Determining K+ Channel Activation Curves from K+ Channel Currents Often Requires the Goldman–Hodgkin–Katz Equation

Potassium ion current in nerve membrane, IK, has traditionally been described by IK = gK(V − EK), where gK is the K ion conductance, V is membrane potential and EK is the K+ Nernst potential. This description has been unchallenged by most investigators in neuroscience since its introduction almost 60 years ago. The problem with the IK ∼ (V − EK) proportionality is that it is inconsistent with the unequal distribution of K ions in the intra- and extracellular bathing media. Under physiological conditions the intracellular K+ concentration is significantly higher than the extracellular concentration. Consequently, the slope conductance at potentials positive to EK cannot be the same as that for potentials negative to EK, as the linear proportionality between IK and (V − EK) requires. Instead IK has a non-linear dependence on (V − EK) which is well described by the Goldman–Hodgkin–Katz equation. The implications of this result for K+ channel gating and membrane excitability are reviewed in this report.

]. At room temperature RT/F = 25 mV. Throughout the remainder of this review RT/F is replaced by 25. The value of P K derived from the results in Figure 1 is 1.25 × 10 −2 cm s −1 as compared to 1.2 × 10 −3 cm s −1 for frog node of Ranvier, the preparation in which the GHK result was fi rst demonstrated (Frankenhauser, 1962). This comparison suggests that K + is 10 times more permeable in squid giant axons relative to frog nerve (see Summary). One question which arises from these results is what is the GHK equivalent at the single channel level? K ions are believed to pass through K + selective channels in a single fi le manner along a row of three K + binding sites (Hodgkin and Keynes, 1955;Zhou et al., 2001). This result is schematically illustrated in the top left inset of Figure 2. Two sites are shown to be occupied by a K ion which is consistent with the view of several groups (Hodgkin and Keynes, 1955;Kohler and Heckmann, 1979;Shumaker and MacKinnon, 1990;Clay, 1991) that a K + channel contains at most a single "vacancy" and that movement of K + through the channel can be viewed as movement of the "vacancy" through the channel. The prediction of the model is (Clay, 1991) I aqN

INTRODUCTION
Potassium ion current -the primary repolarization mechanism in nerve membrane -has often been described by where g K is the K + voltage-and time-dependent conductance, V is membrane potential, and E K is the K + Nernst potential. This description was given by Hodgkin and Huxley (1952) in their classic work on squid giant axons. A central assumption of their analysis is that I K is linearly proportional to the driving force (V − E K ). Several studies since Hodgkin and Huxley (1952) including work on squid giant axons (Frankenhauser, 1962;Binstock and Goldman, 1971;Siegelbaum et al., 1982;Clay, 1991;Taglialatela and Stefani, 1993) have shown that I K has a non-linear dependence on (V − E K ) for physiological conditions which is well described by the Goldman-Hodgkin-Katz equation (Goldman, 1943;Hodgkin and Katz, 1949), referred to here for brevity as GHK. The purpose of this review is to describe the GHK result and its implications for both the voltage dependence of g K and models of membrane excitability.

GHK ANALYSIS
An example of a K + current-voltage relation from squid giant axons is given in Figure 1 (taken from Clay et al., 2008). This result has a curvature in the outward direction with increasing membrane depolarization from the rest level. The curve labeled GHK where P K is the membrane's permeability to K + , F is the Faraday constant, R is the gas constant, T is the absolute temperature, and E K , the Nernst potential, is given by RT F Determining K + channel activation curves from K + channel currents often requires the Goldman-Hodgkin-Katz equation from the GHK equation for −150 < V < 100 mV and so the simpler GHK result is used for the rest of this review in particular the GHK voltage dependence which is given by (3)

VOLTAGE DEPENDENCE OF CHANNEL GATING -SIGNIFICANCE OF GHK
Two primary types of voltage-gated K + channels in nerve membrane (Jan and Jan, 1997) are the delayed rectifi er (Kv2; Kv3) and the rapidly inactivating K + channel (Kv4; Kv1.4) often referred to as I A . An example of recordings of the latter are shown in the right hand panel of Figure 3. These results were obtained from Cajal-Retzius cells in the early postnatal rat brain (Mienville et al., 1999). Similar results have been observed from many preparations. The I A component is rapidly activated with a depolarizing voltage step followed by a somewhat slower time course of inactivation. Activation is so much more rapid than inactivation that the peak current (I A,p in Figure 3) can be used as an approximation of the steady-state current which would occur at any given voltage if inactivation were absent. Those results have traditionally been normalized by the driving force (V − E K ) as indicated by the open circles in Figure 3. One of the primary arguments of this review is that this procedure should be replaced by one in which normalization by GHK[(V − E K )] is used instead. The analysis is illustrated in Figure 3. The open circles were multiplied by (V − E K ) to remove the linear normalization. The results thus obtained were divided by GHK[(V − E K )] -Eq. 3 above. This set of points (fi lled circles) clearly saturate with depolarization. In particular the results for V = 20, 30 and +40 mV are virtually identical. Those results were averaged and that average value was used to normalize all the points. I K activation curves such as those in Figure 3 have traditionally been modeled by the Boltzmann equation, (1 + exp(−(V − V 1/2 )/k)) −1 , as illustrated by the dashed curve (Figure 3) where V 1/2 is the voltage of half-maximal activation and k represents its steepness. A better fi t to either set of results -the open or the closed circles -can be obtained using a model consisting of several Boltzmann equations. An alternative approach is provided by Hodgkin and Huxley (1952). They modeled K + channel gating (the delayed rectifi er) by n(V, t) = −(α n + β n )n(V, t) + α n where 0 ≤ n ≤ 1, and α n and β n are voltage-dependent. They raised the n parameter to the 4th power to account for the sigmoidal time-dependent rise of I K following a voltage step. An n 3 model has been used for I A activation kinetics (Campbell et al., 1993). The activation curve for I A in this model is given by n V n n n ∞ = + 3 3 ( ) ( /( )) α α β with α n and β n for the curve (solid line) in given in the legend of Figure 3. A similar analysis of I A over a broader voltage range is illustrated in Figure 4 (Mienville and Barker, 1997). The activation curve obtained with GHK normalization clearly saturates with depolarization. In contrast normalization by (V − E K ) becomes problematic as increasingly large depolarizations are used (open circles, Figure 4). This argument has some circularity, although voltage-gated channels have been clearly shown to be steeply voltage-dependent (Sigworth, 2003), a dependence which can be demonstrated by other procedures such as deactivation, or "tail" current analysis (Zagotta et al., 1994). The steep voltage dependence is revealed from a family of K + currents by GHK normalization.

GHK NORMALIZATION -SIGNIFICANCE FOR MODELS OF EXCITABILITY
Some modeling studies have been carried out concerning the role of I A in neuronal excitability (Connor and Stevens, 1971;Rush and Rinzel, 1995), but this issue has not yet been fully resolved (Khaliq and Bean, 2008). Consequently, the effect of GHK normalization of I A on excitability cannot yet be determined. In contrast the delayed rectifi er I K is clearly signifi cant for repolarization of the action potential as demonstrated by Hodgkin and Huxley (1952). That analysis uses a linear dependence for I K and (V − E K ). A similar result  (Figure 5 legend). This modifi cation is suffi cient to describe both the I K activation curve and the response of the axon to long duration depolarizing current pulses (Clay et al., 2008). The revised model fi res only once (middle panel, Figure 6) which is consistent with experimental results ( bottom panel, Figure 6). In contrast the Hodgkin and Huxley (1952) model fi res an unending train of action potentials in response to a sustained depolarizing current pulse (Figure 6, top panel). The mechanism underlying this result in the revised model is described in Clay et al. (2008).

FIGURE 1 | Squid axon I K has a non-linear dependence on (V − E K ) which is well described by GHK -Eq. 1 in the text -curve labeled GHK
A second example of the signifi cance of GHK normalization for models of excitability concerns neurons from the suprachiasmatic nucleus (SCN) in the mammalian brain. These cells fi re spontaneously at a relatively low rate, typically 2-8 Hz (Pennartz et al., 2002;Jackson et al., 2004;Belle et al., 2009) which is well mimicked by a recently published ionic model (Sim and Forger, 2007) as shown in the inset of Figure 7. A key component of the model is the description of I K by Bouskila and Dudek (1995). They obtained the I K activation curve using normalization by (V − E K ) -open circles in Figure 7. Normalization by GHK[(V − E K )] gives the fi lled circles in Figure 7. The activation curve is made steeper and shifted leftward on the voltage axis, as with I A from Cajal-Retzius cells and I K in squid axons (Figures 3 and 5, respectively). This modifi cation in the SCN model removes spontaneous fi ring thereby yielding a quiescent preparation having a rest potential of −36 mV (Figure 7 inset). The revised model fails to produce an action potential even in response to strong depolarizations (results not shown). The mechanism for this result concerns the relative position and steepness of the "foot" of the I K activation curve (Figure 7). Removal of I Na (addition of TTX) in the original SCN model (Sim and Forger, 2007) produces quiescence with a rest potential at −62 mV. The revised model rests at −36 mV even with retention of I Na . The foot of the I K curve in the original model has a larger value at −40 mV compared to the revised version. Even at −60 mV this curve is signifi cantly greater than 0 (Figure 7). The foot of the curve in the revised version has a smaller value than the original at −40 mV and is essentially 0 at −60 mV. The revised model allows the "leak" current to depolarize the membrane potential to a level where I Na is almost completely FIGURE 3 | GHK analysis of I A . The records in the right were taken from (Mienville et al., 1999). The scales correspond to 100 ms and 400 pA. The open circles in the graph correspond to peak I A at the respective potential on the   Clay K + channel activation curves in nerve inactivated at rest. The original I K curve repolarizes the membrane potential to a level following an action potential at which I Na inactivation is incomplete. Consequently, repetitive spontaneous fi ring of action potentials occurs in the original but not in the revised model. The squid and the SCN results can be summarized by saying that normalization of I K by GHK reduces excitability in models for both preparations. The GHK modifi cation produces agreement between  (Clay et al., 2008) with the modifi ed I K activation curve given in Figure 5. The revised model fi res once and only once regardless of pulse duration or amplitude, similar to experimental observations, as illustrated in (C). (Vandenberg and Bezanilla, 1991). The open circles were obtained using divalent-free artifi cial seawater. The fi lled circles were obtained under normal divalent cation conditions, Ca 2+ = 10 mM and Mg 2+ = 50 mM. The latter results are well described by (V − E Na ) over the range of potentials spanned by an action potential (−70 to +40 mV) because of partial, voltage-dependent block of I Na by Ca 2+ and Mg 2+ . experiment and theory for squid axons but not for SCN neurons. Further modifi cations in the SCN model will be necessary to bring the model in agreement with experimental results.

A CAUTIONARY NOTE CONCERNING GHK NORMALIZATION
External factors can sometimes mask the GHK voltage dependence of the current-voltage relation. An example is provided by the Na + current in squid axons (Vandenberg and Bezanilla, 1991). The current-voltage relation of this channel is fundamentally described by GHK[(V − E Na )] as illustrated in Figure 8 (open circles). The curvature of this relation is in the opposite direction to that of GHK[(V − E K )] because the external Na + concentration is signifi cantly higher than Na i + . Na + channels are blocked in a voltagedependent manner by extracellular divalent cations (Yamamoto et al., 1984) which are present in signifi cant amounts in seawater (Ca 2+ = 10 mM and Mg 2+ = 50 mM), giving a current-voltage relation for physiological conditions which is nearly linear over the voltage range spanned by an action potential (−70 to +40; fi lled circles in Figure 8). Normalization of peak I Na results by GHK[(V − E Na )] would give an incorrect description of the Na + channel activation curve in squid. Normalization of those results by (V − E Na ) as originally carried out by Hodgkin and Huxley (1952) is serendipitously correct because of divalent cation block of I Na . A similar result may occur for other types of channels. For example, delayed rectifi er K + channels (but apparently not I A K + channels) are partially blocked by physiological levels of Na i + for V > 0 mV (Bezanilla and Armstrong, 1972) which can confound the activation curve analysis for high-voltage-activated channels such as Kv3.1 and Kv3.2 (Hernandenz-Pineda et al., 1999). In particular, Na i block of I K can give an apparent saturation of channel activation using normalization by (V − E K ), a result which would not be found in the absence of Na i + . At some point in the experimental description of an ionic current in any preparation a measurement of the current-voltage relation of the channel is required to determine the appropriateness of GHK for those results. Sim and Forger (2007) model. (B) Prediction of the model when GHK normalization is used to obtain the g K − V curve from the Bouskila and Dudek (1995) measurements of I K described to the right. Main panel: description of I K by Bouskila and Dudek (1995). They normalized their results with (V − E K ) -open circles. The curve describing those results is given by 1/(1 + exp(−(v − 14)/17)). Those points were multiplied by (V − E K ) with E K = −97 mV, and then normalized by the GHK procedure to give the fi lled circles. The curve describing those results is given by (α n /(α n + β n ) 4 with α n = −0.01(V + 30)/(exp(−0.08(V + 30)) − 1) and β n = 0.125 exp(−(V + 40)/30). This description of I K predicts quiescence in the Sim and Forger (2007)

SUMMARY
The results reviewed here are surprising for some investigators. The relation I = g(V − E) with E ≠ 0 as is the case for most ions under physiological conditions has dogmatic status in neuroscience. The simplicity of this equation may be one of the reasons it has been largely unchallenged. Experimental observations have shown that the current-voltage relation for ion channels rectifi es in a manner determined by the respective extra-and intracellular concentrations of the ion in question. The strongest evidence for this result for K + channels are the single channel recordings in Apysia sensory neurons by Siegelbaum et al. (1982)  = . In many preparations the extracellular medium can be exchanged for one in which this condition is obtained, i.e., an increase in K o + to closely match K i + . Depolarizing voltage steps from a relatively negative holding potential as was used for the I A results in Figures 3 and 4 would elicit inward K + currents for these conditions for V < 0 mV. The activation curve from those results could be obtained using linear voltage normalization without introducing the error described in this report when normalizing K + currents in physiological conditions by (V − E K ). Alternatively, as noted above, tail currents could be used for this analysis without increasing K o + . Tail currents are usually measured at a fi xed potential (the holding potential) which would eliminate the GHK nonlinearity in the analysis of the g K − V curve. However, this approach is not feasible for some types of K + channels, in particular I A . Tail currents from I A are typically too small to be reliably measured given the strong inactivation of this component which occurs with membrane depolarization. Consequently, normalization of a family of currents by GHK[(V − E K )] is particularly relevant for I A .
The terms permeability and conductance have been widely used throughout the neuroscience literature. Permeability, P, has units of cm/sec. A comparison of Eqs 1 and 2 demonstrates that at the single channel level P K ∼ aqN K F −1 which also has units of cm/sec with a being a factor describing collisions of K ions with the membrane, q is the elementary electronic charge, N K is the K + channel density, and F is the Faraday. A single value of P is suffi cient and appropriate for the current-voltage (I-V) relation of K + channels for physiological conditions. The non-linearity of the I-V result is given by the other terms either in Eq. 1 or 2. A single value of conductance, G, cannot be used for the I-V curve for physiological conditions. The slope conductance can be used, but that number changes with voltage. For example, the delayed rectifi er channel in squid axons is typically described as having a single channel conductance of 20 pS for physiological conditions (Llano et al., 1988). Those results were obtained for V = +50 mV. According to the GHK voltage dependence (Eq. 3), the single channel (slope) conductance at V = −50 mV would be 2 pS. A single value of G can be used for equimolar conditions, i.e., K K K o i For those conditions I K = P K F(FV/RT)K + = G K V, or G K = P K F 2 K + /(RT). At the single channel level, G K = γ K N K , where γ K is the single channel conductance. From the above discussion concerning permeability, this expression is comparable to aqN K FK + / (RT), so that γ K ≈ aqK + /(25 mV), given that RT/F = 25 mV at room temperature. This expression is only an approximation primarily to give a sense of the units of the various terms involved. The detailed mechanism of ion translocation through K + selective channels at the molecular level of channel proteins is a topic of on-going research (Berneche and Roux, 2001;Fowler et al., 2008; and many other reports referenced in these studies).
One fi nal point concerning the utility of the GHK equation for models of membrane excitability is that it permits a straightforward determination of I K when K o + = 0, conditions which are problematic for I K ∼ (V − E K ) since E K is undefi ned for K o + = 0 (Clay, 1998 exp (V/25) when K o + = 0. The observation that P K in squid axons is 10 times higher than in frog nerve may be of historical interest. The action potential of squid giant axons has a robust after-hyperpolarization potential (AHP) (Hodgkin and Huxley, 1952), whereas frog nerve does not (Frankenhauser and Huxley, 1964). The difference in P K 's provides an explanation for these results.
The primary thrust of this review concerns the use of GHK normalization to obtain ion channel activation curves for the delayed rectifi er I K and for I A . A number of groups (DeFazio and Moenter, 2002;Boland et al., 2003;Van Hoorick et al., 2003;Persson et al., 2005;Nakamura and Takahashi, 2007;Johnston et al., 2008;Dementieva et al., 2009;Sculptoreanu et al., 2009) have recently used the procedure outlined here and in a previous report from this laboratory (Clay, 2000) to obtain these results. One of the purposes of this review is to encourage others to do the same. An accurate description of K + channel activation curves is important for models of neuronal excitability as illustrated by the above examples -squid giant axons and SCN neurons -and for models of K + channel gating. Model building for the excitability properties of neurons in the mammalian brain is an evolving discipline. The analysis given here is one issue which should be kept in mind when constructing models of this type.