BIPN 100 - B1 Passive Membrane Properties

Membrane Dynamics

Neurons are excitable cells with a plasma membrane consisting of a phospholipid bilayer.
- The membrane is semipermeable, meaning ions require ion channels to cross.
Ion channels are membrane proteins with selective permeability.
- Leak channels: Pores that remain open.
- Voltage-gated ion channels: Open/close in response to changes in membrane potential.
- Ligand-gated ion channels: Open in response to chemical signals.

Resting Membrane Potential (Vm)

Membrane potential (Vm): Difference in electrical charge between the inside and outside of a cell.
- Measured in volts (V) or millivolts (mV).
- Voltage is measured as a relative difference between the inside and outside.
Electrophysiology: Method to measure membrane potential.
- A recording electrode is inserted into a neuron, and the voltage is compared to a reference electrode outside the neuron.
- Stimulating electrodes can inject current.
Ion concentrations determine Vm.
- $[K^+]$ , $[Na^+]$ , and $[Cl^-]$ are key determinants of neuron membrane potential.
- Typical ion concentrations:
  - Extracellular fluid (mM): $K^+ = 5$ , $Na^+ = 145$ , $Cl^- = 108$ , $Ca^{2+} = 1$
  - Intracellular fluid (mM): $K^+ = 150$ , $Na^+ = 15$ , $Cl^- = 10$ , $Ca^{2+} = 0.0001$
Resting membrane potential (RMP, Vrest): Vm when a cell is at rest.
- Steady state: no net movement of charge.
- RMP in most neurons is -60 to -70 mV.
Na+-K+ ATPase maintains Vrest.
- Sodium-Potassium Pump (Na+-K+-ATPase) builds up $[Na^+]$ in ECF and $[K^+]$ in ICF.
- Pumps 3 $Na^+$ out and 2 $K^+$ in.
- Mechanism:
  - Antiport: carrier protein that moves substances in opposite directions.
  - Active transport: Na+-K+-ATPase hydrolyzes ATP to move ions against their concentration gradients.
Two factors determine Vm:
1. Electrochemical Gradient: uneven distribution of ions across the cell membrane.
  - Electrical driving force: attraction and repulsion between charged particles.
  - Chemical driving force: diffusion, driving ions from high to low concentration regions (due to entropy and the 2nd Law of Thermodynamics).
2. Differences in membrane permeability: ability for ions to pass through the membrane.
  - Membrane proteins have selective permeability.
  - Permeability is determined by the number of ion channels.
  - At rest, the membrane is more permeable to $K^+$ due to a higher number of $K^+$ leak channels compared to $Na^+$ leak channels.

Calculating Eion

Equilibrium Potential (Eion): Membrane potential that exactly opposes the concentration gradient.
- Electrical and chemical forces are equal and opposite, resulting in no net movement of ions.
Electrochemical Potential (μion): Sum of electrical and chemical potentials.
- $\mu{ion} = RT \ln \frac{[ion]{inside}}{[ion]{outside}} + zF (E{inside} – E_{outside})$
 - z = ion’s charge ( $K^+ = 1$ , $Na^+ = 1$ , $Ca^{2+} = 2$ , $Cl^- = -1$ )
 - E = voltage
- At equilibrium: $\mu_{ion} = 0$
- $- RT \ln \frac{[ion]{inside}}{[ion]{outside}} = zF (E{inside} – E{outside})$
- Nernst Equation:
- $E{ion} = \frac{RT}{zF} \ln \frac{[ion]{outside}}{[ion]_{inside}}$
 - Calculates the equilibrium potential.
 - Represents a voltage across the membrane with no net flow of the ion.
Nernst Equation at 37°C (body temperature):
- $E{ion} = \frac{61}{z} \log \frac{[ion]{outside}}{[ion]_{inside}}$
Calculating Driving Force (Fion)
- $F{ion} = Vm – E_{ion}$
- Driving forces on an ion depend on the difference between Vm and Eion.
- If F_{ion} > 0, the driving force is outward.
- If F_{ion} < 0, the driving force is inward.
- If $F_{ion} = 0$ , there is no net driving force.
 *Example for K+.

Calculating Vm

Membrane potential (Vm) for a membrane permeable to one ion is Eion
- $E_{K+} = -90 mV$
- $E_{Na+} = +60 mV$
Goldman-Hodgkin-Katz (GHK) Equation: Calculates membrane potential (Vm) resulting from the contribution of equilibrium potentials (Eions) of all ions, as a function of permeability
- Permeability ~ relative ion contribution to membrane potential. Usually expressed as a ratio e.g. $P{K+}$ 1: $P{Na+}$ .05 : $P_{Cl-}$ .45
The GHK Equation:
$Vm = 61 \log \frac{PK [K^+]{out} + P{Na} [Na^+]{out} + P{Cl} [Cl^-]{in}}{PK [K^+]{in} + P{Na} [Na^+]{in} + P{Cl} [Cl^-]_{out}}$
The number of leak ion channels determines the permeability of the membrane to specific ions in a neuron at rest
If the membrane is permeable to only one ion, the GHK equation becomes the Nernst equation.
At Vrest conditions:
- Balance (K+ out of the cell matches Na+ into the cell))
- $F{ion} = G{ion}$
- cations going out of the cell = cations going into the cell ( $IK = -I{Na}$ )
- neuron is in steady-state (not in equilibrium for Na+ or K+)
- III. Active Transport
 Set up and maintain the ion gradients
Due to asymmetry, add a small amount of (-) to Vm
Pumps are too slow to generate rapid repolarization

Electrical Current

Electrical Current (I): Flow of electrical charge carried by an ion.
- Measured in amperes (amps).
- Ion movement produces electrical signals.
- I is dependent on Fion (driving force on an ion) and permeability
- $I = \frac{V}{R} = VG$
Current is dependent on Fion.
- $F{ion} = Vm – E_{ion}$
- At $E_{ion}$ , net $I = 0$
Conductance (G): Ease with which ions flow across a membrane.
- Units: siemens (S).
- Conductance is determined by the number of open ion channels.
- Stimuli alter permeability, causing ions to flow with electrochemical forces.
- $G = \frac{1}{R}$
Resistance (R): Difficulty with which ions flow across a membrane.
- Units: ohms (Ω).
- Resistance is determined by the number of closed ion channels.
- Stimuli alter permeability, causing ions to flow with electrochemical forces.
- Different types of resistance in neurons:
  - Membrane resistance (Rm).
  - Cytoplasm resistance (Ri).
Ohm’s Law: States that current flow is directly proportional to the electrical potential difference between two points and conductance.
- $V = IR$
- $I = VG$
- $G = \frac{I}{V}$
Ionic Current (Iion) Is the number of ions (amount of charge) crossing the membrane
- Force ( $F{ion}$ ) * $F{ion} = (Vm – E{ion})$
- Conductance $G_{ion}$
Current Equation
- $I{ion} = G{ion} (Vm - E{ion})$
$I_{ion}$ requires:
- a driving force \F_{ion}
- a pathway ( $G_{ion}$ )
  *Interpreting Voltage-Current Plots

Changes to Vm

Depolarization: Increase in Vm.
- The membrane becomes more permeable to $Na^+$ .
- Inward $I_{Na+}$ with the electrochemical gradient.
- Vm above RMP.
Repolarization and Hyperpolarization: Decrease in Vm.
- The membrane becomes more permeable to $K^+$ .
- Outward $I{K+}$ with the electrochemical gradient or inward $I{Cl-}$ .
- Hyperpolarization: Vm below RMP.