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Department of Life Sciences and the Zlotowski Center for Neurosciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Correspondence: Address reprint requests to Ofer Yifrach, Tel.: 972-8-647-9172; E-mail: ofery{at}bgumail.bgu.ac.il.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONRESULTS AND DISCUSSIONCONCLUDING REMARKSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTS AND DISCUSSIONCONCLUDING REMARKSACKNOWLEDGEMENTSREFERENCES |
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; Sigworth, 1994; Yellen, 1998; Bezanilla, 2000). The voltage sensitivity of these channels arises from voltage-induced displacement of charges across the membrane electric field, a phenomenon first noted by Hodgkin and Huxley (1952). These voltage-sensing charges were later found to be alternating basic amino acids that are uniquely arranged at the S4 transmembrane helix, a motif common to all voltage-activated cation channels (Noda et al., 1984; Stuhmer et al., 1989; Papazian et al., 1991; Liman et al., 1991). In the Shaker voltage-dependent K+ channel, for example, the gating charge corresponds to nearly 12–14 electron charge units traversing the membrane electric field (Schoppa et al., 1992; Seoh et al., 1996; Aggarwal and MacKinnon, 1996). Similar number of gating charges move across the membrane electric field when skeletal muscle Na+ channels are opened (Hirschberg et al., 1995). Additional contributions to the voltage sensitivity of these channels may arise, however, from cooperative interactions along the activation pathways of such channels (Papazian et al., 1991; McCormack et al., 1991; Schoppa et al., 1992; Tytgat and Hess, 1992; Zagotta et al., 1994; Smith-Maxwell et al., 1998a). Such interactions give rise to cooperativity in channel gating transitions. In the following, cooperativity in channel function is interpreted in a simple, intuitive, and traditional way. We ask how the switch of one subunit from the closed to the open state affects the gating transitions of its neighboring subunits (may be more than one transition for each subunit). In other words, considering cooperativity in channel gating addresses the coupling between the channel's subunits. The extreme case is the two-state model in which all four subunits of the channel switch in a concerted manner—all at once—from the closed to the open state. Alternatively, subunits switching from closed to the open state may occur in an independent fashion, in which channel subunits do not "sense" the conformational state(s) of their adjacent subunits. Cooperative interactions along the reaction coordinate of channel gating may originate from intersubunit interactions in sequential KNF-type (Koshland et al., 1966) transitions of the channel, from MWC-type (Monod et al., 1965) concerted transition(s) along the channel activation pathway or from combination of both (for detailed discussion, see Sigworth, 1994; Zagotta et al., 1994). An estimate of the magnitude of these interactions is the effective gating charge (Z), obtained by fitting the voltage-activation data of a channel to a two-state Boltzmann equation, derived assuming equilibrium of the channel between the closed and open states. As already pointed out, the essence of Z is intuitively reminiscent of the Hill coefficient used to describe the magnitude of cooperativity in ligand binding allosteric systems (Almers, 1978; Islas and Sigworth, 1999). In what follows, is a rigorous demonstration that the two-state voltage-dependent Boltzmann equation is analogous to the original Hill equation. A framework to understand the meaning of the Hill coefficient for channel gating transitions is suggested. Using three particular and simple gating schemes for channel activation, the relation of nH to the magnitude and nature of cooperativity in channel gating transitions is demonstrated.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONRESULTS AND DISCUSSIONCONCLUDING REMARKSACKNOWLEDGEMENTSREFERENCES |
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), the magnitude of cooperative interactions along the channel activation pathway may be assessed by the slope of such a relation at moderate-range open probabilities (Smith-Maxwell et al., 1998b). An estimate for the slope of the G–V curve at these latter values is frequently obtained by fitting activation data to a Boltzmann equation, derived assuming a simple two-state model for channel activation. According to this model, the equilibrium between the closed (C) and open (O) states of the channel is voltage-dependent (K(V)) and is given by
 | (1) |
 | (2) |
),
 | (3) |
the fractional binding saturation function, is the fraction of sites occupied with the substrate (S), nH is the Hill coefficient, and K is the apparent binding constant of the substrate to the enzyme. This equation was derived assuming an infinitely cooperative case, i.e., where n substrate molecules bind simultaneously to the enzyme. In practice, however, by fitting experimental binding data with the Hill equation, a value between 1 and the total number of binding sites (n) is frequently observed for nH. nH, therefore, is an index of cooperativity in ligand binding. Whereas nH = n reflects cooperative and concerted ligand binding to the enzyme, nH = 1 reflects noncooperative (independent) sequential binding. Although the last statement is true in the case of many allosteric enzymes, the relation of nH to cooperativity in ligand binding is not always correct, as nH depends on the specific details of the ligation reaction (see below).
The analogy between the Hill equation and the Hill form of the Boltzmann equation is not surprising, bearing in mind the underlying assumption of the Boltzmann equation of one concerted transition of all channel subunits (or protomers) from the closed to open state. The similarity between the two equations is not just in structure but also in essence; conformational transitions within a protein may be driven by changes in chemical potential (logS, in the case of ligand-binding systems) or by changes in electrical potential (ZUFV/RT, in the case of voltage-gated ion channels). Given that the unitary gating charge of the channel is known, fitting channel activation data to the Hill form of the Boltzmann equation may yield a value for the Hill coefficient for channel gating transitions. In a simple way, this allows for comparison of the magnitude of steady-state cooperativity in channel gating transitions between different subtypes of the same channel or between different cation channels, irrespective of differences in the nominal gating charge of the channels. Hill coefficient values for gating transitions of ion channels may be also compared to those derived for other allosteric enzymes like hemoglobin, aspartate transcarbamylase and the GroEL chaperonin. It should be stressed, however, that although it is a common practice to compare Hill values of different proteins, such comparisons, as will be evident later on, are more informative when comparing channels that gate according to similar activation mechanisms. Fitting the Shaker K+ channel activation data to the Hill form of the Boltzmann equation (assuming a unitary gating charge of 3) yields a value slightly above 1 for the Hill coefficient (data not shown) (Yifrach and Mackinnon, 2002). Taking into account the tetrameric organization of K+ channels (MacKinnon, 1991), this low nH value obtained for the Shaker K+ channel may reflect the fact that the activation pathway of this channel involves many independent subunit transitions (Zagotta et al., 1994; Schoppa and Sigworth, 1998). A possible functional explanation for this low Hill value for the Shaker K+ channel will be discussed later.
nH depends on the Adair-like model parameters
The form of the Hill equation for channel gating transitions (Eq. 2) is based on the assumption that no intermediate states exist along the channel gating reaction, or in other words, that all four channel's subunits switch at once from the closed to fully open state. This, of course, is far from reality for almost all channels studied thus far. Gating reactions of ion channels are much more complicated than predicted by a two-state scheme, requiring many closed and (sometimes) open states (Sigworth, 1994; Zagotta et al., 1994; Bezanilla, 2000). For a channel that gates according to a multistates activation scheme, how reliable is it then to estimate the magnitude of cooperativity in gating transitions using a two-state Hill equation? A similar reservation was made by Adair with respect to the use of the Hill equation to assess the extent of cooperativity in oxygen binding to hemoglobin (Adair et al., 1925). It should be stressed, however, that plotting a channel's open-probability data according to the Hill form of the Boltzmann equation is a legitimate procedure. It is instead the interpretation of the Hill coefficient that requires detailed analysis. Accordingly, and in analogy to ligand binding allosteric systems (Levitzki, 1978), expressions for nH that depend on the Adair-like constants for particular channel gating schemes can be derived. This allows more precise understanding of the meaning of the Hill coefficient.
A general definition for a Hill coefficient for gating transitions may be derived from Eq. 2 as follows:
 | (4) |
; see Eq. 32 therein), derived for the general case of sequential transitions along a channel's activation pathway. The analogy is interesting when considered from historical perspective. In Almers' original derivation, the total channel gating charge (qT) was multiplied by a fractional number that is dependent on the number and nature of intermediate states along the gating reaction (Almers, 1978). This number is, in any sense, a scaled Hill-like number. Only several years later, upon cloning of the first voltage-activated channels, was the tetrameric organization of these channels realized (Noda et al., 1984; Tanabe et al., 1987; MacKinnon, 1991). This fact, combined with more accurate determination of the total gating charge per channel (Schoppa et al., 1992; Seoh et al., 1996; Aggarwal and MacKinnon, 1996), allowed, by inference, determination of the unitary gating charge per subunit (ZU). Incorporating ZU in Eq. 2 gives nH its common traditional view, i.e., number of subunits/protomers (normally, 1 
nH n).
The general definition of the Hill equation given here (Eq. 4) is also reminiscent of the equation derived by Sigg and Bezanilla describing the dependence of the mean activation charge, 
qA
(the actual gating charge that is energetically coupled to channel opening), on gating charge displacement (Sigg and Bezanilla, 1997). In that study, the following expression for qA was derived following a detailed and thorough statistical mechanics analysis,
 | (5) |
Z and Zl are the equilibrium gating charge displacement and the latent gating charge (the effective charge movement during transitions among open states), respectively. It can be easily seen that Eqs. 4 and 5 may be combined to yield the following expression for nH,
 | (6) |
Zl = 0,
 | (7) |
; Sigg and Bezanilla, 1997; Islas and Sigworth, 1999). The magnitude and direction of displacement of the Q(V) curve along the voltage axis with respect to the P(V) curve can be compared. In the simplest case, the two curves will coincide in the case of two-state gating model for which nH = ZT/ZU for the entire voltage range. For the Shaker voltage-gated K+ channel that gates according to an activation pathway involving many closed states and one open state (Zagotta et al., 1994; Schoppa and Sigworth, 1998), the Q(V) curve is shifted to the left along the voltage axis with respect to the P(V) curve. This observation combined with results demonstrating that the latent gating charge for this channel is very low (Islas and Sigworth, 1999) may explain the low value of nH obtained at moderate-range open probabilities for which (1–Q(V))/(1–P(V)) is always smaller than 1. For this case, nH will equal n, the total number of channel subunits, only at an Almers' range of very low open probabilities achieved at extremely negative voltages. For the general case, analysis using Eq. 5 above (or Eq. 6 within the context of the present analysis) may yield additional insight into the meaning of nH (Sigg and Bezanilla, 1997).
The definition for the Hill coefficient for channel gating transitions (Eq. 4) makes intuitive sense—nH is a scaled slope of an open probability function (
P/V) relative to the slope of a reference open probability function (ZF/RT)(P(1–P)), derived for a gating scheme that assumes independent subunit transitions. It can be shown that for an independent gating scheme of the sort 
the slope of the open probability function with respect to voltage (P/V) is equal to the scaled multiplication of the probabilities of the channel to be closed and to be opened (P(1–P)(ZF/RT)). For the scheme above, C and O are the respective subunit conformations in the closed and open states and K(V) is the equilibrium constant for subunit transitions. It is also assumed that both the CO and OO states are conductive. Thus, a Hill coefficient value of 1 will be obtained for nH if the slope of the open probability function (P/V) is equal to that of the independent, noncooperative case. Deviations of the Hill value from 1 may indicate the existence of cooperativity in channel gating transitions—a reflection of coupling between the channel's subunits. Using this empirical definition of cooperativity in channel gating transitions, one may derive explicit expressions for nH for different channel gating schemes (different P schemes).
nH for a KNF-type gating model
Consider, for example, the following simple gating scheme for a homodimeric channel undergoing two sequential KNF-type subunit (or protomer) transitions (Koshland et al., 1966), as shown in Fig. 1 A. In this gating scheme, it is assumed that the channel is open only when both subunits are in the open state (O state). Using the principle of microscopic reversibility and Boltzmann-type transitions for channel subunits/protomers (Eq. 1), one can derive expressions for P, 1–P, and P/V for this gating scheme and substitute them into Eq. 4 to obtain the expression for nH,
 | (8) |
). Usually, however, gating isotherms of voltage-activated channels are parameterized based on the voltage midpoint of the gating transition (V1/2) where P = 1/2 and on the slope of the gating isotherm around this midpoint (Z). The Hill coefficient at an open probability of half (nH (V = V1/2)) may be derived for such a gating model by combining Eq. 8 with a limiting condition derived from P = 1/2 to yield the expression
 | (9) |
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 | (10) |
The dependences of both nH at half-activation and V1/2 on the Adair-like model parameters, as well as the relation between these two quantities, can be better realized by graphing the three-dimensional surfaces of nH and V1/2 as a function of both KA and KB, using identical scales (Fig. 2). The main diagonal trajectory in both surfaces, where KA = KB, separates the two domains of negative and positive coupling between channel subunits. Upon comparison of nH and V1/2 surfaces, it may be seen that the simple sequential KNF-type gating scheme (Fig. 1 A) permits different relations between nH and V1/2. For example, increasing KB alone along the outlined trajectory in Fig. 2 (such as might be achieved upon mutations, for example), would shift the midpoint activation voltage (V1/2) to more negative potentials (the activation curve is shifted to the left along the voltage axis open-state stabilization effect) and would increase the slope of the activation curve (i.e., nH increases). Other relations between nH and V1/2 are expected for different trajectories crossing these surfaces (see Fig. 2 legend for further discussion).
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), described in Fig. 3 A. In this scheme, sequential and independent voltage-dependent transitions, probably reflecting charge movements transitions between closed states and between open states (KC(V) and KO(V), respectively), are separated by voltage-independent pore opening transitions (L). We assume, for simplicity, that a similar gating charge moves through both closed and open transitions and that the open probability function for this scheme is given by Po = (O1 + O2 + O3)/(C1 + C2 + C3+ O1 + O2 + O3). Using the Hill definition outlined in Eq. 4, the following expression is thus obtained for nH(V)
 | (11) |
)) are shown in Fig. 3 B. The extreme points of nH(V) at any given c can be obtained by taking the partial derivative of Eq. 11 with respect to voltage (nH(V)/V). This yields the following expression for 
the maximal scaled slope of the MWC open probability function obtained at Vmax, the activation voltage at this maximal slope:
 | (12) |
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depends on the ratio of equilibrium constants for charge movement transitions in the closed and open states and not on the allosteric equilibrium constant L (Karlin, 1967). A Hill value of 2 is obtained for very low values of c (KO >> KC), whereas a value of –2 is obtained for high c-values (KO << KC). Thus, for a channel that gates according to the classical MWC scheme (Fig. 3 A), reflects the magnitude of cooperativity in channel opening transitions which is solely determined by c. It follows from the above analysis that the maximal slope of the open probability function does not necessarily coincide with the slope at the midpoint of activation (nH (V = V1/2)). The direction and magnitude of the departure of from nH (V = V1/2) depends on the relative values of L and c. The expression for nH (V = V1/2) may be obtained by combining Eq. 11 with the limiting condition derived from P = 1/2 to yield
 | (13) |
is always greater than nH at half-activation and that nH (V = V1/2) will equal only at the limits where the allosteric constant L is very high and the ratio of charge movement equilibrium constants in the closed and open states, c, is extremely low, in which case a Hill coefficient of 2 is obtained. For the MWC gating scheme analyzed here, rather than nH (V = V1/2) is a more natural parameter of homotropic interactions, since the former is a measure of the maximum degree of cooperativity under specific conditions (Rubin and Changeux, 1966). The dependences of and Vmax on c are compared in Fig. 4. The symmetry of the graph around KO = KC reflects the symmetric nature of the MWC gating scheme. It may be seen that for c-values indicating open state stabilization, i.e., c < 1 (KO > KC), is increased, whereas the voltage at this maximal slope is shifted to more negative voltages.
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).
Combined MWC and KNF gating model
Sequential KNF-type or concerted MWC-type gating schemes were previously employed to analyze K+ and Na+ channels activation data. Alternative gating schemes, in which sequential but independent transitions at voltage-sensing domains (K(V) transitions between closed states) are followed by a concerted voltage-independent late pore opening transition (L), are, however, often considered (Sigworth, 1994; Zagotta et al., 1994). The simplest version of such a gating scheme is 
(Scheme 3). Assuming that only the end state is conductive, the above analysis can be applied to show that, for high L-values, one obtains the expression for nH at half-activation of
 | (14) |
The approach presented here, for analyzing cooperativity in channel gating transitions using the empiric Hill coefficient definition (Eq. 4), is general and may be applied to other more realistic or more complicated gating schemes. Such analysis would yield, as demonstrated for the three simple gating schemes considered here, a dependence of the magnitude of cooperativity in gating transitions (nH) on the equilibrium constants of the specific gating scheme discussed. Such analysis could provide mechanistic insight to account for experimental observations that may complement insight gained from gating currents measurements (Sigg and Bezanilla, 1997; Islas and Sigworth, 1999).
Cooperativity in gating transitions of voltage-dependent potassium channels
The theoretical range of values of nH for a protein with n subunits exhibiting positive cooperativity is 1 < nH n. For many allosteric proteins, however, nH << n. In the case of the Shaker voltage-gated potassium channel, a Hill coefficient slightly above 1 is obtained, a low value as compared to those obtained with other tetrameric proteins like hemoglobin or glyceraldehyde-3-phosphate dehydrogenase (nH values of 2.8 and 2.5, respectively; Edelstein, 1971; Kirschner et al., 1971). The low value of nH for the Shaker channel may reflect the fact that for this channel, gating occurs through many independent transitions. One hint to a functional explanation for the low Hill value for gating transitions of the Shaker channel may be provided by the recently observed correlation between the midpoint (V1/2) and slope (Z) of gating transitions of wild-type and variants Shaker K+ channel proteins (Yifrach and MacKinnon, 2002). It was found that the further the transition midpoint is shifted to the left upon mutation, as compared to the wild-type protein (i.e., an open state stabilization effect), the steeper the slope of the gating transition. At extremely negative V1/2 values, slope values close to the total number of gating charges of the channel are obtained (nH approaches n). This relation between V1/2 and Z (= ZU nH (V = V1/2)) may be generally explained even by simple gating models incorporating cooperativity in different ways (see, for example, analysis of the KNF and MWC gating models in Figs. 2 and 4, respectively). Detailed analysis (Yifrach and MacKinnon, 2002), however, has shown that this observation may be explained within the framework of a well-developed gating model that is a generalized version of Scheme 3, above (Zagotta et al., 1994; Schoppa and Sigworth 1998). It was concluded that such correlation between the midpoint and slope of the gating isotherms is expected if mutations of the Shaker channel primarily affect the late cooperative pore-opening transition (L transition of Scheme 3). This conclusion implies for a possible explanation for the low Hill value obtained for the Shaker K+ channel: Tuning the channel to be more cooperative (increased L and nH values) would shift the activation curve of the channel to the left, thus bringing V1/2 closer to the resting membrane potential. It would also narrow the voltage range in which transition from the closed to open state occurs. Under such circumstances, potassium channels may be opened even in response to subtle fluctuations (noise) in the resting membrane potential. Uncoordinated opening and closing of K+ channels, which reflects the inability to regulate the channel in such an infinitely cooperative case, would have a dramatic effect on the shape of action potentials.
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CONCLUDING REMARKS |
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TOPABSTRACTINTRODUCTIONRESULTS AND DISCUSSIONCONCLUDING REMARKSACKNOWLEDGEMENTSREFERENCES |
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The Hill equation was originally deduced in 1910 to account for the sigmoidal kinetics of oxygen binding to hemoglobin. This equation made no assumptions about the molecular mechanisms giving rise to cooperativity. Mechanistic models that account for cooperative phenomena and regulation of enzymes were later suggested by Monod and Koshland and their associates in the early 1960s. In recent years, these allosteric models have been widely employed to analyze the steady-state activation of different kinds of voltage-gated channels. The finding that concepts of classical enzymology are also applicable to the study of channel gating, together with the analogous natures of the Hill and Boltzmann equations demonstrated here, point to a unified conceptual framework for understanding the functional behavior of channels and enzymes.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONRESULTS AND DISCUSSIONCONCLUDING REMARKSACKNOWLEDGEMENTSREFERENCES |
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Submitted on February 1, 2004; accepted for publication April 8, 2004.
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REFERENCES |
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(B) Double-mutant cycle representation of the KNF model, above (A). The "mutation" in this cycle is a transition of a subunit from the closed to the open conformation. Cooperativity arises when the free energy associated with this "mutation" depends on the conformational state of the adjacent subunit. Such cooperativity is a function of the magnitude of intersubunit interactions between the monomers in the different conformations and is manifested by KB/KA.
derived using the condition of half-activation (PO = 1/2). Unitary gating charge of 3 was assumed for Z. The main diagonal trajectory where KA = KB separates positive ((KB/KA) > 1) and negative ((KB/KA) < 1) coupling domains to the right and left of this diagonal, respectively. Three extreme scenarios may be envisaged: 1), along the main diagonal trajectory; increasing KA and KB upon mutations, for example, would shift the midpoint activation voltage to the left to more negative voltages; however, nH would not change. 2), The trajectory in bold corresponds to a case in which increasing KB only would result in a shift of the midpoint activation voltage to the left in parallel with steeper nH slopes. 3), Opposite dependence is observed between nH and V1/2 for a case in which only KA is changed upon mutations.
It is assumed that identical gating charge Z moves across the membrane electric field in transitions between closed and between open states. (B) Dependence of nH on voltage at different values of c (= KC/KO). Graphs were generated according to 