FIGURE 8 (A) Measurement of ion currents through a single channel by ''clamping'' a patch of membrane with a micropipette. (B) Apparatus for reconstituting vescicles containing channels into a planar lipid bilayer. The current-voltage (i-V) converter measures the small membrane currents by determining the voltage and developed across a very large resistor.

FIGURE 8 (A) Measurement of ion currents through a single channel by ''clamping'' a patch of membrane with a micropipette. (B) Apparatus for reconstituting vescicles containing channels into a planar lipid bilayer. The current-voltage (i-V) converter measures the small membrane currents by determining the voltage and developed across a very large resistor.

A recording of the activity of a single ion channel is illustrated in Fig. 9. (Note that the ordinate is in pA [i.e., 10"12 A] and the abscissa is in milliseconds.) The openings are characterized by abrupt upward deflections from the baseline that reach a plateau and then abruptly close. An important characteristic of single-channel activity is that the openings and closings are stochastic or random processes. That is, when a channel will open and how long it will remain open under fixed conditions are independent of its past history (e.g., how long it was closed before this opening or how long it was open the last time it was open); the channel does not have a memory. This characteristic enables one to analyze single-channel activity using statistical or probability

theory. For example, by analyzing a large number of opening and closing events, such as those shown in Fig. 9, one can determine the open-time probability of the channel, Po; that is, the likelihood that the channel will be found in the open or conducting state at any given moment or the fraction of time the channel is open. Furthermore, if the channel can only randomly switch between two states due to thermal motion, then the transition between these states is given by:

k-i where the ks are rate constants analogous to those employed in chemical kinetics. In this instance, ki is the probability of transitions from the closed to the open state in unit time so that (l/ki) is the mean duration of the closed states or the mean closed time (rc). Likewise, k-i is the probability of transitions from the open state to the closed state in unit time, so that (l/k-i) is the mean open time (ro). It follows that the open-time probability will equal the mean open time divided by the mean total time; hence, Po = ro/

Now, because of the random nature of the openings and closings of a channel, the mean duration of either the open or closed states is determined only by the rate constants acting away from that state. Thus, by analogy with a simple first-order chemical reaction, we can write d(open)/dt = k-i(open), where the term in parentheses, roughly speaking, refers to the duration of a single channel in the open state. Integrating this equation, we obtain:

which states that once a channel opens the likelihood or probability that it will remain open decays exponentially with a rate determined by k-1 (or to); the probability that a channel, once opened, will remain open for t msec or longer is given by e-t/To.

We can determine the mean open time (to) by examining a large number of open events of a single channel and sorting them into "bins" based on their durations. For example, the first bin could contain the number of events when the channel was open for at least 0.5 msec; the second bin could contain all of the open events lasting at least 1 msec; the third, all of the open events lasting at least 1.5 msec; and so on in 0.5-msec increments. We can then construct a histogram as shown in Fig. 10, where the ordinate is the number or frequency of events per 0.5 msec and the abscissa is open time (t, in msec). If the channel conforms to the two-state model (i.e., either fully closed or fully open),

Time (msec)

FIGURE 10 Histogram of number of open events versus cumulative duration of opening for a single channel observed over a long period of time. The curve corresponds to an exponential decay where to ~ 2 msec. Thus, ~63% of the openings had durations less than 2 msec and ~37% had longer open times.

Time (msec)

FIGURE 10 Histogram of number of open events versus cumulative duration of opening for a single channel observed over a long period of time. The curve corresponds to an exponential decay where to ~ 2 msec. Thus, ~63% of the openings had durations less than 2 msec and ~37% had longer open times.

then the midpoints of the histogram bars can be joined by a curve described by a single exponential decay with increasing time and k-1 or to can be derived by simple curve-fitting procedures. A similar approach can be employed to determine the mean closed time, tc. If, however, the channel's behavior is more complicated, the open-time and closed-time histograms will not conform to single exponentials. For example, a channel could reside in one of three possible states—very closed, not quite as closed but still not open, and open. In this case, one exponential might describe the open-time probability histogram but two might be needed to describe the closed-time histogram.

Finally, by determining single-channel activity when the membrane is clamped at a number of different electrical potential differences one can obtain valuable information regarding the conductance and ionic selectivity of the channel. For example, Fig. 11 illustrates the relation between the size of ionic currents flowing through a single channel (ic) and the membrane potential, Vm, when the solution facing the outer surface of the channel contains 15 mmol/L KCl and that facing the inner surface contains 150 mmol/L KCl. Note that in this example, the relation between ic and Vm is linear (or ohmic) and can be described by a relation analogous to Eq. 13:

where gc, the slope, is the conductance of the single channel. (Recall that the accepted convention is that the ic (pA) T 30

FIGURE 11 Relationship between single channel current, ic, and the electrical potential difference across the channel, Vm, in the presence of asymmetric KCl solutions. EK and Ea are the menst equilibrium potentials for K+ and Cl- and Vr is the electrical potential difference across the channel when current flow is zero (i.e., the "reversal" or "equilibrium" potential for the channel under the given conditions). gc is the conductance of the channel.

would have been described by the equation iK = gK(Vm — EK) and the intercept on the abscissa would be where

FIGURE 11 Relationship between single channel current, ic, and the electrical potential difference across the channel, Vm, in the presence of asymmetric KCl solutions. EK and Ea are the menst equilibrium potentials for K+ and Cl- and Vr is the electrical potential difference across the channel when current flow is zero (i.e., the "reversal" or "equilibrium" potential for the channel under the given conditions). gc is the conductance of the channel.

outer or extracellular solution is considered ground [zero potential] so that Vm is the electrical potential of the inner compartment with respect to that of the outer compartment and that the flow of cations [i.e., a positive current] from the inner or intracellular compartment to the outer compartment results in an upward or positive deflection.)

Now, using the Nernst equation (11), we can calculate the equilibrium potentials for K+ (i.e., EK) and Cl— (i.e., ECl); recall that EK is the value of Vm at which there is no ionic flow if the membrane is permeable to K+ but impermeable to Cl—, and Ecl is the value of Vm at which there is no ionic flow if the membrane is permeable to Cl— and impermeable to K+. As indicated on Fig. 11, for this tenfold concentration ratio, Ek = —60 mV and Ea = 60 mV. Note, however, that the observed value of Vm at which the current is zero (referred to as the zero-current or reversal potential, Vr) is —50 mV. The fact that Vr is not equal to either EK or ECl indicates that the channel is not exclusively permeable to either K+ or Cl—. Moreover, the observation that Vr is much closer to EK than to ECl indicates that the channel is much more permeable to K+ than to Cl—. The actual ratio of PK/PCl can be obtained using the expression for the diffusion potential (Eq. 8), which can be written as follows:

Vr = {[(PK/PCI) - 1]/[(P*/P«)+ 1]} x 60 log{(KCl)o/(KCl),-}

Solving this equation for the condition, (KC1)o = 15 mmol/L, (KCl), = 150 mmol/L, and Vr = -50 mV yields (Pk/Pc) = 11.

Clearly, if the channel were ideally permselective for K+, the current-voltage relationship shown in Fig. 11

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