For the fitting of the gating-current time courses, the amplitudes derived from the eigenvectors of K(_hnbsp;) and the charge movements were scaled by the number of channels. At the 5-kHz filtering bandwidth at which the gating currents were usually recorded, we expected that any charge component decaying faster than ~100 µs was likely to be distorted, given that the measured step response of the recor ding system required a few sample intervals to settle (Schoppa and Sigworth, 1998a). In the simulations, this was accounted for by constraining the rates of the expon ential relaxations: in each component with a time constant shorter than 100 µs, the time constant was fixed to 100 µs, and the amplitude of the component was appropriately adjusted to maintain the correct amount of charge.

The probability density functions fitted to the single-channel open and closed dwell time histograms were calculated using the methods that have been described previously (Colquhoun and Hawkes, 1981). For the calculations of the open times, a correction was performed for missed closed events, using d escribed methods (Crouzy and Sigworth, 1990).

For the simple characterization of activation time courses, we sometimes fitted a single exponential function to simulated time courses in the same manner as was done for the experimental data (Schoppa and Sigworth, 1998a). Briefly, an exponential function was fitted to the time course, starting at the time at which the relaxation reached 50% of its final value. The resulting time constant _a and delay _a parameters have simple interpretations in the case th at all transitions have negligible reverse rates.

Using Data Obtained from Different Patch Recordings

It has been reported that Shaker channels exhibit variabilities in gating between different patches (Zagotta et al., 1994b). Indeed, in our records, WT and V2 channels displayed patch-to-patch variabilities in several gating properties, including the voltage dependences and kinetics of channel opening (Fig. 1, A and B). F or both channels, the variabilities in equilibrium P_o and in the kinetics of channel opening (as reflected in the activation time constant _a) co rresponded to a 5-10-mV voltage shift (Fig. 1 C). The extent of the variabilities was larger than would be expected from drift in the pipette voltage offset. This offset was corrected at the beginning of each recording; it was found to change by no more than 2-3 mV by the end of the experiment, when the offset was reevaluated. One possible source of the variability in the macroscopic currents of V2 is its modal single channel behavior (Schoppa and Sigworth, 1998b). The macroscopic current time course could reflect the modal behavior if factors exist (e.g., second messengers) that shift the properties of the entire population of channels that contribute to the macroscopic current.

View larger version (25K): [in this window] [in a new window]   Fig. 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 a 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.

Since it was impossible to obtain all of the types of data from a single patch recording, the patch-to-patch variabilities implied that no single set of parameters could account for all of the data simultaneously. To account for the variabilities, we allowed the rate values (at 0 mV) to differ by as much as 20% between patches, but the charge values were the same for all the simulations. This magnitude of variation accounts for the ~10-mV voltage shifts in P_o and _a shown in Fig. 1. We considered this a satisfactory approach since our interest in the modeling was to discriminate between different fundamental mechanisms of channel gating rather than to determi ne rate constants to high precision. Most gating mechanisms could be quite easily ruled out by simple qualitative criteria or if they produced extremely poor fits of the data (e.g., the fits of Scheme 2+2 to the equilibrium data in Fig. 11); allowing small variations in rate constants between patches did not obscure our ability to differentiate models. In fact, the number of instances that the rates differed from the values given i n the appropriate tables is quite small. These are explicitly noted in the legends of Figs. 5, 10, 16, and 18.

View larger version (28K): [in this window] [in a new window]   Fig. 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).

View larger version (17K): [in this window] [in a new window]   Fig. 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 val ues for N-1(0) varied slightly from those in Table I; for the two patches, N-1(0) was 17,000 and 14,000 s1.

View larger version (31K): [in this window] [in a new window]   Fig. 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 N-1(0) and N(0) were 520 s1 and N = 280 s1. For V2, N was changed to 1,100 s1. (B) The same deviations in the fits are shown in a comparison of the a 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.

View larger version (27K): [in this window] [in a new window]   Fig. 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 s1, 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 a and a p arameters (symbols, lines) derived from the measured/simulated currents for different Vp. WT's and V2's experimental a and a values reflect the mean ± SEM from one to four experiments. The a values derived from V2's measured/simulated currents reflect the fast time constant in fits of these currents to the sum of two exponentials.

View larger version (18K): [in this window] [in a new window]   Fig. 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 s1) to account for WT's kinetics in this patch; being the slowest "forward" rate 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 (s1), for WT: N-1 = 540,000, N-1 = 60, N = 19,000, N = 12, c = 22, d = 1,000; for V 2: N-1 = 290,000, N-1 = 3,300, N = 10,200, N = 32, c = 54, d = 600.

Finding Optimal Parameter Estimates

For previously described gating models, parameter estimates have typically been found by using a search algorithm that minimizes the error between the fitted curves and the data. In our initial modeling attempts, we employed the simplex search algorithm (Nelder and Mead, 1965; Press et al., 1992) to optimize fits, but obtained disappointing results. One problem was a bias in the fitting toward current traces that were larger in magnitude, since these yielded the largest error values. A simple weighting scheme improved things somewhat, but did not solve the problem that fits to individual traces often accounted well for certain features of the time course but not for others. For example, good fits of the rising phase and the final value of an ionic time course would be obtained (since these features account for most of the data points), but the delay would be poorly represented. The delay, however, was often the more important feature of the current for constraining models, since it reflects many more rates than the rest of the current.

Some attempts were made at using an appropriate error weighting function to avoid these problems. In our experience, however, determining the appropriate function was very tedious, and it was more expedient to perform the fitting by simply setting parameters manually and determining the goodness of fit by visual inspection. Our success in deriving a set of parameters with nonautomated methods can be attributed to the availability of good initial estimates for each of the rate constants.

The complexities surrounding the weighting of the errors in the fits, as well as the presence of variabilities between different patch experiments, made it difficult to provide meaningful confidence limits for the different parameter estimates that we give. However, we emphasize that each of the parameter estimates in our model is highly constrained. As we will describe below, we are generally able to identify particular current measurements that isolate each transition, and thus tightly constrain each of the parameters in the model. A good example of how making modest changes in the parameters affects the fits is illustrated in Fig. 10, C and D.

	RESULTS

Top Abstract Introduction Methods Results Discussion References

Several tenable classes of activation gating models are illustrated in Fig. 2. Since Shaker channels have a symmetrical structure composed of four identical subunits (MacKinnon et al., 1991; Kavanaugh et al., 1992; Li et al., 1994), a credible hypothesis for a gating model is one in which many of the transitions correspond to Shaker's four subunits moving one subunit at a time, and with the subunits acting equivalently. The symmetry can also be exploited in the modeling, since it reduces the number of different parameters that have to be constrained. Having many transitions correspond to the equivalent movement of single subunits has been a feature of many of the published gating models for Shaker (Schoppa et al., 1992; McCormack et al., 1994; Zagotta et al., 1994b), and, for largely philosophical reasons, we also favor this formulation (see also below). Thus, in each of the models in Fig. 2, the channel opens after each of the four subunits undergoes at least one transition between different states that reflect the conformation of each of the individual subunits. These subunit states are designated S₀, S₁, S₂, and S₃, and we will refer to transitions among these states as "subunit transitions." Some of the models in Fig. 2 have one or two additional transitions that follow the subunit transitions. These presumably reflect the concerted action of the four subunits. The naming of each of the models follows the assigned number of subunit transitions and the number of subsequent concerted transitions. For example, in Scheme 1+2, the channels open after one set of four subunit transitions and two concerted transitions.

View larger version (21K): [in this window] [in a new window]   Fig. 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.

In each of the models, the subunit transitions occur in sequence with each other instead of independently since sequential subunit movement better accounts for the long delay in the channel opening time course. Sequential subunit transitions also better account for the presence of a rising phase in the "on" gating currents, assuming that different subunit transitions have different rates (Zagotta et al., 1994b).

All of the models in Fig. 2 have a single open state. Two pieces of evidence in favor of single open state models have been presented previously by Hoshi et al. (1994) and Zagotta et al. (1994a). The first is Shaker's single-exponential open dwell-time distributions, which are best explained by a single open state. The second is the shape of Shaker's voltage dependence of open probability P_o. The activation curve becomes increasingly steep at low P_o (see Fig. 11 B), reaching an asymptotic steepness corresponding with the channel's total charge movement (Seoh et al., 1996); this property is inconsistent with the existence of multiple open states with voltage-dependent transitions among them (Sigg and Bez anilla, 1997).

Our strategy for the modeling here will be to consider the different models in Fig. 2 systematically, starting with the most simple models and moving to more complicated models, as they are required by the data. The modeling will be divided into four stages, as summarized here.

Stage I. In the first stage, we point out a number of observations that suggest that the correct activation model is more complicated than all but three of the classes of models shown in Fig. 2.

Stage II. We formally model Scheme 2+2, which is an example of the class of models Scheme 2+2. We first use the kinetic data at the voltage extremes outlined in the previous two papers to derive starting estimates for the rates in the model, and then model these same data. We find that Scheme 2+2 accounts for the kinetic details quite well. Then, as an additional test, we model the equilibrium voltage dependence of channel opening and charge movement relations. We find that Scheme 2+2 accounts poorly for some features of the equilibrium data. The deviations in the fits suggest that the correct model must have a larger total charge movement, which can be provided only by adding more transitions to the scheme.

Stage III. We consider ways of adding more transitions to Scheme 2+2. We consider two possibilities, one that adds a single concerted transition with a large charge (Scheme 2+3), and another that adds an additional subunit transition (Scheme 3+2). Predictions of these two models for V2's Q-V relation indicate that Scheme 3+2 is a better solution. In this stage, we also refit all of the kinetic and equilibrium data considered in Stage II, to obtain a set of parameters for Scheme 3+2.

Stage IV. In the last stage of the modeling, we compare the behavior of Scheme 3+2 with other types of macroscopic current measurements, including kinetic measurements at intermediate voltages. These experiments act as independent tests for the robustness of Scheme 3+2.

Stage I: Evidence Against Simple Models

Several lines of evidence suggest that models that are more simple than Scheme 2+2 in Fig. 2 are inadequate.

Schemes 1+0, 1+1, and 1+2. Zagotta et al. (1994a) used the magnitude of the delay in the activation time course to derive a minimum estimate of the total number of gating transitions. By fitting the current to a sequential model with equal forward rates, these authors estimated that the Shaker channel undergoes a minimum of five transitions. We performed a similar analysis on Shaker's ionic current time courses, but with currents measured across a broader test voltage rang e (between 13 and +147 mV) and, also while using a more negative holding potential (133 mV). More negative holding potentials load channels into the earliest closed states, and thus provide a more reliable estimate of the total number of transitions. Fits of the currents at +27 mV to a sequential model with equal forward rates yielded an average minimum estimate of seven transitions (the values in three patches were six, seven, and eight). Seven transitions were also required to account for the ionic currents at +67 and +147 mV (one patch each). This lower bound of seven transitions rules out all models in the classes of Schemes 1+0, 1+1, and 1+2, which have no more than six transitions.

Scheme 2+0. This model has an activating channel undergo eight transitions. However, evidence against Scheme 2+0 is provided by the estimates of the voltage dependences of different rates (Schoppa and Sigworth, 1998a), which indicate that there are at least three types of transitions that can be differentiated by the magnitudes of their associated charges. Scheme 2+0, however, has only two types of transitions. An assumption in this argument is that all four of a given type of subunit transition (for example, S₁ S₂ in Scheme 2+0) have equivalent charge movements. While we cannot rule out models that invoke "symmetry breaking" in the movement of charge, we prefer models that do not require this added complexity.

Scheme 3+0. Evidence against Scheme 3+0 is provided by three observations made in the first paper (Schoppa and Sigworth, 1998a) that suggest that the final gating transition is qualitatively different from earlier transitions, including the second to last transiti on.

1

1

1

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Schemes 2+1 and 3+1. Finally, there is one argument against the models Schemes 2+1 and 3+1, which each have a single concerted transition. WT's "off" gating currents after large depolarizations show a rising phase and a slow decay (Bezanilla et al., 1991; Zagotta et al., 1994a). These features imply that the final two transitions that determine the deactivation kinetics (Schopp a and Sigworth, 1998a) have slower reverse rates than earlier transitions. Because the magnitudes of these rates differ from those of earlier transitions, we suggest that the final two transitions represent qualitatively different transitions. The simplest model producing this variety of transition types has the last two transitions be concerted ones. Models with only one concerted transition would require that the movement of one of the four subunits (S₁ S₂ in Scheme 2+1, for example) be much slower than the others. The model of Zagotta et al. (1994b) includes this sort of symmetry breaking to describe the slow reverse rate of the final transition in their model, which is otherwise like Scheme 2+0. However, in the absence of data that indicate that there is symmetry breaking in the rates of the final two transitions, we favor a more simple interpretation of these slow reverse rates.

View larger version (7K): [in this window]

Stage II. Modeling Scheme 2+2

Scheme 2+2 has quite a large number of different parameters that must be constrained (16 for the transitions in the activation path, 28 in total). To expedite the modeling, we consider separately data that reflect different subsets of transitions, following an approach similar to that of Vandenburg and Bezanilla (1991) in modeling activation gating for the squid sodium channel. In A, we model data that reflect transitions near the open state. In B, we model data that reflect earlier transitions, while fixing the parameter estimates of the transitions near the open state to those obtained in A.

(A) Modeling kinetic data that reflect transitions near the open state. We first consider a simplified model (Scheme 0+2)

View larger version (9K): [in this window]

N-1

N-1

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View larger version (38K): [in this window] [in a new window]   Fig. 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 ar e 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: N-1(0) was set to 4,400 s1 an d the value N-1(0) = 300 s1 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).

1

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(B) Modeling kinetic data that reflect S₀  S₁ and S₁ &n bsp;S₂. We next model kinetic data that reflect the subunit transitions S₀  S₁ and S₁  S₂ in Scheme 2+2 . We make use of the complete model while initially constraining the values of the rates of the final transitions to be the same as those just assigned above. In these simulations, the forward and backward rates of each subunit undergoing one of the two subunit transitions are each scaled by a statistical factor that reflects the number of available subunits. For example, for the very first subunit undergoing S₀  S₁, the forward rate is a₁ multiplied by four, and the backward rate is just b₁. Thus, our initial assumption in the modeling is that the four Shaker subunits gate independently of each other, since this is the simplest model. Subunit-subunit interactions are included later as required by the data.

Derivation of starting estimates of the rates for S₀  S₁ and S₁ S₂ We use as starting estimates for the rates a₁ and b₁ of the first subunit transition S₀  S₁ the assigned values ₁ and ₁ for the very first transition in a sequential model, as derived in the previous two papers; that is, we make the very first transition be the first of four S₀  S₁ transitions. This formulation is the simplest, but it is also favored by an apparent paradox that arises when one compares the kinetic description of the first transition with equilibrium data that correspond to the earliest transitions. The ₁ and ₁ estimates derived from ionic and gating current time courses give a charge estimate of z₁ = 0.9 e₀, and a midpoint voltage (V_1/2) for the first transition of 53 mV. However, from the equilibrium q-V relation for WT (Fig. 6), the first 0.9 e₀ of charge that moves in activation occurs at much more negative voltages, below 80 mV. A general way that a transition with a given midpoint voltage can contribute to charge movement at more negative voltages is if the first transition is one of several like transitions with similar midpoint voltages; the charge movement at the most negative voltages would then reflect the sum of the charge associated with several transitions. Fig. 6 illustrates simulations in which we assume that the first transition is one of four S₀  S₁ transitions, and have fixed the charge and the midpoint voltage of S₀  S₁ to that derived from the kinetics. The curve for four transitions (p = 4) accounts quite well for the magnitude of the charge movement at voltages below 90 mV. Models with fewer than four subunits yield charge magnitudes that are too small.

View larger version (18K): [in this window] [in a new window]   Fig. 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 For each curve t he 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 1 and 1. 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.