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Comprehensive Notes on Nernst, Goldman, and Resting Membrane Potential (Chapter 3)

Nernst Equation and Equilibrium Potential

  • Purpose: determine the exact equilibrium potential for a single ion, i.e., the membrane potential at which that ion has no net driving force when the membrane is permeable only to that ion.
  • Equilibrium potential for ion i (Eᵢ): the voltage the inside of the cell would need to be so that ions don’t want to move (balance of electric and concentration (diffusion) forces).
  • Key idea: Eᵢ is the balancing point between electric forces and diffusion forces acting on that ion.
  • Monovalent ion example (37°C):
    • For K⁺: EK = 61.54\,\mathrm{mV} \cdot \log{10}\left(\frac{[K^+]{out}}{[K^+]{in}}\right)
    • If the log term is −1.3, then E_K = 61.54 \times (-1.3) \approx -80\ \mathrm{mV} (as given in the transcript).
    • z = +1 for K⁺.
  • Interpretation: When the membrane is permeable only to K⁺, the resting inside potential would be at E_K (e.g., around −80 mV with the provided concentrations).
  • Temperature effect (brief): The Nernst relationship is temperature dependent through the factor 61.54 mV; higher temperatures speed diffusion and slightly alter the exact numerical factor; general form at temperature T is Ei = \frac{RT}{zF}\ln\left(\frac{[i]{out}}{[i]_{in}}\right), which at 37°C becomes the commonly used 61.54 mV multiplier in base-10 form.
  • Practical note: In practice, neurons are rarely at the exact equilibrium for any single ion; multiple ions contribute to the membrane potential, requiring more comprehensive models (e.g., Goldman).

Equilibrium Potential and Driving Force Concepts

  • Equilibrium potential (Eᵢ): voltage where there is no net ionic flux for ion i if the membrane were permeable only to i.
  • Driving force for ion i: the electrical force that acts on the ion given the current membrane potential, defined as DFi = Vm - E_i.
  • Current for ion i: Ii = gi (Vm - Ei) where gᵢ is the conductance (permeability) for ion i.
  • If the membrane potential equals Eᵢ (DFᵢ = 0), the current for that ion is zero (no net movement driven by gradient/electric forces).
  • Conditions for current flow (from the transcript):
    • There must be charge carriers (ions).
    • There must be a passageway (ion channels open; nonzero conductance).
    • There must be a driving force (concentration gradient and/or voltage).
  • Summary relation: Current flow depends on conductance and driving force: I = g \times (V_m - E)

Potassium Equilibrium: Practice Scenarios

  • Let’s determine the equilibrium potential for potassium when the membrane is only permeable to K⁺. Example scenarios provided:
    • Outside: 4, Inside: 140 → roughly the typical intracellular/extracellular K⁺ values; using the Nernst equation: EK = 61.54\log\left(\frac{[K^+]{out}}{[K^+]_{in}}\right) which yields a negative value (around −80 mV in the transcript).
    • Outside: 40, Inside: 140 → would make the log term more positive, giving a less negative EK (the inside would need to be less negative to balance diffusion).
  • Question: What happens to the equilibrium potential for potassium if extracellular potassium increases? Why?
    • Answer as per transcript: EK becomes more positive (less negative) as [K⁺]out increases, because the ratio [K⁺]out/[K⁺]in increases.
  • Could EK ever be zero? What would that mean?
    • If [K⁺]out equals [K⁺]in, log term is log(1) = 0, so EK = 0 mV. This would mean no net diffusion/electric driving force for K⁺ at that moment.
  • Practical exercise prompts from transcript (also includes a web resource note):
    • How would EK change if intracellular/extracellular concentrations are altered as described? (Example values and outcomes in the transcript: EK ≈ −80 mV with typical values; shifting extracellular K⁺ changes EK linearly on a log scale.)

Sodium Equilibrium: Practice and Implications

  • Similar approach for Na⁺: If membrane is only permeable to Na⁺, the equilibrium potential is given by
    E{Na} = 61.54\,\mathrm{mV} \cdot \log{10}\left(\frac{[Na^+]{out}}{[Na^+]{in}}\right).
  • Example scenarios in the transcript (per practice prompts): Outside: 140, Inside: 10 or 20; Outside: 70, Inside: 10.
  • What happens to the Na⁺ equilibrium when intracellular Na⁺ increases? The driving force decreases toward zero as ENa remains high but Vm becomes less negative, reducing net Na⁺ movement unless channels are open.
  • Condition where E_{Na} would be zero would require [Na^+]out = [Na^+]in.
  • How far from equilibrium is Na⁺ when resting Vm is around −70 mV? Since E_Na ≈ +62 mV, the driving force is substantial (about 132–132 mV when Vm ≈ −70 mV), pushing Na⁺ inward if channels are open.
  • Conceptual takeaway: At rest, Na⁺ has a large potential to enter when channels are open, but resting currents are limited by low Na⁺ permeability relative to K⁺ permeability.

Resting Membrane Potential: Which Ion Controls the Resting State?

  • At rest, many neurons are not at the exact equilibrium for any single ion; the resting potential is a result of multiple ions and their permeabilities.
  • If both Na⁺ and K⁺ could move at rest, why isn’t Vm equal to the sum of their separate equilibrium potentials (−80 mV for K⁺ and +62 mV for Na⁺)?
    • Because the membrane is not at the individual equilibria for all ions simultaneously; it is governed by relative permeabilities and the combined conductance of all ions that are permeable at rest.
  • A qualitative example from the transcript: K⁺ moves a lot; Na⁺ moves a little at rest; thus the resting Vm is closer to EK than to ENa, but not exactly EK due to some Na⁺ permeability and other ions’ contributions.
  • Resting Vm is often around −70 mV in many neurons, a balance primarily shaped by permeabilities rather than a single ion's equilibrium potential.

Goldman-Hodgkin-Katz (GHK) Equation: A More Comprehensive Model

  • Motivation: The Goldman equation accounts for charge, relative concentrations, and permeabilities of multiple ions rather than treating ions one at a time.
  • Two common forms (conceptually):
    • Simplified two-ion form (K⁺ and Na⁺):
      Vm = 61.54\,\mathrm{mV} \cdot \log{10}\left( \frac{PK [K^+]{out} + P{Na} [Na^+]{out}}{PK [K^+]{in} + P{Na} [Na^+]{in}} \right)
    • General GHK form (including Cl⁻ and full ion set):
      Vm = \frac{RT}{F} \ln\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 interpretation: Vm is determined by the balance of driving forces from all permeant ions, weighted by their permeabilities (Pᵢ).
  • Consequence: As permeability to a particular ion changes (e.g., during signaling), Vm shifts toward the corresponding ion’s equilibrium potential.
  • Applications: Predicting resting membrane potential under varying extracellular K⁺, Na⁺, and other ion concentrations; explains why changes in one ion’s permeability can depolarize or hyperpolarize the cell beyond a simple Nernst prediction.

Goldman Equation and Resting Membrane Potential: Practical Insights

  • When extracellular [K⁺] increases, the resting Vm depolarizes (becomes less negative).
  • The Goldman prediction (black line) may deviate from the observed Vm (orange line) because at rest not all current is carried by K⁺; other ions contribute to the resting current.
  • The slope of Vm with respect to extracellular [K⁺] indicates how sensitive Vm is to [K⁺]out: the transcript notes a slope of about 58 mV per tenfold change in the K⁺ gradient for the observed data.
  • Takeaway: The resting membrane potential is not simply EK but is a weighted outcome of multiple ions’ permeabilities and gradients as captured by the Goldman equation.

What Ion Is Actually in Charge at Rest?

  • Given a resting Vm (e.g., −68 mV), and the ions’ equilibrium potentials (e.g., EK ≈ −80 mV, ENa ≈ +62 mV):
    • Driving forces: DFK = Vm − EK ≈ (−68) − (−80) ≈ +12 mV; DFNa = Vm − ENa ≈ (−68) − (+62) ≈ −130 mV.
    • Interpretation: Na⁺ has a large driving force toward ingress (negative driving force with this sign convention), while K⁺ has a smaller driving force outward.
    • Yet, the actual current is dominated by the ion with the larger conductance at rest (permeability). Since K⁺ typically has higher resting permeability, K⁺ often dominates the resting current and helps set Vm closer to EK, even though Na⁺ has a strong driving force when open.
  • The “control” of Vm is a balance between permeability and driving force for all permeant ions, not a simple sum of equilibrium potentials.

Mechanisms Regulating Extracellular Potassium and Brain Homeostasis

  • Why don’t neurons stop functioning when extracellular K⁺ rises after a banana (or diet)? Mechanisms include:
    • Blood–brain barrier: limits movement of potassium between blood and brain interstitial fluid.
    • Astrocyte-mediated potassium spatial buffering: astrocytes take up excess K⁺ via pumps and distribute it through their extensive processes to maintain local [K⁺] and prevent localized depolarization.
    • Potassium pumps to concentrate K⁺ intracellularly (and mechanisms to dissipate K⁺ through astrocyte networks).
  • Overall: The brain has active homeostatic systems to regulate extracellular K⁺ and prevent runaway depolarization of neurons.

Lethal Injection: Ion Context in Practice

  • Three-injection protocol (as described in the transcript):
    1) Sodium thiopental – anesthetic that induces sleep.
    2) Pancuronium bromide – muscle relaxant that paralyzes muscles, including the diaphragm.
    3) Potassium chloride – stops the heart by causing lethal hyperkalemia, leading to cardiac arrest.
  • Conceptual link: Potassium dynamics are central to cardiac and neuronal excitability; dramatic changes in extracellular K⁺ can drastically alter resting potentials and action potential generation.
  • Ethical and policy note: The description provided is from a public source and illustrates how pharmacology can affect neural and cardiac function; the material sits at the intersection of physiology and ethics/policy.

Chapter 3 Learning Objectives (Summary of Topics to Master)

  • Distinguish ion channels versus pumps and know which requires ATP (pumps) and why ATP is necessary.
  • Understand the role of amino acids in protein structure (e.g., in ion channels and pumps).
  • Define electrochemical forces and membrane potential V(m).
  • Distinguish diffusion vs. electrostatic pressure and how they contribute to ion movement.
  • Define equilibrium potential and driving force.
  • Describe conditions under which the membrane reaches an ion’s equilibrium potential.
  • Compute current flow at a given EK (example calculations for K⁺).
  • Explain what happens if an ion is not at its equilibrium potential and what factors influence this.
  • Compare and contrast Nernst potential vs. GHK potential.
  • Explain how resting membrane potential is maintained and why it is typically around −70 mV (and how this varies between neurons).
  • Describe how changes in K⁺ and Na⁺ concentrations alter resting membrane potential and action potentials.
  • Understand the qualitative and quantitative implications of the Goldman equation for resting Vm.