Lecture 4: Equilibrium Potential I

Foundational References and Core Topics in Cellular Electrophysiology

• Primary authoritative texts in quantitative membrane physiology:

  • Quantitative Physiology for Engineers (Feher)

  • Medical Physiology (Boron and Boulpaep)

  • Human Physiology (Widmaier, Raff and Strang)

  • Human Physiology (Silverthorn)

• Core curricular topics in electrochemical membrane transport:

  • Ion Movement

  • Ion Currents

  • Membrane Potential

  • Fick’s Law of Diffusion

  • Ohm’s Law for Drift

  • Nernst-Planck Equation

  • Equilibrium Potential

Structural and Kinetic Properties of Ion Channels

• Primary ionic species conducted across biological membranes:

  • Sodium ($Na^+$)

  • Potassium ($K^+$)

  • Calcium ($Ca^{2+}$)

  • Chloride ($Cl^-$)

• Selectivity and pore biophysics:

  • Selectivity for a specific ion species is governed strictly by the physical dimensions, hydration shell constraints, and amino acid residue charge distributions of the channel pore.

  • Ion permeation is a passive process once the gate is open and does not require direct cleavage or direct dependence on adenosine triphosphate ($\text{ATP}$).

  • Channels are non-static structures; ambient thermal fluctuations drive stochastic, random conformational transitions between open and closed gating states.

• Rate of throughput across channel pores versus biocatalysts:

  • Open ion channels conduct at high flux rates of approximately $\sim 10^8\,\text{ions/sec}$.

  • Classical metabolic enzymes turn over substrates at substantially slower rates of $< 10^7/\text{sec}$.

• Saturation kinetics and pharmacological channel inhibition:

  • Ion conduction is finite and exhibits saturation at elevated substrate concentrations.

  • Conduction can be selectively blocked by specific biological toxins and therapeutic agents:

  • Tetrodotoxin ($\text{TTX}$): Extracted from the puffer fish (Fugu); functions as a selective blocker of the voltage-gated $Na^+$ channel.

  • Tetraethylammonium ($\text{TEA}$): Acts as an exogenous pharmacological blocker that selectively obstructs the voltage-gated $K^+$ channel.

  • Xylocaine (lidocaine): Functions as a small molecule channel blocker.

• Rectification and conductance parameters:

  • Rectifying Channels: Ion channels that permit ion movement more favorably in one direction across the membrane than the opposite direction.

  • Net rate of ion transfer (ionic current, $I$) is determined by channel conductance ($G$), which depends on three primary factors:

  1. Channel permeability ($P$), representing intrinsic structural and biophysical properties of the channel pore.

  2. Concentration of the permeant ion species directly surrounding the channel pore opening, as determined by the transmembrane concentration gradient.

  3. Transmembrane voltage ($V_M$).

Membrane Permeability and Solute Transport Rates

• Definition of membrane permeability ($P$):

  • Defines the rate at which a solute traverses a lipid bilayer membrane.

  • Dimensional units are velocity units: $\text{cm/sec}$.

• Relative baseline permeabilities of biological solutes across pure lipid bilayers:

  • Water: $10^{-4}\,\text{cm/sec}$ to $10^{-3}\,\text{cm/sec}$

  • Urea: $10^{-6}\,\text{cm/sec}$

  • Glucose and Amino Acids: $10^{-7}\,\text{cm/sec}$

  • Chloride ($Cl^-$): $10^{-11}\,\text{cm/sec}$

  • Sodium ($Na^+$) and Potassium ($K^+$): $10^{-13}\,\text{cm/sec}$

• Relationship of permeability to solute flux ($J$):

  • Multiplying the membrane permeability coefficient ($P$) by the solute concentration difference ($[C]$) yields the net flux:     $\text{Flux} = P \times [C]$

  • Flux denotes the physical amount of solute crossing a specified membrane surface area per unit of time (typically expressed in $\text{moles}/(\text{cm}^2\,\text{s})$).

Electrical Foundations of Membrane Biophysics

• Dual physical driving forces operating on biological ions:

  • Diffusion: Driven down chemical concentration gradients (from regions of elevated concentration to regions of lower concentration).

  • Electricity: Driven along transmembrane electrical gradients or electric potential differences ($V_M$).

• Basic electrical definitions and physical units:

  • Voltage ($V$): The electrical driving potential or electromotive "pressure" that propels charged particles (electrons or ions) through a conductive biological or physical circuit; measured in Volts ($\text{V}$).

  • Current ($I$): The net quantity of charge carriers (electrons or ions) traversing a cross-section per unit time; measured in Amperes ($\text{A}$), defined as:     $1\,\text{A} = 1\,\text{coulomb/sec}$

• Fundamental electrostatic constants and equivalences:

  • Elementary charge equivalent:     $1\,\text{coulomb}\,(C) = 6.25 \times 10^{18}\,\text{charges}$

  • Faraday's constant ($F$), representing the electric charge contained within one mole of monovalent charges:     $1\,\text{Faraday}\,(F) = 1\,\text{mole of charges} = 96,500\,\text{coulomb}\,(C)$

Sign Conventions and Polarity of Transmembrane Ionic Currents

• Directional nomenclature of ionic currents:

  • Negative Currents ($\rightarrow$ Inward Currents):

  • Defined by the physical translocation of a positively charged cation into the intracellular volume.

  • Defined alternatively by the exit of a negatively charged anion out of the cell.

  • Canonical example: The influx of $Na^+$ into the cell constitutes an inward, negative current.

  • Electrical consequence on the membrane: Cation influx produces an inward membrane current ($I_m$) that drives depolarization of the membrane potential ($V_m$).

  • Positive Currents ($\rightarrow$ Outward Currents):

  • Defined by the physical movement of a negatively charged anion into the intracellular volume.

  • Defined alternatively by the exit of a positively charged cation out of the cell into the extracellular space.

  • Canonical example: The efflux of $K^+$ leaving the cell constitutes an outward, positive current.

  • Electrical consequence on the membrane: Cation efflux produces an outward membrane current ($I_m$) that drives hyperpolarization of the membrane potential ($V_m$).

Governing Equations of Electrodiffusion: Fick's Law, Ohm's Law, and Nernst-Planck

• Fick’s Law of Diffusion:

  • Chemical diffusion occurs down a concentration gradient (from higher concentration to lower concentration).

  • The rate of diffusion is directly proportional to the magnitude of the concentration gradient, scaled by the diffusion coefficient ($D$):     $J_{diff} = -D \frac{\partial [C]}{\partial x}$

  • Definitions of variables and physical units:

  • $J_{diff}$: Diffusion flux ($\text{moles}/(\text{cm}^2\,\text{s})$)

  • $D$: Diffusion coefficient ($\text{cm}^2/\text{s}$)

  • $\frac{\partial [C]}{\partial x}$: Spatial concentration gradient ($\text{moles}/\text{cm}^3/\text{cm}$)

  • The negative sign indicates mathematically that diffusion flux ($J_{diff}$) flows in the direction of decreasing concentration.

  • The total diffusion rate across a membrane depends on five fundamental parameters:

  1. Concentration gradient across the barrier

  2. Permeability of the membrane to the solute

  3. Surface area of the membrane available for exchange

  4. Molecular weight and geometric dimensions of the solute molecule

  5. Diffusion distance (membrane thickness, $x$)

• Ohm’s Law for Electrodiffusive Drift:

  • Solvated ions experience an electrostatic mechanical force generated by the interaction between their ionic valence charge and the prevailing biological electric field.

  • Drift flux ($J_{drift}$) through an electric field is expressed by:     $J_{drift} = -\mu z [C] \frac{\partial V}{\partial x}$

  • Definitions of variables and physical units:

  • $J_{drift}$: Drift flux ($\text{moles}/(\text{s}\,\text{cm}^2)$

  • $\mu$: Electrophoretic mobility coefficient ($\text{cm}^2/(\text{V}\,\text{s})$)

  • $z$: Valence charge of the ion (dimensionless integer)

  • $[C]$: Ion concentration ($\text{moles}/\text{cm}^3$)

  • $\frac{\partial V}{\partial x}$: Electric potential gradient across space ($\text{V}/\text{cm}$)

  • The negative sign demonstrates that positively charged ions drift down the electric potential gradient (from higher to lower potential) directly proportional to the electric field magnitude, with a proportionality constant equal to $\mu z [C]$.

• The Nernst-Planck Equation:

  • Transmembrane ion transport is simultaneously governed by both the transmembrane electric field and chemical concentration gradients.

  • The physiological origins of this dual driving force comprise:

  1. Differential intracellular and extracellular ion concentrations, maintained by primary active transport (e.g., pumps) and passive equilibrium distributions. Chemical concentration gradients drive diffusion from high to low concentrations.

  2. A nonzero electric field established across the lipid bilayer due to localized microscopic charge separation. Negative ions move toward excess positive charges, and positive ions move toward excess negative charges.

  • The total ion flux ($J$) is the exact linear summation of electrodiffusive drift flux and chemical diffusion flux:     $J = J_{drift} + J_{diff} = -\mu z [C] \frac{\partial V}{\partial x} - D \frac{\partial [C]}{\partial x}$

The Nernst Equation and Thermodynamic Equilibrium Potential

• Biophysical definition of equilibrium potential:

  • When the net cross-membrane current for a specific permeant ion is zero ($I = 0$), the biological membrane is at thermodynamic rest with respect to that ion.

  • At this equilibrium state, the chemical driving force (concentration gradient) and the electrical driving force (transmembrane electric field) are exactly equal in magnitude and oppositely oriented.

  • The transmembrane voltage corresponding to this zero-net-flux state is termed the Equilibrium Potential ($E_i$) or reversal potential of the ion.

• Formulation of the Nernst Equation:

  • Expresses the thermodynamic equilibrium potential as a function of the transmembrane concentration ratio:     $E_i = \frac{RT}{zF} \ln\left(\frac{[C_i]_{out}}{[C_i]_{in}}\right)$

  • Definitions of equation constants and terms:

  • $R$: Universal gas constant

  • $T$: Absolute thermodynamic temperature (Kelvin)

  • $z$: Valence charge of the ion species

  • $F$: Faraday’s constant ($96,500\,\text{coulomb/mole}$

  • $[C_i]_{out}$: Extracellular concentration of ion $i$

  • $[C_i]_{in}$: Intracellular concentration of ion $i$

• Simplified base-10 logarithmic formulations (treating $\frac{RT}{F}$ as a constant):

  • At room temperature ($T = 20^\circ\text{C}$):     $E_i = \frac{58\,\text{mV}}{z} \log_{10}\left(\frac{[C_i]_{out}}{[C_i]_{in}}\right)$

  • At mammalian physiological body temperature ($T = 37^\circ\text{C}$):     $E_i = \frac{62\,\text{mV}}{z} \log_{10}\left(\frac{[C_i]_{out}}{[C_i]_{in}}\right)$

• Primary physiological variables governing equilibrium potential:

  1. Valence charge on the ion ($z$), which dictates responsiveness to electrical driving forces and fixes the mathematical sign.

  2. Ratio of external to internal ion concentrations ($\frac{[C_i]_{out}}{[C_i]_{in}}$), which dictates the magnitude and direction of the chemical driving force.

  3. Absolute temperature ($T$), which scales thermal kinetic energy and diffusion.

• Worked physiological example for Potassium ($K^+$):

  • Physiological input parameters:

  • $[K^+]_o = 4.5\,\text{mM}$

  • $[K^+]_i = 155\,\text{mM}$

  • $T = 37^\circ\text{C}$

  • $z = +1$

  • Step-by-step calculation using the body-temperature Nernst formulation:     $E = \frac{62\,\text{mV}}{z} \log_{10}\left(\frac{[C]_{out}}{[C]_{in}}\right)$     $E_{K^+} = \frac{62\,\text{mV}}{1} \log_{10}\left(\frac{4.5}{155}\right)$     $E_{K^+} = -95\,\text{mV}$

  • Interpretation: The potassium equilibrium potential ($E_{K^+}$) of $-95\,\text{mV}$ represents the exact intracellular-negative electrical potential required to counter and balance the outward chemical diffusion of $K^+$ down its steep concentration gradient.

Linear Mathematical Modeling of the Nernst Equation

• Transformation to linear slope-intercept format ($y = m \cdot x + b$):

  • Algebraic expansion using logarithmic quotient rules at $T = 20^\circ\text{C}$:     $E_i = \frac{58\,\text{mV}}{z} \log_{10}\left(\frac{[C_i]_{out}}{[C_i]_{in}}\right)$     $E_i = \frac{58\,\text{mV}}{z} \cdot \left(\log_{10}[C_i]_{out} - \log_{10}[C_i]_{in}\right)$     $E_i = \left(\frac{58\,\text{mV}}{z}\right) \cdot \log_{10}[C_i]_{out} - \left(\frac{58\,\text{mV}}{z}\right) \cdot \log_{10}[C_i]_{in}$

  • Component mapping to the linear equation $y = m \cdot x + b$:

  • Dependent variable ($y$): $E_i$

  • Slope ($m$): $\frac{58\,\text{mV}}{z}$

  • Independent variable ($x$): $\log_{10}[C_i]_{out}$

  • Constant y-intercept ($b$): $-\left(\frac{58\,\text{mV}}{z}\right) \cdot \log_{10}[C_i]_{in}$

• Physical interpretation of slope:

  • For a monovalent cation ($z = +1$), the slope of the line equals precisely $58\,\text{mV}$ per 10-fold (one decade) difference in extracellular ion concentration.

• Quantitative parametric analysis for potassium ($E_K$):

  • Linear relationship expression:     $E_K = \frac{58\,\text{mV}}{z} \cdot \log_{10}[K^+]_{out} - \frac{58\,\text{mV}}{z} \cdot \log_{10}[K^+]_{in}$

  • Empirical calculation for a 10-fold change in extracellular potassium with $[K^+]_{in} = 140\,\text{mM}$ (black trace):

  • Range: $\Delta [K^+]_{out} = 100\,\text{mM} \rightarrow 10\,\text{mM}$

  • At $[K^+]_{out} = 100\,\text{mM}$: $E_K = -8.48\,\text{mV}$

  • At $[K^+]_{out} = 10\,\text{mM}$: $E_K = -66.48\,\text{mV}$

  • Calculated potential change:       $\Delta E_K = -8.48\,\text{mV} - (-66.48\,\text{mV}) = 58\,\text{mV}$

  • Effect of shifting intracellular potassium concentrations ($[K^+]_{in}$):

  • Standard intracellular baseline ($[K^+]_{in} = 140\,\text{mM}$, black line): Passes through $-8.48\,\text{mV}$ at $[K^+]_{out} = 100\,\text{mM}$.

  • Increased intracellular concentration ($[K^+]_{in} = 180\,\text{mM}$, red line): Increases the magnitude of the negative constant term $b$, shifting the linear response downward to more negative potentials for any given extracellular potassium concentration.

  • Decreased intracellular concentration ($[K^+]_{in} = 100\,\text{mM}$, blue line): Decreases the magnitude of the negative constant term $b$, shifting the line upward toward less negative (more depolarized) potentials across all extracellular concentration values.

  • Invariance of slope: Because the slope depends strictly on $\frac{58\,\text{mV}}{z}$, altering intracellular concentration shifts the intercept without altering the slope, generating parallel response lines across all conditions.

Summary of Equilibrium Electrophysiology Principles

• Transmembrane ion flux (current) is propelled by the summation of chemical and electrical driving potentials.

• Macroscopic ionic currents carry distinct polarities and signs based on valence and vector direction:

  • Inward currents (negative sign) represent either cation entry into the cell or anion exit out of the cell, driving membrane depolarization.

  • Outward currents (positive sign) represent either cation exit out of the cell or anion entry into the cell, driving membrane hyperpolarization.

• An individual ion species’ equilibrium potential across the membrane is mathematically determined by the ratio of its external to internal concentrations as formulated by the Nernst Equation.

• The equilibrium potential represents the critical reversal voltage: the exact membrane potential at which net transmembrane current for that ion is zero and across which the direction of net flux reverses.

• The Nernst Equation behaves as a linear equation exhibiting a constant slope of $\frac{58\,\text{mV}}{z}$ at $20^\circ\text{C}$ (or $\frac{62\,\text{mV}}{z}$ at $37^\circ\text{C}$) for every 10-fold alteration in concentration gradient.