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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 c^o_{Na} = c^i_{Na} \rightarrow V_{Na}=0 \rightarrow V_m = 0 \rightarrow I=0
      • 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
      • \frac{P_{Li}}{P_{Na}} = \frac{c^o_{Na}}{c^o_{Li}}
    • 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
    • I = γ(VmVn)
    • I is the open channel current, γ the open channel conductance, Vn is the reversal potential.
    • If screening of ion is perfect, then Vn is the Nernst potential
    • Otherwise Vn is weighted sum of ions that can permeate
  • Model for gate
    • \tilde{s}(t): random variable of state of gate (open/closed), either 0 or 1
      • average of \tilde{s}(t) = x
    • \tilde{g}(t): random variable of conductance 0 or γ
      • Based on \tilde{s}(t), E[\tilde{g}(t)]=\gamma x=g
    • \tilde{i}(t): random variable of single channel current, 0 or I
      • E[\tilde{i}(t)]=Ix=\gamma (V_m - V_n) x = g(V_m - V_n)
  • Assume cells have N channels that are identical but statistically independent
    • If N is large, total conductance is about the mean = Ng
    •  G = \frac{N}{A} g (specific conductance)
    • Same with current: J = \frac{N}{A} g(V_m-V_n)
  • Model for state of channel
    • First order reversible reaction for probability gate is open
    • x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta}
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