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# Activation of shaker potassium channels. I. Characterization of voltage-dependent transitions.

### Author information

- 1
- Department of Cellular and Molecular Physiology, Yale University School of Medicine, New Haven, Connecticut 06520, USA.

### Abstract

_{N}. (

*A*,

*top*) Inward macroscopic ionic currents elicited by a fixed prepulse to +7 mV followed by test pulses to various hyperpolarizations between −93 and −193 mV. Filtering at 15 kHz. Patch w448. (

*bottom*) A single exponential (

*dashed curves*) poorly accounts for the decay of the tail currents at −153 and −193 mV. (

*B*) An estimate of β

_{N}was obtained from the faster time constant τ

_{f}in fits of the tail currents at

*V*≤ −153 mV to the sum of two exponentials (Eq. ). Values reflect five different experiments, and estimates for β

_{N}(0) and

*q*

_{βN}were derived from each of the patches (

*solid lines*). These estimates are similar to β

_{N}(

*dashed line*) derived from fits of Scheme to the tail current and C

_{N-1 }occupancies in Fig. . (

*C*) Voltage dependence of the relative amplitude of the τ

_{f}component in fits of the tail currents in the same five patches to Eq. . The superimposed dashed curve reflects the amplitude

*A*

_{f(III) }of the faster of two tail current relaxation components predicted by Scheme , assuming values for β

_{N}and β

_{N-1 }derived from the fits in Fig. . For α

_{N–1 }= 0, the amplitude is given by Here, λ

_{f}is the faster relaxation eigenvalue that approaches −β

_{N}at very negative voltages.

_{N-1}. (

*A*) Separation of the reactivation time course into fast and slow components. Selected current traces in Fig.

*A*are shown for

*t*

_{h}between 100 μs and 1 ms. The time course of the fast component, reflecting channels reopening from C

_{N-1}, and the slow component, reflecting channels reopening from all closed states that precede C

_{N-1}, were approximated by fits of Eq. (

*smooth curves*). In the fitting, the value for α

_{N}was fixed to the mean value derived from the reactivation time courses in Fig.

*D*. From the slow component, we derived an estimate of the delay δ

_{a}(shown for

*t*

_{h}= 1 ms;

*dashed curve*). (

*B*) The occupancies in C

_{N-1}derived from the reactivation time courses, as well as tail currents (

*C*), were fitted to Scheme (

*superimposed smooth curves*) to obtain estimates of β

_{N-1}and β

_{N}. The occupancy

*p*

_{N-1}in C

_{N-1 }for different amplitude

*V*

_{h}and duration

*t*

_{h}was obtained from fits of Eq. to the reactivation time course as

*p*

_{N−1}=

*A*

_{f}/ (

*I*

_{inst}+

*A*

_{f}+

*A*

_{s}); that is, taking

*A*

_{f}to reflect the amplitude of current due to the return of channels from C

_{N-1}to the open state. This expression for

*p*

_{N-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: β

_{N}(0) = 150 s

^{−1},

*q*

_{αN }= −0.57 e

_{0}, and β

_{N-1 }(0) = 320 s

^{−1},

*q*

_{βN-1 }= −0.30 e

_{0}. In the fitting, α

_{N}was fixed to mean estimate value for α

_{N}(Fig.

*D*); all channels were assumed to reside in the open state at the beginning of the test pulse.

_{N}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

*V*

_{h }= −153 mV. The displayed currents correspond to different hyperpolarization durations

*t*

_{h }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

*t*

_{h}= 70 μs and 1 ms have been fitted to a single exponential to estimate τ

_{a}(

*smooth curves*). (

*B*) The dependence of the derived values for τ

_{a }on the hyperpolarization duration

*t*

_{h}, shown for different hyperpolarization amplitudes

*V*

_{h}. Note that for each

*V*

_{h}, the fastest τ

_{a}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

*t*

_{h }= 100 μs with two different simulated current traces reflecting Eq. . 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 (τ

_{f}) values at different voltages were fitted to an exponential (

*solid curves*) to estimate the voltage dependence of α

_{N}, yielding estimates for α

_{N}(0) and

*q*

_{αN}for each of three patches (α

_{N}(0) = 7,600, 6,800, and 6,500 s

^{−1}and

*q*

_{αN}= 0.19, 0.16, and 0.18 e

_{0}). For each patch, τ

_{f }was obtained from currents measured after a 150-μs hyperpolarization to −153 mV. The reactivation time course for

*t*

_{h}= 150 μs includes a small slow component (accounting for, on average, 14% of the time course) as well as the dominant fast component; the τ

_{f}values reflect the fast time constant in fits of the reactivation time course to the sum of two exponentials:

_{1}and β

_{d}. (

*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 β

_{1}, the τ values derived from the currents in

*A*were fitted to an exponential (

*solid curve*), yielding β

_{1}(0) = 200 s

^{−1}and

*q*

_{β1}= −0.48 e

_{0}. (

*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 β

_{1}at −93 mV (0.77 ms) estimated in

*B*, consistent with β

_{d}being similar to β

_{1}at −93 mV.

_{i}→ 0 for all i, the mean latency to arrive at the open state

*O*

_{N}is given by 3

*P*

_{o}and charge movement

*Q*. Relative

*P*

_{o}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.

*A*). Each plotted value reflects one to eight experiments. Relative

*Q*estimates (○) were obtained as described previously (), 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.

_{f1 }and C

_{f2}. (

*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 τ

_{r}(

*solid curves*). At +127 and +147 mV, the time constants are τ

_{r }= 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

*P*

_{o}induced by these voltage steps (10 and 6%). (

*B*) The rate

*f*

_{1}from C

_{f1 }to the open state was estimated by fitting an exponential (

*solid curve*) to the τ

_{r }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

*P*

_{o}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

*P*

_{o}apparently saturates near 0.9.

*P*

_{o}predicted by different models. (

*A*) Fits to the time course of

*P*

_{o}(

*solid curve*) from the

*n*

^{4}scheme Note that the order of rate constants in this scheme does not influence the time course. A fit of Eq. , taking only points where

*P*

_{o}≥ 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

*P*

_{o}≥ 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

*P*

_{o}from Scheme , in which each of four independent subunits undergoes two transitions, with the forward rate constants

*a*

_{1}and

*a*

_{2}equal to 1 and 4, respectively. The fit (

*dotted curve*, fitted for

*P*

_{o}≥ 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

*t*

_{l}that was computed by use of a recursive subroutine as the expectation value over all possible paths in Scheme of the sum of dwell times in states in each path. (

*D*) Fits to the time course from Scheme with

*a*

_{1}and

*a*

_{2}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*).

*Shaker*'s gating process in this paper is reasonable comes from comparing the first-pass estimates of various rates obtained here (in Tables and ) with the results of the modeling in the third paper in this series (). There, we consider a number of different branched models that are similar to Scheme . We derive starting rate estimates for the transitions in the models from each of the rate estimates obtained here; the rate values that then yield the best fits of the data are quite similar to these starting estimates, being at most a factor of 2–3 different.

_{1}. (

*A*) Macroscopic ionic currents at −13 and +37 mV were fitted to a single exponential (

*smooth curves*) to estimate an activation time constant τ

_{a}. Extrapolating the fitted exponential to zero current yielded an estimate of the activation delay δ

_{a}. 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 τ

_{on}

*.*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 τ

_{a}(▪) and τ

_{on }(○) 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 α

_{1}, obtained by fitting the τ

_{a}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.

*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 τ

_{a}= 1.04. The gating current time courses for each of the conditions (

*bottom*), however, differed. For C

_{1}→ C

_{2}, C

_{2}→ C

_{3}= 1, the decay of the currents (

*dashed curves*) was not well described by a single exponential. For C

_{3}→ C

_{4}= 1, the decay of the current (

*solid curve*) was well described by a single exponential (

*dotted curve*), but the fitted time constant τ

_{on}= 0.69 was faster than τ

_{a}. For C

_{0}→ C

_{1 }= 1, the decay time constant τ

_{on}= 1.09 was similar to τ

_{a}

*.*(

*E*) Exponential fits of the activation time courses and on gating currents for Scheme , with the forward rate constants

*a*

_{1}and

*a*

_{2}equal to 1 and 4, respectively, or with

*a*

_{1}= 4 and

*a*

_{2}= 1. The two cases yielded identical activation time courses (

*top*), which were fitted to an exponential with τ

_{a}= 1.12 (

*dotted curve*). The two cases, however, yielded different gating currents (

*bottom*). The decay of the current for the case of

*a*

_{1}= 1 and

*a*

_{2}= 4 (

*solid curve*) was well described by a single exponential with a time constant τ

_{on}= 1.05, similar to τ

_{a}. The case of

*a*

_{1}= 4 and

*a*

_{2}= 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 τ

_{on}= 0.51 that was much faster than τ

_{a}. In all simulations, each transition carried an equivalent charge movement.

_{p}. (

*A*) The upper half of the macroscopic ionic current time course at +87 and +147 mV was fitted to one or two exponentials (Eq. ), respectively, to estimate τ

_{a}and δ

_{a}(

*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 τ

_{a}and δ

_{a}taken from current measurements in two different patches (

*left*and

*right*). The τ

_{a}values display shallower voltage sensitivities at high voltages. The voltage dependence of the rate α

_{p}was estimated from the τ

_{a}values at

*V*≥ +87 mV (

*solid lines*at

*V*≥ +87 mV). The δ

_{a}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 δ

_{a}values at

*V*≤ +67 and ≥ +87 mV (

*dashed lines*). These charge values will be used in a following paper ().

^{+}(Fig.

*A*). One criterion to determine that the tail current time course entirely reflects the channel closing rate β

_{N}is that it should be well fitted by a single exponential, but even at the most negative voltages, the tail currents were poorly fitted by a single exponential. Assuming that the complicated channel deactivation time course reflects channel reopenings from the last closed state in the activation path, as reflected in Scheme , estimates of β

_{N}could nevertheless be obtained by fitting the tail currents to the sum of two exponentials: 8

*q*

_{βd}. (

*A*) Values of δ

_{a}taken from the slow component in the reactivation time courses (shown in Fig.

*A*), for hyperpolarizations of different amplitude

*V*

_{h}and duration

*t*

_{h}. For each

*V*

_{h}, the time course of the accumulation of the delay for increasing

*t*

_{h}was approximated by a fit of the δ

_{a}values to a single exponential (

*solid curves*), yielding τ

_{acc}. The fitted exponential was constrained to begin at zero. The amplitude was fixed to have the value of δ

_{a}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

*V*

_{h}= −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

*V*

_{h}= −93 mV come from the same patch (w448). (

*B*) Values of τ

_{acc}for different

*V*

_{h}were fitted to an exponential (

*solid curve*) to estimate a charge

*q*

_{βd }= −0.24 e

_{0}. Values reflect pooled measurements made in four different patches.

*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 (τ

_{1}, τ

_{2}, and τ

_{3}) 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 τ

_{3}values were fitted to an exponential (

*solid line*) to estimate the rate

*d*from C

_{iN}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 τ

_{0}of the single exponentials fitted to the open times display little voltage dependence. Also, the large discrepancy between τ

_{0}and the reciprocal of the channel closing rate β

_{N}(

*solid line*) indicates that nearly all of the closures at depolarized voltages are to states that are not in the activation path. The τ

_{0}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 C

_{iN }was derived by fitting an exponential (

*solid curve*) to estimates of the frequency of C

_{i}closures, corresponding to the τ

_{3}component in the closed dwell-time distributions. Frequency estimates were obtained from the measured channel open times normalized by the relative contribution of the C

_{i}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

*A*

_{1},

*A*

_{2}, and

*A*

_{3 }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.

_{i}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. ;

*solid curve*), yielding the indicated τ

_{a}, τ

_{s}, and relative

*A*

_{s}values. The dashed curve reflects just the fast component in Eq. . The difference in the amplitude of the two curves reflects

*A*

_{s}. Patch w265. (

*B*) The time constants τ

_{s}obtained in the fits of Eq. 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 τ

_{s}values were fitted to single exponential (

*solid curve*). The derived charge estimate

*q*= −0.14 e

_{0}indicates that the time constant of the slow component is nearly voltage independent. (

*C*) The relative magnitude of the slow exponential component

*A*

_{s}/(

*A*

_{f }+

*A*

_{s}) is also nearly voltage independent. These values were fitted to an exponential, yielding

*q*= 0.07 e

_{0}.

_{0 }into C

_{N-1}, and then into C

_{iN-1}and C

_{iN}, before reaching the open state O

_{N}. From which particular set of closed states the channel can enter the C

_{i}states remains unclear. Scheme does not include transitions between C

_{iN}and C

_{f}, because we prefer a simple model for these states.

_{N}close first into C

_{f2}and then into C

_{f1}), but we choose Scheme because of its simplicity.

*q*

_{αd},

*q*

_{βd}, and β

_{d}(0) that we ascribe to intermediate transitions should be taken as quite rough approximations; these are shown in Scheme with parentheses. The charge parameters

*q*

_{αd}and

*q*

_{βd}were derived from the delay in the channel opening time course, which reflects the composite properties of many transitions. These parameters are also somewhat difficult to interpret because transitions with the fastest forward rates contribute little to the delay. Thus, these charge parameters are not likely to reflect the partial charges associated with the most rapid transitions. However, the number of rates with partial charges similar to

*q*

_{αd}and

*q*

_{βd}is likely to be quite large. Zagotta et al. (

*b*) estimated that a minimum of five transitions are required to account for the delay in the ionic current time course; our own analysis gives a minimum estimate of seven transitions (Schoppa et al., 1998

*b*), implying that at least six transitions contribute significantly to the delay.

### Publication type, MeSH terms, Substance, Grant support

#### Publication type

#### MeSH terms

- Animals
- Biotransformation/genetics
- Biotransformation/physiology
- Electrophysiology
- Ion Channel Gating/genetics
- Ion Channel Gating/physiology
- Kinetics
- Membrane Potentials/physiology
- Mice
- Mice, Neurologic Mutants
- Mutation
- Oocytes/metabolism
- Patch-Clamp Techniques
- Potassium Channels/genetics
- Potassium Channels/metabolism*
- Xenopus laevis

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