Resting Potential Of Biomembranes: Exhaustive Study Notes

Overview of the Resting Potential

• Definition: The resting potential is defined as the electric potential difference ($\Delta \phi$) between the interior and exterior of a cell when the cell is in a state of physiological rest.

• Charge Distribution: The internal surface of the cell membrane carries a net negative charge relative to the external surface.

• Magnitude: The resting potential varies across different cell types, typically ranging from $-50\,mV$ to $-100\,mV$.

• Key Components of Study:

  • Equilibrium membrane potential.

  • Steady-state membrane potential.

  • The electrogenic role of the sodium-potassium ($Na^+ - K^+$) ion pump.

  • Properties of the resting potential in damaged cells.

Measurement of the Resting Potential

• Apparatus: Measurement is conducted using microelectrodes and an amplifier.

  • Glass Microelectrode (Probe): Inserted directly into the cell's cytoplasm.

  • Reference Microelectrode: Placed in the extracellular fluid surrounding the cell.

• Observation Scenarios:

  • Both electrodes outside: No potential difference is detected ($0\,mV$).

  • Both electrodes inside: No potential difference is detected ($0\,mV$).

  • Probe inside, reference outside: A negative resting potential is recorded.

• Subject Selection: Large cells are preferred for these measurements to minimize damage; a common example is the giant axon of the squid.

Nature of the Resting Potential: Bernstein’s Hypothesis

• The Hypothesis: Proposed by Julius Bernstein, this theory suggests that the cell membrane is exclusively permeable to potassium ions ($K^+$) at rest.

• Equilibrium Model: Under this hypothesis, the resting potential is equivalent to the equilibrium potential for potassium ions.

• The Nernst Equation: Used to calculate the potential based on concentration gradients:     $\Delta \phi = \phi_i - \phi_e = \frac{kT}{e} \ln \left( \frac{c_{K,e}}{c_{K,i}} \right)$

  • Where $k$ is the Boltzmann constant, $T$ is absolute temperature, $e$ is elementary charge, $c_{K,e}$ is extracellular potassium concentration, and $c_{K,i}$ is intracellular potassium concentration.

• Balancing Forces for $K^+$:

  • Concentration Gradient: Since $[K^+]$ is higher inside the cell, diffusion moves $K^+$ out of the cell.

  • Electric Field: Since the cell interior is negative, the electric field pulls $K^+$ back into the cell.

  • Equilibrium Condition: Potential is reached when these two opposing flux tendencies perfectly balance, or when the membrane is permeable only to potassium.

Discrepancies and Limitations of Bernstein’s Hypothesis

• Incompleteness: The potassium hypothesis is incomplete because the membrane is actually permeable to other ions, primarily sodium ($Na^+$) and chloride ($Cl^-$).

• Experimental Variance: Theoretical values calculated using the Nernst equation for potassium often differ significantly from actual measured experimental data.

• Sodium Non-Equilibrium: At rest, the membrane is slightly permeable to sodium ions. Unlike potassium, both the electric gradient and the concentration gradient for sodium drive the ion into the cell, meaning sodium is not in equilibrium.

Quantitative Experimental Data: Squid Axon and Frog Muscle

Giant Axon of the Squid

• Intracellular Concentrations (In):

  • $[Na^+] = 50\,mM$

  • $[K^+] = 400\,mM$

  • $[Cl^-] = 40 \text{ to } 150\,mM$

• Extracellular Concentrations (Out):

  • $[Na^+] = 440\,mM$

  • $[K^+] = 20\,mM$

  • $[Cl^-] = 560\,mM$

• Calculated Nernst Potentials:

  • $Na^+$: $+56\,mV$

  • $K^+$: $-77\,mV$

  • $Cl^-$: $-34 \text{ to } -68\,mV$

• Measured Resting Potential: $-60\,mV$

Frog Muscle Cell

• Intracellular Concentrations (In):

  • $[Na^+] = 10.4\,mM$

  • $[K^+] = 124\,mM$

  • $[Cl^-] = 1.5\,mM$

• Extracellular Concentrations (Out):

  • $[Na^+] = 109\,mM$

  • $[K^+] = 2.25\,mM$

  • $[Cl^-] = 77.5\,mM$

• Calculated Nernst Potentials:

  • $Na^+$: $+60\,mV$

  • $K^+$: $-103\,mV$

  • $Cl^-$: $-101\,mV$

• Measured Resting Potential: $-90\,mV$

Ionic Fluxes and the Hodgkin-Katz Theory

• Core Concept: The resting potential is a "steady state" potential, not an "equilibrium" potential. It is the net result of the simultaneous fluxes of $K^+$, $Na^+$, and $Cl^-$.

• Potassium Dynamics at Rest: For a squid axon, the resting potential ($-60\,mV$) is less negative than the $K^+$ equilibrium potential ($-77\,mV$). Consequently, the electric force is too weak to keep $K^+$ in, resulting in a net flux of potassium leaking out of the cell.

• Sodium Dynamics at Rest: The resting potential ($-60\,mV$) is far from the $Na^+$ equilibrium potential ($+56\,mV$). Both driving forces push sodium into the cell, resulting in a net leakage of sodium into the cell.

• Charge Flux Density ($J$): The total charge flux density across the membrane is the sum of the cation fluxes minus the anion fluxes:     $J = J_K + J_{Na} - J_{Cl}$

  • By convention, $J$ is positive when ions move into the cell.

Steady State Conditions and Membrane Permeability

• Steady State Definition: At steady state, the net charge flux density is zero ($J = 0$), meaning no net current flows across the membrane. This prevents the membrane potential from changing.

• Chloride Influence: In many resting cells, the chloride distribution is close to equilibrium ($J_{Cl} \approx 0$). Therefore, the steady state condition simplifies to:     $J_K + J_{Na} = 0$

• Permeability ($P$) and Driving Force ($X$): Flux density is proportional to these two factors:     $J \sim PX$

• Balance of Factors:

  • Potassium: High permeability ($P_K \text{ is large}$) but small net driving force ($X_K \text{ is small}$) because electric and concentration gradients oppose each other.

  • Sodium: Low permeability ($P_{Na} \text{ is small}$) because sodium channels are closed, but large net driving force ($X_X \text{ is large}$) because both gradients act in the same direction.

  • Result: It is possible for $P_K X_K \approx P_{Na} X_{Na}$, allowing for a steady state where leaks cancel each other out.

The Goldman Equation

• Function: Calculates the steady-state resting potential ($\Delta \phi$) for a membrane permeable to multiple ions.

• Equation (Potassium and Sodium):     $\Delta \phi = \frac{kT}{e} \ln \left( \frac{P_K c_{K,e} + P_{Na} c_{Na,e}}{P_K c_{K,i} + P_{Na} c_{Na,i}} \right)$

• Full Equation (Including Chloride):     $\Delta \phi = \frac{kT}{e} \ln \left( \frac{P_K c_{K,e} + P_{Na} c_{Na,e} + P_{Cl} c_{Cl,i}}{P_K c_{K,i} + P_{Na} c_{Na,i} + P_{Cl} c_{Cl,e}} \right)$

  • Note: For chloride ($Cl^-$), the concentration terms are reversed (extracellular in denominator, intracellular in numerator) because of its negative charge.

• Dominant Ion: The ion with the highest permeability contributes most significantly to the resting potential.

• Relationship to Nernst: If $P_K \gg P_{Na}$ and $P_K \gg P_{Cl}$, the Goldman equation reduces to the Nernst equation for potassium.

• Squid Axon Permeability Ratio: $P_K : P_{Na} : P_{Cl} = 1 : 0.04 : 0.45$.

The Sodium-Potassium ($Na^+ - K^+$) Ion Pump

• Role: The pump maintains the resting potential by compensating for the continuous leakage of potassium out of the cell and sodium into the cell.

• Mechanism: It uses active transport to move $Na^+$ out of the cell and $K^+$ back into the cell.

• Electrogenic Property: The pump generates a net outward flux of positive charge. This directly makes the internal electric potential more negative.

• Importance: It is essential for maintaining the ion concentration gradients that enable the resting potential in the first place.

Resting Potential of Damaged Cells

• Increased Permeability: In damaged cell membranes, permeability to all ions increases, and transport processes lose their specificity.

• Decrease in Potential: The absolute value of the resting potential decreases, meaning it becomes less negative (moves closer to zero).

• Severely Damaged State: If the membrane is severely compromised, the ion distribution becomes governed primarily by the concentration of fixed (non-diffusible) anions within the cell.

• Donnan Equilibrium: In such cases, the membrane potential reaches the Donnan equilibrium potential.