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