Semi-Permeable Membranes and Resting Membrane Potential

Semi-Permeable Membranes and Resting Membrane Potential (BI2432)

Overview of Resting Membrane Potential

Learning Outcomes

1. Understand the relationship between voltage, charge, and capacitance.

2. Understand the membrane integration cycle.

3. Representing a cell as an equivalent electric circuit.

4. Knowledge of electrophysiological recording techniques.

5. Understand battery voltages set up by ion concentration gradients.

6. Ability to predict resting membrane potential ($V_m$) from reversal potentials ($E_{rev}$) and fractional conductances ($f_x$).

7. Using Multiple I-V relationships to deduce resting membrane potential.

Key Components of Semi-Permeable Membranes

• Lipid Bilayer: Itself is impermeable to charged ions (which are hydrophilic). The arrangement of a lipid layer sandwiched between two ionic solutions acts as a capacitance ($C$). It separates and accumulates charge, producing a membrane potential ($V_m$) between the separated opposite charges.

• Pump Proteins (and other transporters): These consume metabolic energy to establish ion concentration gradients. The primary example is the $Na^+/K^+\text{ ATPase}$, which moves ions against their concentration gradients.

• Ion Channel Proteins: These allow ions to cross the membrane much faster than pumps. They rapidly harness the energy stored in concentration gradients to send signals and change the voltage across the membrane.

Physics of the Membrane Potential

Fundamental Principles

• Charge Accumulation: Current ($I$) consists of moving ions. As ions cannot pass directly through the lipid bilayer, opposite charges pile up on either side of the membrane. These charges spread out along the surface, repelling like charges and attracting opposite charges on the other side.

• Total Ion Fraction: The ions that establish the membrane potential make up only a tiny fraction of the total number of ions within the cell.

• Steady-State: A steady-state, unchanging resting membrane potential occurs when there is zero net current flowing onto the intracellular surface of the membrane capacitance (and off the extracellular surface), resulting in no net change in charge ($Q$).

Mathematical Relationships

The membrane potential is determined by the charge stored divided by the capacitance:

$V_m = \frac{Q_{tot}}{C}$

Where:

• $Q = I \times t$ (Charge equals current multiplied by time).

• $I$ is the current injected by ion channels while they are open.

• $C$ is the membrane capacitance provided by the phospholipid bilayer.

The Membrane Integration Cycle

The Mechanism

The membrane integration cycle runs continuously across all patches of the membrane at all times. It follows this sequence of information flow:

1. Concentration Gradients: Selectively permeable ion channels, combined with ion concentration gradients, act as minute batteries.

2. Current Injection: When channels open, they inject current ($I$) into the cell.

3. Capacitance Integration: The membrane capacitance ($C$) integrates these currents.

4. Charge Accumulation: As current flows for a short time ($t$), charge ($Q$) piles up onto the existing charge ($Q_0$).     * Net charge: $Q_{tot} = I \times t - Q_0$

5. Voltage Change: The accumulated charge results in a new membrane potential ($V_m$).

Equivalent Circuit Representation

In an electrical circuit mapping:

• Conductance ($g$): Represents the channel pore conductance.

• Capacitor ($C$): Represents the phospholipid bilayer.

• Current ($I$): Represents the flow of ions through channels.

• Voltage ($V_m$): Represents the potential difference generated across the membrane.

Ion Concentrations and Reversal Potentials

Typical Ion Concentrations (Mammalian Neuron)

Reversal (Equilibrium) Potential

• Definition: The membrane potential at which there is no net flow of ions across a membrane. This is the point where the chemical gradient and electrical gradient are equal and opposite.

• Direction of Force: The net driving force on an ion is the difference between the chemical and electrical gradients.

• Nernst Rule of Thumb: A $10$-fold concentration gradient for a monovalent cation gives approximately $60\,mV$ of battery voltage at $37\,^\circ C$.     * For monovalent anions, multiply by $-1$.     * For divalent cations ($Ca^{2+}$), the value is approximately $30\,mV$.

Electrophysiological Recording Techniques

Recording Modes

1. Extracellular Recording: Uses metal, silicon, or saline-filled glass electrodes placed outside the cell.

2. Intracellular (Sharp) Recording: Uses an electrolyte-filled hollow glass "spear" with a tip diameter less than $0.1\,\mu m$ to impale the cell.     * Drawbacks: Becomes leaky as the membrane seals around the glass; high resistance and noise due to the small tip.

3. Patch-Clamp Recording: Uses a clean glass pipette with a wider tip ($\approx 1\,\mu m$). The membrane is sucked into the pipette, creating a tight seal.     * Giga-seal: A seal resistance greater than $10^9\,\Omega$ ($1\,G\Omega$). This ensures negligible current leakage, reduces noise, and allows for the detection of minute currents (picoamps, $10^{-12}\,A$) from single channels.

Patch-Clamp Configurations

• Cell-Attached: Pipette is sealed against the intact membrane.

• Inside-Out (Excised): A patch of membrane is pulled away, exposing the intracellular surface to the external media.

• Whole-Cell: The membrane patch is ruptured, allowing the pipette to access the entire intracellular space. Allows for voltage or current clamp.

• Outside-Out (Excised): The membrane reseals with the extracellular surface facing the external media.

• Perforated Patch: Uses antibiotics (e.g., nystatin) to form small pores in the membrane rather than rupturing it.

Voltage Clamp vs. Current Clamp

• Voltage Recording (Current Clamp): The researcher records the voltage while injecting constant current or defined pulses. The membrane potential is allowed to vary freely in response to inputs.

• Voltage Clamp: The researcher controls (clamps) the voltage at a constant level or imposes specific command waveforms.     * Mechanism: To hold the voltage constant, the amplifier injects a "clamp current" ($I_{clamp}$) that is equal and opposite to the biological current ($I_{bio}$).     * $I_{clamp} = -I_{bio}$     * Result: No net charge进入 cell means no change in voltage.

Determining Resting Membrane Potential (RMP)

The Golden Rule

The membrane potential is always driven toward the reversal potential of the open conductances.

• If $V_m$ is below (more negative than) $E_{rev}$, the current is depolarizing.

• If $V_m$ is above (more positive than) $E_{rev}$, the current is hyperpolarizing.

• $Na^+$ currents depolarize ($E_{Na}$ is positive); $K^+$ currents hyperpolarize ($E_{K}$ is negative).

Calculations Using Conductance

For a cell with multiple leak channels, $V_m$ is a weighted sum of the reversal potentials, where the weighting is the fractional conductance ($f_x$):

$V_m = f_1E_1 + f_2E_2 + … + f_nE_n$

$f_x = \frac{g_x}{g_{total}}$

Example Calculation:

• $g_K = 5\,nS$, $E_K = -90\,mV$

• $g_{Na} = 5\,nS$, $E_{Na} = +60\,mV$

• $g_{total} = 10\,nS$; $f_K = 0.5$; $f_{Na} = 0.5$

• $V_m = 0.5(-90) + 0.5(60) = -45 + 30 = -15\,mV$

The Goldman-Hodgkin-Katz (GHK) Equation

Used for channels permeable to multiple monovalent ions, considering their permeabilities ($p$):

$E_{rev} = 60 \log_{10} \left( \frac{P_K[K^+]<em>o + P</em>{Na}[Na^+]<em>o + P</em>{Cl}[Cl^-]<em>i}{P_K[K^+]_i + P</em>{Na}[Na^+]<em>i + P</em>{Cl}[Cl^-]_o} \right)$

• Intracellular chloride is on the numerator because of its negative charge.

• The equation simplifies to the Nernst equation when only one ion is permeable.

• Relative Permeabilities (Squid Axon): $P_K : P_{Na} : P_{Cl} = 1 : 0.04 : 0.45$.

Biological Components and Variability

Contribution of Pumps to RMP

• Pumps (like $Na^+/K^+\text{ ATPase}$) move only $20\text{-}100$ ions/s, whereas channels move $4 \times 10^7$ ions/s.

• The $Na^+/K^+\text{ ATPase}$ is electrogenic (3 $Na^+$ out for 2 $K^+$ in), creating a hyperpolarizing effect.

• Under normal conditions, the pump contributes only a few millivolts directly to $V_m$. Blocking the pump with ouabain, vanadate, or ATP removal only causes immediate depolarization of a few mV.

• The primary role of the pump is indirect: maintaining the concentration gradients that set the battery voltages.

Diverse Resting Potentials

• Glial Cells: $-90$ to $-80\,mV$

• Skeletal Muscle (Frog): $-90\,mV$

• Mammalian Skeletal Muscle/Cardiac Muscle: $-80\,mV$

• Cortical Pyramidal Neurons: $-75$ to $-55\,mV$

• Smooth Muscle Cells: $-60$ to $-50\,mV$

• Thalamic Relay Neurons: $-55\,mV$

• Retinal Photoreceptors: $-40\,mV$

• Red Blood Cells: $-10\,mV$

Key Conductance Contributors

• $K^+$ Leak (K2P): Dominates RMP in many cells, pulling it toward $-90\,mV$.

• Non-specific Cation Leak (NALCN): Present in many neurons; $E_{cat} \approx 0\,mV$. These depolarize the RMP above $E_K$.

• Chloride Channels: Large contribution in skeletal muscle ($80\%$ of resting conductance) and squid giant axon.

• Inward Rectifier $K^+$ (KIR): Opens with hyperpolarization.

• Hyperpolarization-activated Cation (HCN): Slower opening; $E_{rev} \approx -30\,mV$.

Summary and Electrical Modeling

The Equivalent Circuit of a Cell

• All channels are connected in parallel with the membrane capacitance.

• Passive components include leak conductances ($g_K$, $g_{Cl}$, $g_{cation}$) and their respective batteries.

• Total current is the sum of all individual ionic currents:     $I_{total} = I_K + I_{Cl} + I_{Na} + …$     $I_x = g_x(V_m - E_x)$

Stability of RMP

A stable resting potential requires:

1. Zero net current.

2. Self-correcting currents: If $V_m$ is deflected, the resulting currents must push it back (e.g., inward current at more negative $V_m$, outward current at more positive $V_m$).

Coursework Information (ICA2)

• Topic: Neurophysiology and Neurotechniques Coursework.

• Deadline: 17/03/2026.

• Components:     1. PowerPoint poster (Group work).     2. Individual written poster abstract (max 300 words) and peer evaluation.

• Preparation Session: 20/01/2026.