Membrane Resting Potentials and Action Potential Mechanisms

General Objectives and Study Guide

• Key Determinations: Understanding what determines membrane potentials and the driving forces that cause ions to move across the membrane.
• Cellular Composition: Explaining the similarity between an electronic capacitor and the cell membrane/composition.
• Excitation and Phases: Defining cell membrane excitation, identifying the known action potential phases, and describing each phase in detail.
• Equilibrium Potential: Defining the ion equilibrium potential (ion resting potential) and describing Donnan reactions.
• Mathematical Models: Describing the Nernst equation and the Goldman-Hodgkin-Katz (GHK) equation, including which membrane states/processes can be calculated with each.
• Physiological Laws: Explaining the "All-or-None Law" and the concept of threshold potential.
• Impulse Propagation: Describing the local current flow transmission mechanism, the saltatory conduction process, and factors affecting the speed of impulse propagation.

Electric Properties of the Membrane: The Capacitor Model

• Lipid Bilayer and Permittivity: The main component of the plasma membrane is a lipid bilayer with inserted membrane proteins for material transport. The lipid bilayer is characterized by low electric conductivity (low permittivity).
• Capacitor Principle: Biological membranes work on the principle of a capacitor. The lipid bilayer acts like an insulator (dielectric) separating two conducting media: the extracellular fluid and the cell cytoplasm. This leads to the accumulation of opposite charges on both surfaces of the membrane.
• Capacitor Definition: A capacitor is an electronic component that stores electric charge, typically made of two conductors (plates) separated by a dielectric material. When connected to a power source, one plate accumulates a positive charge ($+Q$) and the other a negative charge ($-Q$).
• Capacitance ($C$): This is the amount of electric charge stored in the capacitor at a voltage of $1\text{ Volt}$.
• Units: Capacitance is measured in Farads ($F$).
• Capacitance Formula:$C = \frac{Q}{V}$
  • $C$: Capacitance in Farads ($F$).
  • $Q$: Electric charge in Coulombs ($C$) stored on the capacitor.
  • $V$: Voltage between the capacitor's plates in Volts ($V$).
• Physical Model: In a cell, the model involves two plates with a non-conductive medium (the bilayer) approximately $3\text{ nm}$ thick between the extracellular space and the cytosol.

Ohm’s Law and the Electrical Model of the Membrane

• Ohm’s Law: Current is directly proportional to voltage and inversely proportional to resistance.     $I = \frac{V}{R}$
  • $I$: Current, measured in Amperes ($Amp$).
  • $V$: Electromotive force, measured in Volts ($Volt$).
  • $R$: Resistance, measured in Ohms ($Ohm$).
• Biological Application: The biological membrane is an electric circuit containing:
  • Capacitance ($C_m$): Provided by the lipid bilayer.
  • Ohmic Resistance ($R_m$): Provided by the ionic protein channels.
• Membrane Potential Regulation: The capacitance and ohmic resistance define the selective permeability to various ions. This determines the non-equal redistribution of ions, the formation of chemical and electric gradients, and the resulting potential difference known as the membrane potential.

Membrane Potential and Resting Conditions

• Measurement: Membrane potential ($V$) is measured as the difference between the inside and the outside:     $V = V_{\text{inside}} - V_{\text{outside}}$
• Reference Point: Like distance, potential is relative. For a cell, the reference point is always the outside of the cell ($V_{\text{outside}} = 0$).
• Resting Membrane Potential Range: In most resting neurons, the potential difference is about $-30\text{ mV}$ to $-90\text{ mV}$. The negative value indicates the inside is more negative than the outside.
• Polarity Terminology:
  • Polarized: When there is a potential difference across the membrane.
  • Depolarized: When the membrane potential becomes more positive than the resting potential.
  • Hyperpolarized: When the membrane potential becomes more negative than the resting potential.
• Energy Storage: The resting potential determines energy storage ($E_m$) which is essential for the regulation of cell homeostasis. This potential exists even without external stimuli due to charge separation produced by ion redistribution under chemical and electric gradients.

Ion Equilibrium and the Nernst Potential

• Model System: Imagine a container separated into compartments A and B by a membrane. Side A has $100\text{ mM KCl}$ and Side B has $10\text{ mM KCl}$.
• Scenario 1 (Permeable to all): If the membrane is permeable to both $K^+$ and $Cl^-$, ions distribute equally; no potentials or gradients form.
• Scenario 2 (Selective Permeability): If the membrane is permeable only to $K^+$:
  • $K^+$ diffuses from High (Side A) to Low (Side B) concentration.
  • $Cl^-$ anions are left behind on Side A.
  • This charge separation creates an electrostatic force (voltage) that pulls $K^+$ back toward Side A.
• Electrochemical Equilibrium: This state is reached when the diffusion force (acting A to B) is fully counterbalanced by the electrical force (acting B to A). At this point, net diffusion is zero.
• Definition: The voltage generated at this equilibrium is the equilibrium potential or Nernst potential.

Mathematics of Nernst Potential

• Diffusion Force ($\Delta G_D$):$\Delta G_D = RT \ln\left(\frac{[K^+]_o}{[K^+]_i}\right) = RT \times 2.303 \log_{10}\left(\frac{[K^+]_o}{[K^+]_i}\right)$
• Electrostatic Force ($\Delta G_E$):$\Delta G_E = zFE_m$
• Nernst Equation (at equilibrium $\Delta G_D = \Delta G_E$):$E_k = -\frac{RT}{zF} \ln\left(\frac{[K^+]_o}{[K^+]_i}\right)$
  • Constants:
    • $R$ (Gas Constant): $8.315\text{ J mol}^{-1}\text{ K}^{-1}$.
    • $T$ (Absolute Temp): $293\text{ K}$ (at $20^{\circ}\text{C}$).
    • $Z$ (Valency): Charge on the ion (e.g., $1$ for $K^+$).
    • $F$ (Faraday’s Constant): $96480\text{ C mol}^{-1}$.
    • $[K^+]_o, [K^+]_i$: Concentrations outside and inside cell.
• Specific Equilibrium Potentials at $37^{\circ}\text{C}$:
  • $K^+$: Outside $5\text{ mM}$, Inside $100\text{ mM}$, Ratio $1:20$, $E_x = -80\text{ mV}$.
  • $Na^+$: Outside $150\text{ mM}$, Inside $15\text{ mM}$, Ratio $10:1$, $E_x = 62\text{ mV}$.
  • $Ca^{2+}$: Outside $2\text{ mM}$, Inside $2 \times 10^{-4}\text{ mM}$, Ratio $10,000:1$, $E_x = 123\text{ mV}$.
  • $Cl^-$: Outside $150\text{ mM}$, Inside $13\text{ mM}$, Ratio $11.5:1$, $E_x = -65\text{ mV}$.

Goldman-Hodgkin-Katz (GHK) Equation

• Application: Used to calculate the membrane potential when the membrane is permeable to multiple ions simultaneously ($K^+$, $Na^+$, $Cl^-$).
• Formula:$V_m = \frac{RT}{F} \ln\left(\frac{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}}\right)$
• Significance: It accounts for the varying permeabilities ($P$) of different ions, which determines the actual resting potential better than the Nernst equation for single ions.

Donnan Equilibrium (Gibbs-Donnan Effect)

• Definition: Refers to the uneven distribution of charged particles across a semipermeable membrane when one side contains an impermeable solute.
• Mechanism: In living cells, impermeable anionic colloids (proteins and organic phosphates) stay inside the cytosol. These cannot cross the membrane.
• Result: The presence of these non-diffusible anions creates a permanent high concentration of solutes inside, leading to more ions being inside the cell than outside at equilibrium.

Excitable Tissues and Stimulus Response

• Types of Excitable Tissues: Nerve, muscle, and secretory tissues.
• Excitation: A rapid response to a stimulus marked by physical-chemical changes, primarily a change in the electrical properties of the cell membrane.
• Stimulus Types: Mechanical influence, Thermal signals, Chemical signals, and Electric impulses (rarely).
• Action Potential Definition: The potential difference produced by ion distribution in the excited site of the cell membrane.
• The Stimulus-Response Model & Reflex Arc:
  1. Receptor: Converts stimulus to impulse.
  2. Sensory Neuron (Afferent): Transmits impulse to CNS.
  3. Relay Neuron (Interneuron): Conducts impulse within the CNS.
  4. Motor Neuron (Efferent): Transmits impulse from CNS to an effector (muscle/gland).
  • Reflex Arc: Directly relays sensory to motor pathways in the spine, bypassing conscious brain decision-making for survival.

Action Potential Generation and the All-or-None Law

• Calmar-Squid Evidence: Research on the giant axon of the squid showed that excitation involves a drop in membrane resistance from $1000\text{ Ohm/cm}^2$ (resting) to $25\text{ Ohm/cm}^2$ (excited).
• Threshold Potential: To produce an action potential, the stimulus intensity must exceed a critical value, typically around $-55\text{ mV}$.
• All-or-None Law: If the threshold is reached, an action potential occurs in its entirety and at full strength. If not, no action potential occurs. The amplitude is independent of stimulus strength.
• Phases of the Action Potential:
  1. Resting State: Approximately $-70\text{ mV}$; maintained by the $Na^+/K^+$ pump.
  2. Depolarization: Stimulus reaches threshold; voltage-gated $Na^+$ channels open. $Na^+$ influx causes the membrane potential to rise ($+30\text{ mV}$ to $+40\text{ mV}$).
  3. Repolarization: Inactivation of $Na^+$ gates and opening of voltage-gated $K^+$ channels. $K^+$ efflux restores negativity.
  4. Undershoot (Hyperpolarization): Potential briefly becomes more negative than resting level because $K^+$ gates are slow to close.
  5. Refractory Period: Time during which a second action potential cannot be initiated, ensuring one-way propagation.

Impulse Propagation Mechanisms

• Local Current Flow: In unmyelinated fibers, an action potential in one segment causes local currents. The outer surface current flows from non-excited to excited segments; the inner current flows oppositely. This stimulates adjacent voltage-gated channels.
• Saltatory Conduction (Myelinated Axons):
  • Myelin Sheath: Formed by Schwann cells; acts as an ion-impermeable insulator.
  • Nodes of Ranvier: Gaps in the myelin (every $1$ to $2\text{ mm}$) where the bare axon is exposed and contains high concentrations of $Na^+$ and $K^+$ channels.
  • Mechanism: The impulse "jumps" from node to node, which is much faster than continuous travel.
• Speed of Propagation:
  • Range: $0.5$ to $120\text{ m/sec}$.
  • Size Factor: Larger diameter increases speed due to lower resistance.
  • Myelination Factor: A myelinated fiber has a $6$-fold increase in speed compared to a non-myelinated fiber of the same diameter (e.g., $1\text{ m/sec}$ to $6\text{ m/sec}$).
  • Human Example: A nerve impulse travels from head to toes in approximately $0.02\text{ seconds}$.

Synaptic Transmission (Between Neurons)

• Synapse: The physical gap (synaptic cleft) between pre-synaptic and post-synaptic neurons.
• Chemical Transfer Process:
  1. Action potential reaches the axon terminal, opening voltage-gated $Ca^{2+}$ channels.
  2. $Ca^{2+}$ influx triggers vesicle fusion and neurotransmitter release via exocytosis.
  3. Neurotransmitters cross the cleft and bind to ligand-gated channels on the post-synaptic membrane.
• Neurotransmitter Types:
  • Excitatory (e.g., Noradrenaline): Opens ligand-gated $Na^+$ channels (depolarization).
  • Inhibitory (e.g., GABA): Opens ligand-gated $K^+$ or $Cl^-$ channels (hyperpolarization).
• Termination: Neurotransmitters are either recycled by reuptake pumps or degraded by enzymes.
• Neuromuscular Junction: A specific type of synapse where an axon terminal meets a muscle fiber.