Resting Membrane Potential and Ionic Equilibrium

Chemical Principles of Membrane Potentials

• Ion Definition and Key Biological Ions:

  • An ion is an atom or molecule carrying a net electrical charge.

  • Potassium ion: $\text{K}^+$ (the predominant positively charged ion inside a resting neuron).

  • Sodium ion: $\text{Na}^+$ (predominantly present in extracellular fluid).

  • Chloride ion: $\text{Cl}^-$ (a major negatively charged anion).

• Concentration Notation:

  • Square brackets $[\ ]$ denote "concentration of" a specified chemical species (e.g., $[\text{K}^+]$ represents potassium ion concentration).

• Diffusion Properties:

  • Diffusion is the passive movement of a substance from an area of high concentration to an area of low concentration ($[\text{high}] \rightarrow [\text{low}]$).

  • Diffusion is energy-free, occurring spontaneously down a concentration gradient.

• Voltage and Electrostatic Forces:

  • Voltage ($V$) represents an electrical potential difference resulting from a separation of opposite charges across a membrane or distance.

  • Opposite electrical charges attract each other, whereas like charges repel.

  • A positively charged ion, such as $\text{K}^+$, experiences an electrostatic force pushing it away from a positively charged environment and pulling it toward a negatively charged environment.

Theoretical Model of Resting Membrane Potential Generation

• Selective Permeability Premise:

  • Inside a resting neuron, $\text{K}^+$ is the most abundant cation relative to the extracellular fluid.

  • In a theoretical membrane model, assume the lipid bilayer contains $\text{K}^+$ leak channels allowing $\text{K}^+$ to enter and exit freely, while remaining totally impermeable to all other ions (such as $\text{Cl}^-$).

• Temporal Evolution of Membrane Potential ($t_1 \rightarrow t_2 \rightarrow t_3$):

  • Time Phase 1 ($t_1$):

  • Cellular setup: High initial concentration of $\text{K}^+$ and $\text{Cl}^-$ inside the cell ($[\text{K}^+]_{\text{in}}$ high) and low concentration outside ($[\text{K}^+]_{\text{out}}$ low).

  • Charge neutrality: Positive $\text{K}^+$ charges balance negative $\text{Cl}^-$ charges on both sides.

  • Net voltage ($V_{\text{in vs. out}}$): $0\,mV$.

  • Time Phase 2 ($t_2$):

  • Driven by the concentration gradient, $\text{K}^+$ diffuses out of the cell through $\text{K}^+$ leak channels.

  • Impermeable $\text{Cl}^-$ anions cannot follow and remain trapped inside.

  • Inside concentration ($[\text{K}^+]_{\text{in}}$) becomes highish, while outside concentration ($[\text{K}^+]_{\text{out}}$) becomes lowish.

  • Leaving positive charge creates an excess negative charge inside (e.g., $-2$ relative charge) and an excess positive charge outside (e.g., $+2$ relative charge).

  • Voltage ($V_{\text{in vs. out}}$): Drops below zero (e.g., $-2\,mV$).

  • Time Phase 3 ($t_3$) - Electrochemical Equilibrium:

  • As more $\text{K}^+$ diffuses out, internal negative charge increases, exerting an attractive electrostatic force pulling $\text{K}^+$ back inside.

  • Two opposing forces reach exact balance:

    • Force 1: The concentration gradient (force of diffusion) driving $\text{K}^+$ outward.

    • Force 2: The electrical potential difference (electrostatic attraction) driving $\text{K}^+$ inward.

  • When these opposing forces become equal in magnitude, net flux of $\text{K}^+$ ceases, establishing electrochemical equilibrium.

Quantitative Determination of Equilibrium Potential via the Nernst Equation

• The Nernst Equation:

  • The equilibrium potential ($E_x$) for a single monovalent ion $X$ permeable across a membrane is calculated using the Nernst equation:     $E_x = 58 \log\left(\frac{[X]_2}{[X]_1}\right)$

  • Parameter definitions: $E_x$ is expressed in millivolts ($mV$), $[X]_2$ represents extracellular ion concentration, and $[X]_1$ represents intracellular ion concentration.

  • Constant value: The coefficient $58$ incorporates fundamental physical constants ($R$, $T$, $F$) calculated at physiological room temperature.

• Logarithmic Relationship and Slope Properties:

  • The Nernst equation demonstrates that for every tenfold ($10\times$) change in the concentration gradient ratio of $\text{K}^+$, the equilibrium potential changes by exactly $58\,mV$.

  • Linear plotting of membrane potential $V_{1-2}$ (in $mV$) versus $\log\left(\frac{[\text{K}^+]_2}{[\text{K}^+]_1}\right)$ yields a line with a slope of $58\,mV$ per tenfold change:

  • Concentration ratio $1:1$ ($\log(1) = 0$): Membrane potential $V_{1-2} = 0\,mV$.

  • Concentration ratio $1:10$ ($\log(0.1) = -1$): Membrane potential $V_{1-2} = -58\,mV$.

  • Concentration ratio $1:100$ ($\log(0.01) = -2$): Membrane potential $V_{1-2} = -116\,mV$.

Experimental Validation in Giant Squid Axons

• Experimental Hypothesis:

  • If $\text{K}^+$ is the primary ion determining resting membrane potential, manipulating external potassium concentration ($[\text{K}^+]_{\text{out}}$) should predictably shift the resting potential along the predicted Nernstian slope.

• Hodgkin and Katz Investigation (1949):

  • Subject model: Giant squid axons.

  • Baseline physiological measurement: Measured resting membrane potential was $-65\,mV$

  • Measured resting membrane potentials across varying extracellular potassium concentrations ($[\text{K}^+]_{\text{out}}$):

  • $3.5\,mM\,\text{K}^+$: Resting membrane potential measured near $-65\,mV$

  • $10\,mM\,\text{K}^+$: Depolarizes slightly to approximately $-60\,mV$

  • $20\,mM\,\text{K}^+$: Depolarizes further to approximately $-50\,mV$

  • $50\,mM\,\text{K}^+$: Depolarizes to approximately $-40\,mV$

  • $200\,mM\,\text{K}^+$: Depolarizes to approximately $-15\,mV$

  • $450\,mM\,\text{K}^+$: Depolarizes to approximately $0\,mV$ to $+5\,mV$

• Key Findings and Conclusions:

  • At elevated external $\text{K}^+$ concentrations, the measured resting membrane potential strictly follows the theoretical Nernst slope of $58\,mV$ per tenfold change in $\text{K}^+$ gradient.

  • At lower concentrations (near normal physiological levels around $3.5\,mM$ to $10\,mM$), the experimental curve bends away from the theoretical Nernst line and flattens near $-65\,mV$

  • Biological significance: This deviation proves that while selective permeability to $\text{K}^+$ is the dominant factor in establishing resting potential, the resting neuronal membrane is also slightly permeable to other ions.

Multi-Ion Dynamics and the Goldman Equation

• Resting Ion Gradients Across Neuronal Membranes:

  • $\text{K}^+$: Concentration is significantly higher inside the neuron than outside ($[\text{K}^+]_{\text{in}} > [\text{K}^+]_{\text{out}}$).

  • $\text{Na}^+$: Concentration is significantly higher outside the neuron than inside ($[\text{Na}^+]_{\text{out}} > [\text{Na}^+]_{\text{in}}$).

  • $\text{Cl}^-$: Concentration is significantly higher outside the neuron than inside ($[\text{Cl}^-]_{\text{out}} > [\text{Cl}^-]_{\text{in}}$).

• Role of Ion Channels:

  • Selective ion permeability is mediated by specialized trans-membrane protein channels embedded in the lipid bilayer.

  • $\text{K}^+$ leak channels remain open at rest, providing high baseline permeability to $\text{K}^+$.

• The Goldman Equation (Goldman-Hodgkin-Katz Voltage Equation):

  • To account for multiple ion species and their relative membrane permeabilities ($P$), membrane potential ($V$) is calculated using the Goldman equation:     $V = 58 \log\left(\frac{P_K [\text{K}]_2 + P_{Na} [\text{Na}]_2 + P_{Cl} [\text{Cl}]_1}{P_K [\text{K}]_1 + P_{Na} [\text{Na}]_1 + P_{Cl} [\text{Cl}]_2}\right)$

  • Parameter definitions:

  • Subscript $2$ indicates extracellular concentration.

  • Subscript $1$ indicates intracellular concentration.

  • $P_K$, $P_{Na}$, and $P_{Cl}$ represent relative membrane permeabilities for potassium, sodium, and chloride, respectively.

  • Note: Because chloride ($\text{Cl}^-$) carries a negative charge, its concentration positions are inverted in the ratio ($[\text{Cl}]_1$ in the numerator and $[\text{Cl}]_2$ in the denominator).

• Functional Implications of Permeability Shifts:

  • Rapidly shifting membrane permeability from $\text{K}^+$ dominance to $\text{Na}^+$ dominance drives the membrane potential from a strongly negative value (near $-65\,mV$) toward a positive potential.

  • Dynamic regulation of relative ion permeability serves as the fundamental mechanism governing action potential generation and neuronal signaling.

  • Injecting hyperpolarizing current makes the membrane potential even more negative than its resting baseline (e.g., hyperpolarizing beyond $-65\,mV$), which does not induce active firing in the axon.

  • In contrast, depolarizing shifts caused by permeability alterations trigger the onset of an action potential.