## Kinetics ofthe Change in Ionic Conductance Following a Step Depolarization

We saw in Chapter 6 that differences in the speed with which the three types of voltage-sensitive gates respond to voltage changes are important in determining the form of the action potential. For instance, the opening of the potassium channels must be delayed with respect to the opening of the sodium channels to avoid wasteful competition between sodium influx and potassium efflux during the depolarizing phase of the action potential. We will now consider how the time-course, or kinetics, of the conductance changes fit into the charged gating particle scheme just presented.

Hodgkin and Huxley assumed that the rate of change in the membrane conductance to an ion following a step depolarization was governed by the rate of redistribution of the gating particles within the membrane. That is, they assumed that the interaction between gating particle and binding site introduced negligible delay into the temporal behavior of the channel. As an example, we will consider the kinetics of opening of the sodium channel following a step depolarization. In formal terms, the movement of gating particles within the membrane can be described by the following first-order kinetic model:

Here, m is the proportion of particles on the outside of the membrane, where they can interact with the binding sites, and 1 - m is the proportion of particles on the inside of the membrane. The rate constant, am, represents the rate at which particles move from the inner to the outer face of the membrane, and bm is the rate of reverse movement. Because of the charge on the particles, a step change in the membrane voltage will cause an instantaneous change in the rate constants am and bm. For instance, a step depolarization would increase am and decrease bm, leading to a net increase in m and therefore a decrease in 1 - m.

The equation governing the rate at which the charges redistribute following a change in membrane potential will be dm/dt = am(1 - m) - bmm (7-5)

In Equation (7-5), dm/dt is the net rate of change of the proportion of particles on the outside face of the membrane. In words, am(1 - m) is the rate at which particles are leaving the inside of the membrane, and bmm is the rate at which particles are leaving the outside surface; the difference between those two rates is the net rate of change in m. If the distribution of particles is stable as it would be if Em had been constant for a long time the rate at which particles move from inside to outside would equal the rate of movement in the opposite direction, and dm/dt would be zero. If the system is suddenly perturbed by a depolarization, a and b would change and the balance on the right side of Equation (7-5) would be destroyed. If the depolarization is maintained, the rate at which the system will approach a new steady distribution of particles will be governed by Equation (7-5).

The solution of a first-order kinetic expression like Equation (7-5) is an exponential function; that is, following a step change in membrane voltage m will approach a new steady value exponentially. The exponential solution can be written m(t) = m^ - m - m0) e~(am +hm)t (7-6)

This equation states that following a change in membrane potential, m will change exponentially from its initial value (m0) to its final value (m J at a rate governed by the rate constants (am and bm) for movement of the gating particles at that new value of membrane potential. The behavior of m with time after a depolarization, as expected from Equation (7-6), is summarized in Figure 7-9. The number of binding sites occupied by gating particles would be expected to be proportional to m, the fraction of available particles on the outer face of the membrane. Thus, if the occupation of a single binding site causes the channel to open and if the coupling between binding of the gating particle and opening of the channel involves no significant delays, the number of open channels would be expected to follow the same exponential time-course as m after a step depolarization.

Because the total membrane sodium conductance is determined by the number of open sodium channels, sodium conductance measured with a voltage clamp would be expected to be exponential as well, given the assumption of a single gating particle leading to opening of the channel. This prediction, along with the actually observed kinetic behavior of gNa, is diagrammed in Figure 7-10. Unlike the predicted exponential behavior, the rise in gNa actually

At normal resting

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Immediately after depolarization m increasing

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Long time after depolarization

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Plasma membrane m o m

Figure 7-9 Change in the distribution of sodium channel gating particles after a depolarization of the membrane. (a) A schematic diagram of the distribution of charged gating particles at the normal resting potential and at different times after depolarization of the membrane. (b) The effect of a step change in membrane potential (top trace) on the rate constants for movement of the gating particles (middle traces) and on the proportion of particles on the outer side of the membrane (bottom trace).

Figure 7-10 The predicted |
Time -> |
-20 mV |

time-course of the change in | ||

sodium conductance | ||

following a depolarizing step |
E -70 mV | |

(dashed line), assuming that | ||

the proportion of open | ||

channels and hence the | ||

total sodium conductance |
Predicted — |
/ / |

is directly related to the |
/ Observed | |

fraction of gating particles |
/ | |

on the outer face of the |

membrane. The solid line shows the observed change in sodium conductance following a step depolarization. exhibited a pronounced delay following the voltage step. The S-shaped increase in gNa would be explained if more than one binding site must be occupied by gating particles before the channel will open. If the binding to each of several sites is independent, the probability that any one site is occupied will be proportional to m and will thus rise exponentially with time after a step voltage change, as discussed above. The probability that all of a number of sites will be occupied will be the product of the probabilities that each single site will bind a gating particle. That is, if there are two binding sites, the probability that both are occupied will be the product of the probability that site 1 binds a particle and the probability that site 2 binds a particle. Because each of these probabilities is proportional to m, the joint probability that both sites are occupied is proportional to m2. Similarly, if there were x sites, the probability of channel opening would be proportional to mx. The actual rise in sodium conductance following a depolarizing step suggested that x = 3 for the sodium channel: three binding sites must be occupied by gating particles before the channel will conduct. Thus, the turn-on ofgNa following a voltage-clamp step to a particular level of depolarization was proportional to m3, and the temporal behavior of m was given by Equation (7-6).

A similar analysis was carried out for the change in potassium conductance following a step depolarization. The results suggested that x = 4 for the voltage-sensitive potassium channel of squid axon membrane. Thus, the gating charges for the potassium channel redistributed after a change in membrane potential according to a relation equivalent to Equation (7-5):

By analogy with the sodium system, n is the proportion of potassium gating particles on the outside of the membrane, 1 - n is the proportion on the inner face of the membrane, and an and bn are the rate constants for particle transition from one face to the other. Equation (7-7) has a solution equivalent to Equation (7-6):

Here, n0 and n^ are the initial and final values of n. The rise in potassium conductance following a step depolarization was found to be proportional to n4; therefore, the potassium channel behaves as though four binding sites must be occupied by gating particles in order for the gate to open. A major difference between the potassium and the sodium channels is that the rate constants, an and bn, are smaller for potassium channels. That is, the sodium channel gating particles appear to be more mobile than their potassium channel counterparts; this accounts for the greater speed of the sodium channel in opening after a depolarization, which we have seen is a crucial part of the action potential mechanism.

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