Schoppa NE , Sigworth FJ .

	Figure 3. Fits of Scheme 0+2′ to selected WT and V2 macroscopic ionic current time courses that reflect the final transitions. For WT, these include (A) tail currents at voltages between −93 and −193 mV (patch w448), and (B) time-dependent occupancies in the last closed state in the activation path CN-1, derived from reactivation time courses. Occupancies in CN-1 are indicated for hyperpolarizations to voltages Vh = −93, −153, and −193 mV as a function of the hyperpolarization duration th. Occupancy estimates were derived from the amplitude of the fast reactivation component, as described in a previous paper (Schoppa and Sigworth, 1998a), and reflect averages from one to four experiments. In the simulations of these data, we set αN-1 = 0 during the test pulse, or, effectively, βN-2 >> αN-1 at V ≤ −93 mV. Scheme 0+2′ also accounts for (C) V2's macroscopic ionic tail currents at voltages between −73 and +27 mV, and (D) V2's channel opening time courses after a prepulse to +7 mV. In D, the prepulse loads most channels into the last closed states, so that the test currents mostly reflect the kinetics of the final two transitions. All V2 data are from the same patch (v329).
	Figure 10. Fits of Scheme 2+2′ to WT and V2 reactivation kinetics. (A) Scheme 2+2′ accounts for WT and V2's reactivation time courses after hyperpolarizations of various amplitudes Vh (between −53 and −193 mV) and duration th. Test voltages were +37 and +67 mV for WT and V2, respectively. All of the displayed data come from the same WT and V2 patch recordings and correspond to the following th values. WT: for Vh = −113 mV, th = 0.5, 0.7, 1, and 2 ms; for Vh = −153 mV, th = 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, and 1 ms; for Vh = −193 mV, th = 0.1, 0.2, 0.3, 0.4, and 0.6 ms. V2: for Vh = −53 mV, th = 0.2, 0.5, 1, 2, and 5 ms; for Vh = −53 mV, th = 0.1, 0.2, 0.5, 1, and 5 ms. In the simulations, for WT, a1 and a2 were each increased by 20% compared with the values in Table II, and the values for βN-1(0) and βN(0) were 520 and 280 s−1. For V2, βN was changed to 1,100 s−1. Data are from patches w448 and v162. (B) Scheme 2+2′ accounts for the delay δa in WT and V2's reactivation time course for different Vh and th. The δa values were derived from the measured and simulated currents from A. (C and D) WT's reactivation time courses were used to place constraints on the sizes of qb1 and qb2. For reactivation time courses measured after Vh = −193 mV, values for qb1 and qb2 that are two or three times as large as the values in Table II predict a reactivation delay that is too long. In D, the three superimposed lines to the left of the squares reflect δa values derived from simulated currents for the qb1 and qb2 values in Table II (the best fit) or qb1 and qb2 values that are two or three times larger. For these simulations, the values of b1(0) and b2(0) were first adjusted to achieve good fits of WT's reactivation time course for Vh = −113 mV (shown by the derived δa values in D).
	Figure 1. Patch-to-patch variabilities are apparent in WT and V2 voltage dependence of Po (A) and ionic current time courses (B). The Po-V plots were taken from two different patches each for WT and V2, and time courses were taken from seven different patches. The test voltages for the currents in B were −13 and +27 mV for WT and V2, respectively. (C) A comparison of the τa values at different test voltages from two WT and V2 patches shows that the variability in the current kinetics is well accounted for by a simple voltage shift.
	Figure 11. Scheme 2+2′ accounts for some but not all of the features of WT's and V2's equilibrium voltage dependence of channel opening and charge movement. WT and V2 plots are shown with ordinates that are either linear (A) or log transformed (B). The discrepancies in the fits are the largest for V2's linearly plotted Q-V relation and for WT's log-transformed values of Po. The model also slightly underestimates the steepness of the Q-V relation at the most hyperpolarized voltages (seen in B). For the linear plot, the values reflect mean ± SEM from one to eight experiments. The log-transformed data reflect single patch experiments (WT patches w158 and w249; V2 patches v206 and v240).
	Figure 5. Fits of Scheme 0+2′ to the equilibrium Po at depolarized voltages for WT (A) and V2 (B). For V2, we have fitted Po-V relations obtained from current measurements made in two patches (v096 and v142). For WT, we have fitted the mean Po-V relation, since its complete Po-V relation was constructed using observations that were made in more than one patch (Schoppa and Sigworth, 1998a). In fitting WT's Po-V relation, we are here only interested in the shape of the Po-V relation at depolarized voltages, but needed to add several early transitions to Scheme 0+2′ to approximate Po at lower voltages. The model used wasin which we have added one set of four subunit transitions to Scheme 0+2′. For the modified model, the charge associated with S0 ↔ S1 was set at 2.55 e0 and its midpoint voltage was −53 mV. The simulations for V2 reflect Scheme 0+2′, but the values for βN-1(0) varied slightly from those in Table I; for the two patches, βN-1(0) was 17,000 and 14,000 s−1.
	Figure 16. Scheme 3+2′ accounts for some but not all of WT's and V2's reactivation time courses. (A) Selected WT and V2 reactivation time courses from Fig. 10 were fitted to Scheme 3+2′. Scheme 3+2′ accounts for the reactivation time courses for the less negative Vh reasonably well, but fails to account for the reactivation time courses for the most negative Vh. In these simulations, for WT, the values of a1, a2, and a3 were each increased by 20% compared with the values in Table III, and the values for βN-1(0) and βN(0) were 520 s−1 and βN = 280 s−1. For V2, βN was changed to 1,100 s−1. (B) The same deviations in the fits are shown in a comparison of the δa values that were derived from the measured and simulated currents in A. These discrepancies reflect the 20% increases in qb1, qb2, and qb3, used to help achieve a sufficiently large total gating charge.
	Figure 18. Fits of Scheme 3+2′ to WT's and V2's ionic current measured after prepulses of different amplitude Vp. The test voltages used were +37 mV for WT and +107 mV for V2. (A) Scheme 3+2′ accounts well for the current time courses measured for wide range of Vp values. These simulations were made while incorporating changes to Scheme 3+2′ identical to those described for Scheme 2+2′ in Fig. 8; that is, we include the transition from the last closed state to the state CiN-1 (and also the transition CiN-1 ↔ CiN). The rates for these additional transitions are those given in the legend to Fig. 8, except that rate of CN-1 → CiN-1 for V2 (at +147 mV) was increased from 6,700 to 8,600 s−1, to account for the relatively large amplitude of the slow activation component observed in this patch recording. Also, for WT, a1, a2, and a3, were each increased by 4% compared with the values in Table III; for V2, each were increased by 10%. Data are from patches w139 and v148. (B and C) The good fits of the ionic currents are also reflected in a comparison of the normalized τa and δa parameters (symbols, lines) derived from the measured/simulated currents for different Vp. WT's and V2's experimental τa and δa values reflect the mean ± SEM from one to four experiments. The τa values derived from V2's measured/simulated currents reflect the fast time constant in fits of these currents to the sum of two exponentials.
	Figure 8. A modified version of Scheme 2+2′ accounts for a slow component in the activation time course. In this model, the channel can enter the state CiN-1 from the last closed state CN-1. A transition between CiN-1 and CiN is also allowed. The indicated rates of the added transitions are for +147 mV, where the slow component is prominent in the ionic current. The partial charges associated with these transitions were set to be identical to those associated with the parallel transitions; e.g., the charges for CN-1 ↔ CiN-1 are the same as for ON ↔ CiN. The rates for the other transitions in the model are nearly identical to the rates for Scheme 2+2′, given in Tables I and II. Rates for V2 are boxed. The one exception is that the rate d for WT had to be increased slightly (from 600 to 1,000 s−1) to account for WT's kinetics in this patch; being the slowest “forward” rate in the alternate path (at +147 mV), the rate d sets the time course of the slow component. The amplitude of the slow component is largely set by the rate of CN-1 → CiN-1, as well as the occupancy in CN-1. Interestingly, the model accounts for the fourfold larger amplitude of the slow component for V2 without a substantial change in CN-1 → CiN-1, suggesting that the difference in WT's and V2's ionic current arises from differential occupancy in CN-1. For reference, the values of the other relevant rate constants at +147 mV are (s−1), for WT: αN-1 = 540,000, βN-1 = 60, αN = 19,000, βN = 12, c = 22, d = 1,000; for V2: αN-1 = 290,000, βN-1 = 3,300, αN = 10,200, βN = 32, c = 54, d = 600.
	Figure 2. Classes of gating models for Shaker potassium channels. For each model, each of four subunits undergoes one, two, or three transitions between subunit states, designated by S0, S1, etc. In some of the models, the channel undergoes one, two, or three additional concerted transitions. The models are named by the number of subunit transitions and additional concerted transitions. For models with no concerted transitions, the channel is taken to be open after the fourth subunit has undergone the last subunit transition; this is indicated by the dashed line to the open state.
	Figure 4. Fits of Scheme 0+2′ to the single-channel closed and open dwell-time histograms at depolarized voltages for WT (A) and V2 (B). For each channel, the closed time histograms are shown on the left, and open times on the right. Solid curves reflect the predictions of Scheme 0+2′ with the values in Table I. The dashed curves on V2's histograms were computed with parameters that are modified from those in Table I in the following way: βN-1(0) was set to 4,400 s−1 and the value αN-1(0) = 300 s−1 was chosen to best fit the closed time histograms. The solid and dashed curves are not always distinguishable. All data are from the same two patches (w265 and v433), except at +107 mV (w266 and v344).
	Figure 6. The magnitude of the average, absolute charge movement per channel at negative voltages was used to test the “n4” scheme as a model of the first gating transitions. The solid curves were computed from a model assuming a number p of equivalently acting subunits, for which the charge movement q is given by\documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \begin{equation*}q=pz_{1}\frac{1}{1+e^{-z_{1}[(V-V_{1})/kT]}}.\end{equation*}\end{document} For each curve the charge z1 and midpoint voltage V1 for the transition were fixed to 0.89 and −53 mV, which are the charge and midpoint voltage values of the very first gating transition derived from the kinetic estimates of α1 and β1. The displayed charge values reflect the same data as in Fig. 1 in Schoppa and Sigworth (1998a), except scaled by the estimate of the single channel charge movement q = 12.3 e0 reported in Schoppa et al. (1992). The error bars are smaller than the symbols for most of the values.
	Figure 9. Scheme 2+2′ accounts for WT's and V2's gating currents at hyperpolarized voltages. Simulations are shown for (A) currents induced by voltage steps between −93 mV and more negative voltages (patches w249 and v240) and (B) currents induced by step hyperpolarizations from intermediate prepulse voltages (patches w217 and v417). In B, the duration of the prepulse was 2 and 20 ms for WT and V2, respectively. A short prepulse was used for WT to minimize the contribution of channels in the open state to the gating current time course.
	Figure 12. Constraints on how to add more transitions to Scheme 2+2′ were provided by the shape of V2's Q-V relation. (A) V2's mean Q-V relation has been superimposed (solid curve) with predictions of Scheme 2+3′, with the concerted transition in the third to last position having a valence of 3.5. This model steepens the Q-V relation compared with the predictions of Scheme 2+2′ (dashed curve). However, it yields a Q-V relation that is too steep at voltages near −40 mV, while remaining too shallow at hyperpolarized voltages (near −80 mV). For the simulations, the parameter estimates for all but the large valenced transition were fixed to be the same as in Scheme 2+2′; the midpoint voltage of the new transition was varied to best account for V2's Q-V relation. (B) Scheme 3+2′ performs better at steepening V2's Q-V relation across the entire voltage range. For these simulations, the parameter values for S0 ↔ S1 and S1 ↔ S2 were those given for the same transitions in Scheme 2+2′ in Table II. For S2 ↔ S3, we assigned a charge (0.88 e0) that is one-fourth of the charge (3.5 e0) assigned to the concerted transition in Scheme 2+3′ in A. Its midpoint voltage was varied to yield the best fits of the data. (C) V2's Q-V relation was also used to obtain a rough estimate of the equilibrium constant for S2 ↔ S3. Three different simulations of Scheme 3+2′ are shown for different values of a factor R (R = 3, 1, and 0.33) that reflect the ratio of the equilibrium of S2 ↔ S3 versus that of S1 ↔ S2. In these simulations, the charge associated with S2 ↔ S3 was assigned to be the same as that associated with S1 ↔ S2, given in Table II.
	Figure 13. Scheme 3+2′ accounts for WT's and V2's macroscopic ionic current time courses across a wide voltage range. Simulations are shown for currents measured at depolarized voltages (A) and at intermediate voltages (B). The holding potential was −93 mV. To allow comparison of Scheme 3+2′ with Scheme 2+2′, the simulations of activation time courses here were made while incorporating changes to Scheme 3+2′ identical to those made for Scheme 2+2′ in Fig. 8; we have added a transition from the last closed state to the state CiN-1 (and also the transition CiN-1 ↔ CiN). The rates for these additional transitions are those given in the legend to Fig. 8. All current traces for WT and V2 come from the same two WT and V2 patches (w312 and v096). (C) The good fits of the ionic currents are also reflected in a comparison of the τa and δa parameters (symbols, lines) derived from the measured/simulated currents. The τa values at high depolarized voltages (V ≥ +67 mV) reflect the fast time constant in fits of the measured and simulated currents to the sum of two exponentials.
	Figure 14. Scheme 3+2′ accounts for WT's and V2's on gating currents across a wide voltage range, between −73 and +47 mV. All current traces come from the same two WT and V2 patches (w212 and v219). Notably, at −33 mV, the model accounts for a slow component that is present in WT's current, but absent in V2's current. The holding potential was −93 mV in these recordings.
	Figure 15. Scheme 3+2′ accounts for WT and V2's gating currents at most hyperpolarized voltages. Simulations are shown for (A) currents induced by voltage steps between −93 mV and more negative voltages, and (B) currents induced by step hyperpolarizations from intermediate prepulse voltages. Data are identical to the traces in Fig. 9. For the most negative test voltage in A (−153 mV), Scheme 3+2′ predicts a current that decays too rapidly, by roughly a factor of 2. This discrepancy reflects the 20% increase in qb1 that was introduced to help achieve a sufficiently large total gating charge (see text).
	Figure 17. Scheme 3+2′ performs much better than Scheme 2+2′ at predicting certain features of WT's and V2's equilibrium voltage dependence of channel opening and charge movement, including the steepness of V2's Q-V relation and the voltage sensitivity of WT's Po at low Po. As in Fig. 11, plots are shown with ordinates that are linear (A) or log transformed (B).
	Figure 19. Fits of Scheme 3+2′ to WT's and V2's off gating currents after a range of prepulses of different amplitude Vp (A) and different duration tp (B). In B, WT's currents were measured after different prepulses to −33 mV that ranged in duration from 30 to 2 ms (bottom to top). The test voltage for all recordings was −93 mV. For WT's currents, the model accounts for the development of a slowly decaying off current with an increase in the amplitude or duration of the prepulse. The model also accounts for V2's faster off current decay kinetics. The arrow on V2's off current after a prepulse to +27 mV in A marks a small inflection in the simulated current that corresponds to a plateau phase. Data from patches w212 and v219 in A, and patch w217 in B.

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