Notes

Membrane Potential & Passive Electrical Properties

Membrane The membrane is a very thin bilipid layer that is absolutely impermeable for ions. The membranes are fluid. Membrane Theory (Bernstein)

• Excitable Cells: Membrane selectively permeable to K+ ions at rest.
• Excitation: Permeability for other ions increases.

Membrane Potential The membrane potential at rest is approx. -80mv (-50 to -90). This value is mainly due to the high concentration of K+ ions inside the cell compared to outside, which move outside the cell following their concentration gradient and bringing positive charges with them. At the same time, a contrary electrical gradient develops, since the inside is negatively charged, K+ ions are subject to an electrical force that pushes them inside (membrane potential). When these two forces are in equilibrium, the net flux of K+ is 0 and the cell is at its resting/equilibrium potential (E).

Equilibrium Potential (E) It is the voltage at which chemical diffusional driving force is balanced by electrical driving force. Nernst Equation can be used to calculate E:

Since membrane potential is not simply due to K+, but also to other ions concentrations, a more comprehensive formula the Goldman-Hodgkin-Katz Equation has been developed. The membrane potential is indeed maintained by the Na+/K+ Pump (ATP) which pumps in 2K+ and pumps out 3Na+. The GHK equation can be used to calculate Vm:

The membrane is mostly permeable to K+ ions at rest, hence EK dictates the Resting Membrane Potential. The Resting Membrane Potential represents the relationship between electric field and potential difference. It is the potential difference between two point separated by distance d:

Hence, E = -80mV/5nm = -16 x 10ˆ6 V/m.

Passive Electrical Properties Resistance The Amplitude of V change is influenced by membrane resistance (R). For the same I, a larger R produces a larger potential difference.

Capacitance The Capacitance is the voltage separation over the passive membrane. The Time course of V change is influenced by membrane capacitance and resistance (positive correlation).

RC Circuit Membranes with ion channels are often schematized as RC circuits.

Cell Size A larger cell usually has a smaller resistance, larger capacitance and larger time constant. A "leaky" membrane has a small resistance, while a thick membrane has a small capacitance.

Length Constant The length constant ($\lambda$) is the point at which $V_m$ has fallen to I/e (or 37%) of its original value.

Measuring and Controlling $V_m$ Patch Clamp The patch clamp allows the active control of current (I) (Current Clamp) or Vm through I injection (Voltage Clamp). Indeed, through a patch clamp it is possible to measure the injected current and calculate $V_m$.

• Current injection to counteract changes in $I_m$ Measure membrane potential.
• Current injection to counteract changes in Vm (goal: Vcommand = $V_m$) Measure current (clamp/fix V).

Action Potential & Ion Channels

Ion Channels and the AP Ion channels are "digital", they are either open or closed. The amplitude of single channel current (i) depends on single-channel conductance (g) and "driving force" ($V_m - E_{ion}$). The probability of a V-gated Channel is dependent on the voltage (Na+, K+). The whole-cell current depends on single-channel conductance, driving force, number of open channels (n), and open probability ($P_o$):

Ion Channels Requirements & Design Challenges

• Selectivity: Ion filter mimics hydration of ion
• High Conductivity (Speed): approx. speed of diffusion in H2O but as a purely passive process.
  • Na+ ions/channel = 10,000,000/s
  • Ions/channel = 1000/ms - 10,000/ms
  • More than 1000 times faster than membrane transporters and pumps. (Na+/K+ pump/ATPase = 100/s).
• Gating: Voltage gating involves V-dependent movement of charged channel structures.
  • Inactivation vs. Closure: during inactivation the channel does not conduct during the stimulus. The inactivation of V-gated ion channels involves charged channel structures (ball).
• Specificity (>90%): K+ channels 99.99% versus Na+ (Ionic Radius: K+ 152pm, Na+ 116pm).

The Action Potential The action potential propagation:

How can we increase conduction speed?

High-Frequency Transmission Short AP duration is required for high-frequency firing and transmission. This characteristic is very important to perform Rate Coding.

Examples (Physiology & Pathology) Hyperexcitability in ALS patient-derived iPS motoneurons due to decreased K+ current. In this experiment, sample of skin from patients with sclerosis were taken. Based on these samples, they made stem cells and transdifferentiated them in motoneurons. They did current clamps to analyze spiking patterns in these cells, which highlighted that these cells presented hyperexcitability compared to control ones. And voltage clamps highlighted a decrease of potassium currents over sodium currents. Hence, one of the problems of these cells may be related to a decreased number of potassium channels.