6.021/Notes/2006-11-22

Single ion channels

• Neher & Sakmann 1970s, Nobel 1991
• Patch clamp
  • Seal pipette against membrane to measure currents (~2pA)
• Distinctive properties
  • Discrete levels of conduction
  • rapid transitions
  • seemingly random
• Nothing like the macroscopic behavior from Hodgkin-Huxley model
• Model
  • Integral membrane protein
  • Selectivity filter to sort out ions
  • Aqueous pore
  • gate that opens/closes to let ion through
• How selective?
  • Li can seemingly substitute for Na
  • Can quantify selectivity
    • Set [math]\displaystyle{ c^o_{Na} = c^i_{Na} \rightarrow V_{Na}=0 \rightarrow V_m = 0 \rightarrow I=0 }[/math]
    • Then replace extracellular Na with same amount of Li
    • If channels substitute Li perfectly for Na, no current will flow
    • Find the amount of extracellular Li that makes the current zero
    • [math]\displaystyle{ \frac{P_{Li}}{P_{Na}} = \frac{c^o_{Na}}{c^o_{Li}} }[/math]
  • Measuring relative permeability of channel to various ions
    • Many different ions can flow through the sodium and potassium channels, some better than sodium and potassium!
• Linear approximation for permeation
  • [math]\displaystyle{ I = \gamma (V_m-V_n) }[/math]
  • I is the open channel current, [math]\displaystyle{ \gamma }[/math] the open channel conductance, [math]\displaystyle{ V_n }[/math] is the reversal potential.
  • If screening of ion is perfect, then [math]\displaystyle{ V_n }[/math] is the Nernst potential
  • Otherwise [math]\displaystyle{ V_n }[/math] is weighted sum of ions that can permeate
• Model for gate
  • [math]\displaystyle{ \tilde{s}(t) }[/math]: random variable of state of gate (open/closed), either 0 or 1
    • average of [math]\displaystyle{ \tilde{s}(t) = x }[/math]
  • [math]\displaystyle{ \tilde{g}(t) }[/math]: random variable of conductance 0 or [math]\displaystyle{ \gamma }[/math]
    • Based on [math]\displaystyle{ \tilde{s}(t) }[/math], [math]\displaystyle{ E[\tilde{g}(t)]=\gamma x=g }[/math]
  • [math]\displaystyle{ \tilde{i}(t) }[/math]: random variable of single channel current, 0 or I
    • [math]\displaystyle{ E[\tilde{i}(t)]=Ix=\gamma (V_m - V_n) x = g(V_m - V_n) }[/math]
• Assume cells have N channels that are identical but statistically independent
  • If N is large, total conductance is about the mean = Ng
  • [math]\displaystyle{ G = \frac{N}{A} g }[/math] (specific conductance)
  • Same with current: [math]\displaystyle{ J = \frac{N}{A} g(V_m-V_n) }[/math]
• Model for state of channel
  • First order reversible reaction for probability gate is open
  • [math]\displaystyle{ x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta} }[/math]