On the theory of ion transport across the nerve membrane. V. Two models for the Cole-Moore K + hyperpolarization delay

Potassium current in the squid giant axon

The squid giant axon was the first preparation to be investigated with the voltage clamp technique over 30 years ago by Cole (1949) and Hodgkin et al. (1952). During the intervening years it has continued to serve as a useful preparation for the development of other new techniques such as internal perfusion (Baker et al., 1962), gating current measurements (Armstrong and Bezanilla, 1974), and patch clamp measurements (Conti and Neher, 1980). It also has served as a useful comparative preparation for investigations of sodium and potassium currents in other excitable membrane preparations. This article has focused on the activation kinetics and the instantaneous current-voltage relation of the potassium component. The squid axon is well suited for studies of IK, because it appears to have only a single type of potassium channel, and the leakage current is relatively small under ideal conditions. The IK component is activated in a sigmoidal manner following membrane depolarization. It deactivates with a single exponential time constant following return of the membrane potential to the holding level, although the deactivation time constant varies with changes in the external potassium concentration. There has not, as yet, appeared a self-consistent model which describes all of these results. The current-voltage relation is a nonlinear function of driving force, which is approximately described by the Goldman-Hodgkin-Katz equation, although a model of the IV based on single file diffusion of ions through a channel is more in tune with the modern view of the ion permeation process (Hodgkin and Keynes, 1955; Hille and Schwarz, 1978; Clay and Shlesinger, 1977, 1983, 1984). Further progress in this area will probably be achieved both by the traditional techniques and by the patch clamp technique. The traditional method is well suited for studying tail current kinetics and the slow inactivation process. The patch clamp technique is well suited for studying the distribution of channels in the membrane and the kinetics of channel gating in steady state conditions.

Dynamics of potassium ion currents in squid axon membrane. A re-examination

The original experiments of Cole and Moore (1960. Biophys. J. 1:161-202.), using conditioning and test membrane potentials to examine the dynamics of the potassium channel conductance in the squid axon, have been extended to test voltage levels by the use of tetrodotoxin to block the sodium conductance. The potassium currents for test voltage levels from -20 to +85 mV were superposable by translation along the time axis for all conditions tested: (a) with depolarizing conditioning voltages; (b) with hyperpolarizing conditioning voltages; and (c) in normal and in high potassium external media. The only deviations from superposition seen were when the internal sodium concentration was abnormally high and the potassium currents showed saturation at high levels of depolarization. Some restoration toward normal kinetics could be obtained by rapidly repeated depolarizations.

Ba2+-induced conductance fluctuations of spontaneously fluctuating K+ channels in the apical membrane of frog skin (Rana temporaria)

We studied the influence of mucosal Ba2+ ions on the recently described (Zeiske & Van Driessche, 1979a, J. Membrane Biol. 47:77) transepithelial, mucosa towards serosa directed K+ transport in the skin of Rana temporaria. The transport parameters G (conductance), PD (potential difference), Isc (short-circuit current, "K+ current"), as well as the noise of Isc were recorded. Addition of millimolar concentrations of Ba/+ to the mucosal K+-containing solution resulted in a sudden but quickly reversible drop in Isc. G and Isc decreased continuously with increasing Ba2+ concentration, (Ba2+)o. The apparent Michaelis constant of the inhibition by Ba2+ lies within the range 40-80 microM. The apical membrane seems to remain permselective for K+ up to 500 microM (Ba2+)o. Higher (Ba2+)o, however, appears to induce a shunt (PD falls, G increases). This finding made an accurate determination of the nature of the inhibition difficult but our results tend to suggest a K+-channel block by K+-Ba2+ competition. In the presence of Ba2+, the power spectrum of the K+ current shows a second Lorentzian component in the low-frequency range, in addition to the high-frequency Lorentzian caused by spontaneous K+-channel fluctuations (Van Driessche & Zeiske, 1980). Both Lorentzian components are only present with mucosal K+ and can be depressed by addition of Cs+ ions, thus indicating that Ba2+ ions induce K+-channel fluctuations. The dependence of the parameters of the induced Lorentzian on (Ba2+)o shows arise in the plateau values to a maximum around 60 microM (Ba2+)o, followed by a sharp and progressive decrease to very low values. The corner frequency which reflects the rate of the Ba2+-induced fluctuations, however, increases quasi-linearly up to 1 mM (Ba2+)o with a tendency to saturate at higher (Ba2+)o. Based on a three-state model for the K+ channel (having one open state, one closed by the spontaneous fluctuation and one blocked by Ba2+) computer calculations compared favorably with our results. The effect of Ba2+ could be explained by assuming reversible binding at the outer side of the apical K+ channel, thereby blocking the open channel in ;competition with K+. The association-dissociation of Ba2+ at its receptor site is thought to cause a chopping of the K+ current, resulting in modulated current fluctuations.

Cole-Moore effect in the frog node

Potassium currents were recorded from the voltage-clamped frog node (Rana esculenta) during various test pulses that followed hyperpolarizing prepulses of different amplitudes and durations. Both the delay in potassium current onset and the shape of the current trace as a function of time were found to be a function of prepulse parameters. This finding is different from the current trace superposition described by Cole and Moore for a specific test pulse, sodium equilibrium potential in the squid giant axon. The Cole-Moore effect, which was found here only under a specific set of conditions, thus may be a special case rather than the general property of the membrane. The implication of these findings to the various excitable membrane potassium channel models, which are based on the Cole-Moore effect, is discussed.

Channel noise in nerve membranes and lipid bilayers

The basic principles underlying fluctuation phenomena in thermodynamics have long been understood (for reviews see Kubo, 1957; Kubo, Matsuo & Kazuhiro 1973 Lax, 1960). Classical examples of how fluctuation analysis can provide an insight into the corpuscular nature of matter are the determination of Avogadro's number according to Einstein's theory of Brownian motion (see, e.g. Uhlenbeck & Ornstein, 1930; Kac, 1947) and the evaluation of the electronic charge from the shot noise in vacuum tubes (see Van der Ziel, 1970).

Fluctuations and noise in kinetic systems. Application to K+ channels in the squid axon

We consider the equilibrium or steady-state noise power density spectrum in the quantity N = Sigma(x) (i=0)a(i)N(i) for an ensemble of independent and equivalent systems each of which can exist in the discrete set of states i = 0, 1, ..., x. N(i) is the number of systems of the ensemble in state i and the a(i)'s are constants. There is a transition rate constant alpha(ij) for an arbitrary transition i --> j; the kinetic equations are linear. There are possible applications to enzyme and biochemical kinetics generally, to membrane transport, muscle contraction, binding on macromolecules, etc. In each case, noise measurements would provide information about the kinetic scheme. The particular application considered here is to K(+) channels or gates (one channel = one system) in the squid axon membrane: a(i)g(K) is the K(+) conductance of a channel in state i and the kinetic scheme is of the Hodgkin-Huxley type (HH). Here we allow an arbitrary set of a(i)'s. This is a generalization of our treatment of K(+) channel noise in an earlier paper. The theory is discussed and some calculations made using Fishman's recent experimental results on K(+) channel noise as a guide. Preliminary indications are that the HH choice of a(i)'s may be oversimplified and that a(0) congruent with 0, a(1) not equal a(0), a(x) not equal a(x-1). Quite possibly the a(i)'s increase from a(0) to a(x), though the early a(i)'s must be relatively small to give the observed induction behavior in g(K)(t). An increase in equal steps is unsatisfactory because this is essentially HH with x = 1 (no induction). More refined experiments may modify these tentative conclusions. In any case, it appears from Fishman's work that noise measurements will probably be very useful in distinguishing between rival models of K(+) channels.