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Ion channels

  • Gate model
    • α: rate of opening
    • β: rate of closing
  • Theory of absolute reaction rates (Arrhenius)
    • potential energy of open ([math]\displaystyle{ E_O }[/math]), closed ([math]\displaystyle{ E_C }[/math]), and intermediate barrier state ([math]\displaystyle{ E_B }[/math])
    • [math]\displaystyle{ \alpha=Ae^{(E_C-E_B)/kT} }[/math]
    • [math]\displaystyle{ \beta=Ae^{(E_O-E_B)/kT} }[/math]
  • If gate has some charge, the potential energy can be function membrane potential
    • If assume simple voltage dependence, get similar looking rates as parameters in Hodgkin-Huxley model
    • As gate moves, its charge moves, so have extra gating current
    • [math]\displaystyle{ q_g }[/math]: gating charge
    • [math]\displaystyle{ i_g = \frac{d}{dt}q_g }[/math]
    • [math]\displaystyle{ \tilde{i}_g = \frac{d}{dt}\tilde{q}_g }[/math]
    • [math]\displaystyle{ \tilde{q}_g }[/math] is 0 or Q
    • [math]\displaystyle{ \tilde{i}_g }[/math] is impulses of amplitude Q up and down
    • At steady state, average of [math]\displaystyle{ \tilde{i}_g = 0 }[/math]
    • [math]\displaystyle{ q_g = Qx }[/math]
    • [math]\displaystyle{ i_g = E[\tilde{i}_g] = E[\frac{d}{dt}\tilde{q}_g] = \frac{d}{dt}E[\tilde{q}_g] = \frac{d}{dt}Qx = Q\frac{dx}{dt} }[/math]
    • To measure gating current, disable the ion channel (ionic current)
      • Have to separate out capacitance current which is much larger
      • Gating current is non-linear while capacitance current is linear so can sum response to equal step up and step down to cancel out capacitive current and leave gating current