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 $$c^o_{Na} = c^i_{Na} \rightarrow V_{Na}=0 \rightarrow V_m = 0 \rightarrow I=0$$
 * 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
 * $$I = \gamma (V_m-V_n)$$
 * I is the open channel current, $$\gamma$$ the open channel conductance, $$V_n$$ is the reversal potential.
 * If screening of ion is perfect, then $$V_n$$ is the Nernst potential
 * Otherwise $$V_n$$ 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 $$\gamma$$
 * 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
 * $$x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta}$$