474 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
12345-110 -100 -90
membrane potential, mV∆F/kTBb
020406080100-120 -100 -80 -60 -40
membrane potential, mVfraction of channels open, %a
Figure 12.18: (Experimental data with fit.) (a)Voltage dependence of sodium channel opening. The current
through a single sodium channel reconstituted in an artificial bilayer membrane was measured under voltage-clamp
conditions while increasing the voltage from hyperpolarized (∆V =− 120 mV)todepolarized (0mV). Channel
inactivation was suppressed; see Figure 12.15. [Data from Hartshorne et al., 1985.] (b)The free energy difference
∆F/kBTbetween open and closed states, computed from (a) under the hypothesis of a simple two-state switch
(Equation 12.27). The curve is nearly linear in the applied voltage, as we would expect if the channel snapped
between two states with different, well-defined, spatial distributions of charge. The slope is− 0. 15 mV−^1.
Suppose that upon switching, a total chargeqmoves a distancein the direction perpendicular
to the membrane. The electric field in the membrane isE≈V/d,wheredis the thickness of the
membrane (see Section 7.4.3 on page 233). The external electric field then makes a contribution to
∆Fequal to−qE,or−qV/d,and so our model predicts that
∆F(V)=∆F 0 −qV/d,orPopen=1
1+Ae−qV /(kBTd), (12.27)
where ∆F 0 is an unknown constant (the internal part of ∆F), andA=e∆F^0 /kBT.
Equation 12.27 gives our falsifiable prediction. Though it contains two unknown constants (A
andq/d), still it makes a definite prediction about the sigmoidal shape of the opening probability.
Figure 12.18a shows experimental patch-clamp data givingPopenas a function ofV.Tosee whether
it obeys our prediction, panel (b) shows the quantity ln((Popen)−^1 −1) = ∆F/kBT. According
to Equation 12.27, this quantity should be a constant minusqV /(kBTd). Figure 12.18b shows
that it is indeed a linear function ofV.From the slope of this graph, Equation 12.27 gives that
q
kBTrd=0.^15 mV
− (^1).
Your Turn 12d
Interpret the last result. Using the fact thatcannot exceed the membrane thickness, find a
bound forqand comment.
d. Kinetics Section 6.6.2 on page 194 also drew attention to the implications of the two-state
hypothesis fornonequilibrium processes: If the probabilities of occupation are initially not equal to
their equilibrium values, they will approach those values exponentially, following the experimental