounding pharmacology and physiology.

Chapter 3 Learning Objectives (Aligned with the Notes)

  • Describe ion channels versus pumps and which requires ATP.
  • Explain the role of amino acids in protein structure relevant to ion channels and pumps.
  • Define electrochemical forces and membrane potential V(m).
  • Distinguish diffusion versus electrostatic pressure.
  • Define equilibrium potential and driving force.
  • Determine when the membrane reaches an ion’s equilibrium potential.
  • Compute the current for a given ion at a given EK or ENa (e.g., what is the current at a specific Vm and g_ion).
  • Explain what happens if an ion is not at its equilibrium potential and what factors influence this.
  • Distinguish between the Nernst and Goldman potentials.
  • Discuss how resting membrane potential is maintained and why it is typically around -70 mV (or neuron-dependent).
  • Explain how changing K+ or Na+ concentrations alters the resting membrane potential and action potentials.
  • Understand the role of permeability changes in altering Vm.