Study Guide: Ion Channels & Membrane Electrophysiology

1. Overview

• Ion flow is governed by:

  • Membrane permeability (via specific ion channels).

  • Concentration gradients (chemical driving force).

  • Electrical driving forces (voltage difference across the membrane).

• Membrane potential ($V_m$$V_m$) arises from:

  • Selective ion channels.

  • Active transport mechanisms (e.g., Na⁺/K⁺ pumps).

  • Impermeant anions and charge separation.

• Equilibrium potential (Nernst) defines the voltage at which ion movement is balanced by the chemical gradient:

  $$E_{\text{ion}}=\frac{+60}{z}{\log }_{10}(\frac{[\text{ion}{]}_{\text{outside}}}{[\text{ion}{]}_{\text{inside}}})$$

  • $z$$z$: ion valence (e.g., +1 for K⁺, +1 for Na⁺, +2 for Ca²⁺, etc.).

  • 60 mV is an approximation at ~37 °C (more precisely 61–62 mV).

2. Key Equations

1. Ohm’s Law

   $$V=I\times R$$

2. Conductance ($g$$g$)

   $$R=\frac{1}{g}⟹V=\frac{I}{g}$$

3. Weighted Sum for Membrane Potential (multiple ions)

   $$V_m=\frac{\sum _i(g_i{E}_i)}{\sum _i{g}_i}$$

   • $g_i$$g_i$: conductance of ion $i$$i$

   • $E_i$$E_i$: Nernst (equilibrium) potential for ion $i$$i$

4. Ion Current

   $$I=g(V_m-E_{\text{ion}})$$

   • Net current depends on the difference ($V_m-E_{\text{ion}}$$V_m - E_{\text{ion}}$).

5. General Flow Equation

   $$\text{Flow}=(\text{concentration})\times (\text{mobility})\times (\text{area})\times (\text{driving force})$$

6. Time Constant

   $$\tau =R\times C$$

7. Chord Conductance

   $$g=\frac{I-I_{\text{rev}}}{V-E_{\text{rev}}}$$

3. Capacitance & RC Properties

• Capacitor stores charge by separating it across an insulating layer.

• The cell membrane behaves as a capacitor:

  • Intracellular/extracellular fluid = conductors.

  • Lipid bilayer = dielectric (insulator).

3.1 Stored Charge

• $Q$$Q$: charge (coulombs)

• $C$$C$: capacitance (farads)

• $V$$V$: voltage (volts)

3.2 Capacitive Current

• Faster change in voltage $\Rightarrow$$\Rightarrow$ more current needed to charge the membrane capacitor.

3.3 Time Constant ($\tau$$\tau$)

• Governs how fast the voltage changes in response to current:

  • Time to reach $\approx 63\mathrm{\%}$$\approx63\%$ of final $V$$V$ after a step current.

4. Measuring & Calculating Membrane Properties

1. Measure the Time Constant ($\tau$$\tau$)

   • Inject a small current step; record time to reach $\approx 63\mathrm{\%}$$\approx63\%$ of the final voltage.

2. Measure Steady-State Voltage Response

   • After current injection, at steady-state:

     $$G=\frac{\mathrm{\Delta }I}{\mathrm{\Delta }V}⟹R=\frac{1}{G}$$

3. Calculate Capacitance

   $$C=\frac{\tau }{R}$$

4. Determine Cell Area

   $$\text{Cell Area}=\frac{C}{1\mu \text{F}/{\text{cm}}^2}$$

   • Typical biological membranes have $\sim 1\mu \text{F}/{\text{cm}}^2$$\sim1\,\mu\text{F}/\text{cm}^2$.

5. Find Specific Membrane Resistance

   $$R_m=R\times (\text{Cell Area})$$

5. Membrane as an Electrical Circuit

• Capacitor: lipid bilayer.

• Resistor: ion channels.

• Battery: Nernst potential (ionic gradient).

5.1 Equivalent Circuit

• Membrane = RC circuit:

  • Capacitance ($C_m$$C_m$) in parallel with membrane resistance ($R_m$$R_m$).

• Ohm’s Law: $\mathrm{\Delta }V=I\times R$$\Delta V = I \times R$.

• Input Resistance ($R_{\mathrm{in}}$$R_{\mathrm{in}}$):

  $$R_{\mathrm{in}}=\frac{\mathrm{\Delta }V}{I}$$

5.2 Time Constant

• $\tau =R_m{C}_m$$\tau = R_m \, C_m$

• Time to charge/discharge the membrane by 63%.

5.3 Length Constant

• For cable-like processes (axon, dendrite):

  $$\lambda =\sqrt{\frac{R_m}{R_i}}$$

  • $R_i$$R_i$: axial (internal) resistance.

  • $\lambda$$\lambda$ determines how far a passive potential spreads before decaying significantly.

6. Voltage-Gated Channels

• Structure:

  • S4 segment contains positively charged amino acids (voltage sensor).

• Ion Selectivity:

  • Channels discriminate ions based on size, charge, hydration.

• Voltage Dependence:

  • Voltage sensors respond to membrane potential changes → channel gates open/close.

6.1 Na⁺ Channels

• Fast activation and fast inactivation.

• Inactivation ensures transient current even with sustained depolarization.

• Refractory Period:

  • After an action potential, channels remain inactivated briefly → cannot reopen immediately.

6.2 K⁺ Channels

• Slower activation (classical Hodgkin-Huxley channels).

• Often do not inactivate (or do so minimally) → can remain open with prolonged depolarization.

• Inward Rectifiers: block outward flow at positive potentials → stabilize resting potential.

6.3 Ca²⁺ Channels

• Depolarization opens Ca²⁺ channels:

  • Often slower inactivation → can prolong action potentials (e.g., cardiac cells).

• Ca²⁺ entry can trigger diverse intracellular events (secretion, contraction, signaling).

7. Current–Voltage (I–V) Relationships

• Plotting current ($I$$I$) vs. voltage ($V$$V$) helps identify:

  • Reversal potential (where net current = 0).

  • Slope conductance = $\mathrm{\Delta }I/\mathrm{\Delta }V$$\Delta I / \Delta V$.

• Chord Conductance ($g$$g$):

  $$g=\frac{I}{V-E_{\text{ion}}}$$

• I–V Plot Regions:

  • Positive slope: typical ohmic or outward K⁺ flow.

  • Negative slope: inward rectification or channel inactivation phenomena.

8. Conductance–Voltage (G–V) Plots

• Derived from I–V data:

  $$G=\frac{I}{V-E_{\text{rev}}}$$

• Sigmoidal activation curve:

  • $G_{\text{max}}$$G_{\text{max}}$: maximum conductance.

  • Half-activation voltage ($V_{1/2}$$V_{1/2}$): voltage at which $G$$G$ = $G_{\text{max}}/2$$G_{\text{max}}/2$.

9. Single-Channel Behavior

• Microscopic currents: discrete channel openings/closings.

• Macroscopic current: sum of many channels.

• Stochastic gating: channels open with certain probability depending on voltage, time, etc.

10. Protocols to Study Na⁺ Activation/Inactivation

1. Activation:

   • Clamp the membrane at various depolarizing steps from a negative holding potential.

   • Measure how quickly and strongly Na⁺ current activates.

2. Inactivation:

   • Use prepulses at different voltages.

   • Then step to a test voltage → measure available Na⁺ current.

   • Plot fraction of channels available vs. prepulse voltage → midpoint of inactivation.

11. Consequences of Na⁺ Channel Inactivation

• Transient Na⁺ current despite sustained depolarization.

• Refractory period ensures:

  • Unidirectional action potential propagation.

  • Finite firing frequency (cannot immediately trigger another AP).

• Prolonged depolarization → channels remain inactivated → no further firing (weakness or paralysis).

12. External Ion Concentrations

• Decreasing $[{\text{Na}}^{+}{]}_{\text{out}}$$[\text{Na}^+]_{\text{out}}$ → smaller action potential amplitude.

• Increasing $[{\text{K}}^{+}{]}_{\text{out}}$$[\text{K}^+]_{\text{out}}$ → more depolarized rest → smaller AP or failure to fire.

13. Pathophysiology: Muscle Weakness & Inexcitability

• Shifts in Resting $V_m$$V_m$:

  • Abnormal channel gating or ion gradients → altered excitability.

• Input Resistance & Time Constant Changes:

  • Affected by channel density or membrane properties → can require bigger/longer stimuli to reach threshold.

• Clinical Correlates:

  • Periodic paralysis, channelopathies → result from subtle gating alterations or ionic imbalances.

14. Axonal Conduction

14.1 Passive Spread (Cable Properties)

• Voltage decays over distance if no regenerative channels are present.

• Larger diameter → lower axial resistance ($R_i$$R_i$) → greater length constant ($\lambda$$\lambda$).

14.2 Unmyelinated Axons

• Continuous conduction:

  • Slower (few m/s in small fibers).

  • Speed often scales with axon diameter (e.g., squid giant axon ~20 m/s).

14.3 Myelinated Axons

• Saltatory conduction:

  • Na⁺ channels clustered at nodes of Ranvier.

  • Myelin → lowers capacitance, increases membrane resistance in internodes.

  • Conduction speeds can reach ~100 m/s in large myelinated fibers.

• Myelin reduces capacitance → less charge is needed to depolarize the next node.

15. Calcium Action Potentials

• Ca²⁺ Channels:

  • Open upon depolarization, often inactivate more slowly → longer-duration APs.

• Replacing Na⁺ with Ca²⁺:

  • Yields slower, longer spikes due to lower channel conductance and slower inactivation.

• Cardiac Action Potentials:

  • Plateau phase from sustained Ca²⁺ influx.

  • Increasing $g_{\mathrm{Ca}}$$g_{\mathrm{Ca}}$ → prolongs plateau.

  • K⁺ currents eventually repolarize the cell.

Summary

1. Membrane potential depends on ion gradients, selective permeability, and ion channel gating.

2. RC properties (resistance $R$$R$, capacitance $C$$C$) govern how quickly a cell’s voltage responds.

3. Time constant ($\tau$$\tau$) is crucial for understanding the speed of electrical responses.

4. Voltage-gated channels underlie action potentials and are finely regulated by voltage sensors.

5. Ion concentrations critically affect excitability and AP amplitude.

6. Myelination optimizes conduction velocity without drastically increasing axon diameter.

7. Inactivation of channels (especially Na⁺) is key to the refractory period and directional AP propagation.

Properties of Voltage-Activated Calcium Channels

1. Subunit Structure :

   • main ${\alpha }_1$$\alpha_1$ subunit

     • has 4 domains

     • the main pore-forming subunit that determines ion selectivity, conductance, and gating properties

   • $\gamma$$\gamma$ :

     • Four-transmembrane-spanning protein, initially found in skeletal muscle calcium channels but also present in the brain (e.g., stargazin).

   • $\beta$$\beta$ :

     • Cytoplasmic protein that binds to the α1 subunit’s intracellular I-II linker, helping with trafficking to the plasma membrane and modulation

   • ${\alpha }_2$$\alpha_2$ :

     • Derived from a single gene and proteolytically cleaved into an extracellular α2 portion and a membrane-spanning δ portion, which stabilizes the channel at the plasma membrane.

   

2. Gate / Voltage Sensor :

   • S4 segment :

     • Contains positively charged amino acids , arginine and lysine , every 3rd or 4th residue

     • unctions as the voltage sensor

   • Depolarization causes deformation in P-loops , creating a pore

3. Selectivity Filter for Calcium $\left(C{a}^{+2}\right)$$\left(\ Ca^{+2} \ \right)$ :

   • Calcium selectivity is determined by the pore loop ( P-loop ) glutamates :

     • Glutamate residues ( E ) are conserved in each of the four domains ( I-IV ) of the α1 subunit

     • These residues form a high-affinity binding site for Ca2+ ,

       • ensuring preferential permeability over other ions

   

   

4. How would you classify or identify different subtypes of calcium channels ?

   • Voltage-gated calcium channels are classified into :

     • High-Voltage Activated ( HVA )

     • Low-Voltage Activated ( LVA ) ( T-type ) channels

   • These subtypes can be distinguished using :

     • Pharmacological blockers ( e.g., ω-Agatoxin IVA for P/Q-type channels )

     • Genetic expression ( different subtypes are present in different tissues )

     • Electrophysiology ( LVA T-type channels activate at lower voltages than HVA channels )

   Channel Type	Clone (CaV)	Function	Blockers
   L-type	CaV1.1 - CaV1.4	Skeletal, cardiac muscle contraction, secretion	DHP (nifedipine), verapamil
   P/Q-type	CaV2.1	Neurotransmitter release (central)	ω-Agatoxin IVA
   N-type	CaV2.2	Neurotransmitter release (peripheral & central)	ω-Conotoxin GVIA
   R-type	CaV2.3	Uncertain	SNX 482
   T-type	CaV3.1 - Cav3.3	Cell excitability	Mibefradil, ethosuximide (controversial)

• L-type blockers:

  • Dihydropyridines (DHPs): Nifedipine, Amlodipine.

  • Non-DHPs: Verapamil (cardiac selective).

• P/Q-type blockers:

  • ω-Agatoxin IVA (funnel web spider toxin).

• N-type blockers:

  • ω-Conotoxin GVIA (cone shell mollusk toxin).

  • Used for neuropathic pain management.

• T-type blockers:

  • Mibefradil, ethosuximide (antiepileptics).

${\alpha }_1$$\alpha_1$ Subunits

1. Q: Explain why the resting $V_m$$V_m$ is usually closer to $E_{\mathrm{K}}$$E_{\mathrm{K}}$ than $E_{\mathrm{Na}}$$E_{\mathrm{Na}}$ in typical neurons. A: The cell is more permeable to $K^{+}$$K^+$ at rest (many K${}^{+}$$^+$ leak channels), so $V_m$$V_m$ lies near $E_{\mathrm{K}}$$E_{\mathrm{K}}$.

2. Q: Sketch/describe an RC charging curve and label $\tau$$\tau$. What if you double $C_m$$C_m$? A: An RC curve rises toward its final voltage, reaching 63% of the total change at $t=\tau$$t=\tau$. Doubling $C_m$$C_m$ doubles $\tau$$\tau$, slowing the voltage response.

3. Q: On an $I$$I$–$V$$V$ plot, identify the slope and reversal potential. A: The slope is conductance ($g$$g$). Where $I=0$$I=0$ is the reversal potential ($E_{\mathrm{rev}}$$E_{\mathrm{rev}}$), usually near $E_{\mathrm{ion}}$$E_{\mathrm{ion}}$ if one ion dominates.

4. Q: How does changing $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ affect AP shape/amplitude? A: Higher $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ → larger driving force for Na${}^{+}$$^+$ → bigger, faster upstroke and more positive peak. Lower $[{\mathrm{Na}}^{+}{]}_{\mathrm{out}}$$[\mathrm{Na}^+]_{\mathrm{out}}$ does the opposite.

5. Q: Absolute vs. relative refractory periods in terms of Na${}^{+}$$^+$, K${}^{+}$$^+$ channel states? A: Absolute: most Na${}^{+}$$^+$ channels are inactivated → no new AP possible. Relative: some Na${}^{+}$$^+$ channels have recovered but K${}^{+}$$^+$ conductance is high → requires stronger stimulus.

6. Q: Why might a muscle fiber at $-90\text{ mV}$$-90\text{ mV}$ fail to fire if channels become inactivated? A: If resting $V_m$$V_m$ is shifted or channels are in a state with fewer available Na${}^{+}$$^+$ channels, normal stimuli might not reach threshold.

7. Q: Raising $[{\mathrm{K}}^{+}{]}_{\mathrm{out}}$$[\mathrm{K}^+]_{\mathrm{out}}$ from 4.5 mM to 25 mM: effect on resting potential? A: Depolarizes it (less negative) because $E_{\mathrm{K}}$$E_{\mathrm{K}}$ becomes more positive, so $V_m$$V_m$ shifts closer to that new $E_{\mathrm{K}}$$E_{\mathrm{K}}$.

8. Q: A short current pulse doesn’t reach threshold, but a longer one does. Why? A: The membrane (RC circuit) needs enough time to charge up to threshold. A short pulse may end before hitting threshold.

9. Q: Multiple myelin wraps: effect on conduction velocity?

   A: Myelin greatly reduces $C_m$$C_m$ (and increases $R_m$$R_m$) in internodes, so fewer charges are needed to depolarize each segment. Conduction is much faster.