# 6.021/Notes/2006-11-22

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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
• $x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta}$