Schoppa NE , Sigworth FJ .

NS-21501 NINDS NIH HHS , R01 NS021501 NINDS NIH HHS

	Figure 8. Estimate of βN-1. (A) Separation of the reactivation time course into fast and slow components. Selected current traces in Fig. 5 A are shown for th between 100 μs and 1 ms. The time course of the fast component, reflecting channels reopening from CN-1, and the slow component, reflecting channels reopening from all closed states that precede CN-1, were approximated by fits of Eq. 9 (smooth curves). In the fitting, the value for αN was fixed to the mean value derived from the reactivation time courses in Fig. 5 D. From the slow component, we derived an estimate of the delay δa (shown for th = 1 ms; dashed curve). (B) The occupancies in CN-1 derived from the reactivation time courses, as well as tail currents (C), were fitted to Scheme SIII (superimposed smooth curves) to obtain estimates of βN-1 and βN. The occupancy pN-1 in CN-1 for different amplitude Vh and duration th was obtained from fits of Eq. 9 to the reactivation time course as pN−1 = Af/ (Iinst + Af + As); that is, taking Af to reflect the amplitude of current due to the return of channels from CN-1 to the open state. This expression for pN-1 is approximate, but is expected to hold in this case. Each of the data points reflects one to four experiments. The parameter estimates obtained were: βN(0) = 150 s−1, qαN = −0.57 e0, and βN-1 (0) = 320 s−1, qβN-1 = −0.30 e0. In the fitting, αN was fixed to mean estimate value for αN (Fig. 5 D); all channels were assumed to reside in the open state at the beginning of the test pulse.
	Figure 5. Estimate of αN from a fast reactivation component. (A, top) WT's macroscopic ionic currents elicited by a triple-pulse stimulus, using a pair of depolarizations to +47 mV separated by a voltage step to a hyperpolarized voltage Vh = −153 mV. The displayed currents correspond to different hyperpolarization durations th between 70 and 1,000 μs. Tail currents during the second pulse were inward since the pipette solution contained 14 mM K+. Data were filtered at 15 kHz. Patch w448. (bottom) The same traces are shown, but expanded and time shifted to align the start of the test pulses. The “upper half” of the reactivating current relaxations for th = 70 μs and 1 ms have been fitted to a single exponential to estimate τa (smooth curves). (B) The dependence of the derived values for τa on the hyperpolarization duration th, shown for different hyperpolarization amplitudes Vh. Note that for each Vh, the fastest τa values appear to approach 100 μs. All data reflect the patch in A. (C) To demonstrate that the fast reactivating component is not a recording artifact, we compare the apparent fast reactivating current in A for th = 100 μs with two different simulated current traces reflecting Eq. 7. The solid and dashed smooth curves, respectively, correspond to x(t) being a single exponential with a fast time constant τ = 100 μs (the good fit) or a slower τ = 430 μs (the poor fit). The total amplitude (−39 pA) associated with x(t) was constrained by fitting the tail current at −153 mV in the same patch to the sum of two exponentials and determining the amount of decrease in current by 100 μs after the beginning of the pulse. The impulse response h(i) was determined by differentiating the step response measured at the 15-kHz bandwidth (see methods). (D) The fast reactivation time constant (τf) values at different voltages were fitted to an exponential (solid curves) to estimate the voltage dependence of αN, yielding estimates for αN(0) and qαN for each of three patches (αN(0) = 7,600, 6,800, and 6,500 s−1 and qαN = 0.19, 0.16, and 0.18 e0). For each patch, τf was obtained from currents measured after a 150-μs hyperpolarization to −153 mV. The reactivation time course for th = 150 μs includes a small slow component (accounting for, on average, 14% of the time course) as well as the dominant fast component; the τf values reflect the fast time constant in fits of the reactivation time course to the sum of two exponentials:\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*}I(t)=A_{f}+A_{s}-(A_{f}e^{-t/{\tau}_{f}}+A_{s}e^{-t/{\tau}_{s}}).\end{equation*}\end{document}
	Figure 6. Estimates of β1 and βd. (A) Gating currents elicited by voltage steps between −93 and −153 mV. The current recorded at −93 mV (top) reflects the on current measured after a voltage step from −133 mV; the currents recorded at the −153-, −133-, and −113-mV test voltages reflect the off current measured after voltage steps from −93 mV. Currents were fitted to a single exponential (smooth curves) to estimate a decay time constant τ. Patch w249. (B) To estimate β1, the τ values derived from the currents in A were fitted to an exponential (solid curve), yielding β1(0) = 200 s−1 and qβ1 = −0.48 e0. (C) The off gating current measured at −93 mV after a 2-ms depolarization to −33 mV was fitted to a single exponential with τ = 0.82 ms. This time constant is similar to the reciprocal of the value of β1 at −93 mV (0.77 ms) estimated in B, consistent with βd being similar to β1 at −93 mV.
	Scheme I.
	Figure 1. Voltage dependence of relative channel open probability Po and charge movement Q. Relative Po estimates (▪) at V ≤ +67 mV were derived from measurements of ionic currents made using a double pulse protocol, in which currents were measured at a fixed amplitude voltage pulse (at −13 or +7 mV) that followed different test pulses. The values at +107 and +147 mV were obtained by measuring the magnitude of the current relaxation elicited by voltage jumps from +67 mV (in experiments similar to those illustrated in Fig. 12 A). Each plotted value reflects one to eight experiments. Relative Q estimates (○) were obtained as described previously (Schoppa et al., 1992), by numerically integrating the on gating current at each voltage. Each value reflects three to five experiments. The vertical dashed lines at −20 and −90 mV mark the boundaries of the defined hyperpolarized (V ≤ 90 mV), activation (−90 mV < V < −20 mV), and depolarized (V ≥ −20 mV) voltage ranges.
	Figure 12. Characterization of the transitions to Cf1 and Cf2. (A, top) Macroscopic ionic currents elicited by voltage steps between pairs of depolarized voltages. The smaller current (at both the prepulse and test voltages) reflects a voltage step from +7 to +127 mV, and the larger current reflects a voltage step from +47 to +147 mV. Data were filtered at 15 kHz. Patch w447. (bottom) The same traces are expanded to show just the current relaxation during the test pulse; zero time is the start of the test pulse. The current relaxations are fitted to a single exponential to estimate the time constant τr (solid curves). At +127 and +147 mV, the time constants are τr = 230 and 180 μs, respectively, and the amplitudes of the fitted exponentials are −32 and −21 pA. These amplitudes can be compared with the size of the rest of the test current (278 and 323 pA, respectively) to yield estimates of the change in Po induced by these voltage steps (10 and 6%). (B) The rate f1 from Cf1 to the open state was estimated by fitting an exponential (solid curve) to the τr values (▪) taken from the current relaxations measured in the experiment in A and from similar measurements made in one other patch. (C) Estimates of absolute Po at voltages between +47 and +147 mV were derived from the mean measured channel closed and open dwell times in the equilibrium single channel activity. The data points reflect measurements made in four patches. Absolute Po apparently saturates near 0.9.
	Figure 2. Fits of a single exponential function and delay (Eq. 4) to the time course of open probability Po predicted by different models. (A) Fits to the time course of Po (solid curve) from the n4 scheme\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*}C^{4}_{0}{\rightarrow}C^{3}_{1}{\rightarrow}c^{2}_{2}{\rightarrow}C^{1}_{3}{\rightarrow}O_{4}.\end{equation*}\end{document} Note that the order of rate constants in this scheme does not influence the time course. A fit of Eq. 4, taking only points where Po ≥ 0.5, is shown as the dotted curve; it yields τ−1 = 0.89 and δ = 1.09. The values expected from the approximate theory, τ−1 = 1 and δ = 1.083, yield the dashed curve. (B) Fits to the time course from the same scheme but with all four rate constants equal to 1. The fit (dotted curve, fitted for Po ≥ 0.5) yielded τ−1 = 0.53 and δ = 2.45; the expected values are τ−1 = 1 and δ = 3 (dashed curve). (C) Fits to the time course of Po from Scheme SII, in which each of four independent subunits undergoes two transitions, with the forward rate constants a1 and a2 equal to 1 and 4, respectively. The fit (dotted curve, fitted for Po ≥ 0.5) yielded τ−1 = 0.89 and δ = 1.38. The expected values are τ−1 = 1 and δ = 1.37 (dashed curve); these were obtained from the mean latency to opening tl that was computed by use of a recursive subroutine as the expectation value over all possible paths in Scheme SII of the sum of dwell times in states in each path. (D) Fits to the time course from Scheme SII with a1 and a2 both equal to 1. The fit (dotted curve) yielded τ−1 = 0.71 and δ = 2.38; the expected values are τ−1 = 1 and δ = 2.55 (dashed curve).
	Scheme II.
	Figure 3. Estimate of α1. (A) Macroscopic ionic currents at −13 and +37 mV were fitted to a single exponential (smooth curves) to estimate an activation time constant τa. Extrapolating the fitted exponential to zero current yielded an estimate of the activation delay δa. Patch w312. (B) The decay of WT's on gating currents at the same voltages was fitted to a single exponential (smooth curves) to estimate the decay time constant τon. Patch w212. For the fitting, a baseline was first calculated from the mean current measured at the end of the voltage pulse (horizontal lines); the fitting began at the time point at which the current had decayed by 20% from the peak value. (C) The values of τa (▪) and τon (○) derived from the fitting have nearly equal values at voltages between −13 and +67 mV. Each data point reflects the average from two to eight experiments. Superimposed curve reflects the average voltage dependence of α1, obtained by fitting the τa values between −13 and +67 mV in seven different patches. (D) Exponential fits of the activation time courses and on gating currents for a five-state sequential scheme, as depicted in the legend for Fig. 2 A. For these simulations, the rates for all but one of the transitions was set to be 4; the slowest transition had a rate constant equal to 1. The position of the slowest transition was varied to yield the different curves. The activation time courses (top) were identical for each of the conditions; these were fitted to an exponential (dotted curve), yielding an activation time constant τa = 1.04. The gating current time courses for each of the conditions (bottom), however, differed. For C1 → C2, C2 → C3 = 1, the decay of the currents (dashed curves) was not well described by a single exponential. For C3 → C4 = 1, the decay of the current (solid curve) was well described by a single exponential (dotted curve), but the fitted time constant τon = 0.69 was faster than τa. For C0 → C1 = 1, the decay time constant τon = 1.09 was similar to τa. (E) Exponential fits of the activation time courses and on gating currents for Scheme SII, with the forward rate constants a1 and a2 equal to 1 and 4, respectively, or with a1 = 4 and a2 = 1. The two cases yielded identical activation time courses (top), which were fitted to an exponential with τa = 1.12 (dotted curve). The two cases, however, yielded different gating currents (bottom). The decay of the current for the case of a1 = 1 and a2 = 4 (solid curve) was well described by a single exponential with a time constant τon = 1.05, similar to τa. The case of a1 = 4 and a2 = 1, however, yielded a more complex gating current decay time course, which, when approximated by a single exponential (dotted curve), yielded a decay time constant τon = 0.51 that was much faster than τa. In all simulations, each transition carried an equivalent charge movement.
	Figure 4. Estimate of αp. (A) The upper half of the macroscopic ionic current time course at +87 and +147 mV was fitted to one or two exponentials (Eq. 6), respectively, to estimate τa and δa (smooth curves). Two exponentials were required at +147 mV to account for a slow relaxation that corresponds to an alternate activation path. Patch w312. (B) Voltage dependence of τa and δa taken from current measurements in two different patches (left and right). The τa values display shallower voltage sensitivities at high voltages. The voltage dependence of the rate αp was estimated from the τa values at V ≥ +87 mV (solid lines at V ≥ +87 mV). The δa values between −13 and +147 mV also become less voltage sensitive at high voltages. Estimates of the charge q that determines the voltage sensitivity of the delay at low and high depolarized voltages, respectively, were obtained by fitting an exponential to the δa values at V ≤ +67 and ≥ +87 mV (dashed lines). These charge values will be used in a following paper (Schoppa and Sigworth, 1998b).
	Scheme III.
	Figure 9. Estimate of qβd. (A) Values of δa taken from the slow component in the reactivation time courses (shown in Fig. 8 A), for hyperpolarizations of different amplitude Vh and duration th. For each Vh, the time course of the accumulation of the delay for increasing th was approximated by a fit of the δa values to a single exponential (solid curves), yielding τacc. The fitted exponential was constrained to begin at zero. The amplitude was fixed to have the value of δa measured from the current elicited by the first of the three voltage pulses in the triple pulse protocol (starting from a −93-mV holding potential). For Vh = −113, −153, and −193 mV, the amplitude was multiplied by a scaling factor to account for the fact that prepulses to voltages more negative than −93 mV yield a slightly longer delay in the channel opening time course. All data except for Vh = −93 mV come from the same patch (w448). (B) Values of τacc for different Vh were fitted to an exponential (solid curve) to estimate a charge qβd = −0.24 e0. Values reflect pooled measurements made in four different patches.
	Figure 10. Single channel activity at depolarized voltages. (A) Traces of single channel currents elicited by voltage pulses from −93 mV to different test voltages. The beginning of the voltage pulse is indicated by the vertical dashed line. Patches w265 and w276. (B) Closed dwell-time histograms from depolarizations in 40-mV increments between −13 and +147 mV. The histograms were fitted to the sum of three exponentials; dashed curves correspond to each of the fitted exponential components. The number of events in each histogram is ≥605. (C) The open dwell-time histograms at the same voltages were well fitted by a single exponential. (D) The time constants (τ1, τ2, and τ3) of the fast, intermediate, and slow exponential components of the closed-time histograms have nearly voltage-independent values near 0.1, 0.3, and 2 ms. The τ3 values were fitted to an exponential (solid line) to estimate the rate d from CiN to the open state. Data points reflect pooled results from several patches. Time constant values from the fits of the histograms shown in B are indicated by the filled symbols. (E) The time constants τ0 of the single exponentials fitted to the open times display little voltage dependence. Also, the large discrepancy between τ0 and the reciprocal of the channel closing rate βN (solid line) indicates that nearly all of the closures at depolarized voltages are to states that are not in the activation path. The τ0 values from the fits of the histograms shown in C are indicated by the filled symbols. (F) An estimate for the rate c from the open state to CiN was derived by fitting an exponential (solid curve) to estimates of the frequency of Ci closures, corresponding to the τ3 component in the closed dwell-time distributions. Frequency estimates were obtained from the measured channel open times normalized by the relative contribution of the Ci closures to the total number of measured closures. Values obtained from the histograms in B and C are indicated by the filled symbols. For the closed dwell-time histograms shown in B, the relative amplitudes A1, A2, and A3 of the fast, intermediate, and slow exponential components are as follows: at −13 mV, 0.74, 0.24, and 0.02; at +27 mV, 0.83, 0.16, and 0.01; at +67 mV, 0.79, 0.20, 0.02; at +107 mV, 0.83, 0.16, and 0.01; and at +147 mV, 0.85, 0.14, and 0.01.
	Figure 11. Ci states can be entered from closed states in the activation path. (A) The upper half of the cumulative first latency histogram at +67 mV was fitted to the sum of two exponentials (Eq. 6; solid curve), yielding the indicated τa, τs, and relative As values. The dashed curve reflects just the fast component in Eq. 6. The difference in the amplitude of the two curves reflects As. Patch w265. (B) The time constants τs obtained in the fits of Eq. 6 to first latency distributions (▪) and macroscopic ionic currents (○) at V ≥ +67 mV have values of 1–3 ms. Only two of the single channel experiments had enough traces that a slow component could be convincingly discerned in the latency histogram. However, 21 of 28 relatively smooth macroscopic ionic current time courses at V ≥ +67 mV (measured in seven patches) displayed an unambiguous slow component. The τs values were fitted to single exponential (solid curve). The derived charge estimate q = −0.14 e0 indicates that the time constant of the slow component is nearly voltage independent. (C) The relative magnitude of the slow exponential component As/(Af + As) is also nearly voltage independent. These values were fitted to an exponential, yielding q = 0.07 e0.
	Scheme IV.
	Scheme V.
	Scheme VI.

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