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Figure 1. . (A) Effect of linear subtraction (LS). 5aa currents during depolarizations from −90 mV to 30 and 90 mV with (black) and without (gray) linear subtraction are compared as described in the text. The faster currents in each panel were recorded with stretch (using −45 mm Hg), the slower currents before and after stretch. Channel currents and capacitive transients are well separated. The traces demonstrate that nonleaky patches were used and that 5aa, with its slow activation, was an ideal mutant for the kinetic analysis undertaken here. Thus, although LS was used routinely (see materials and methods), it was not critical for obtaining accurate 5aa channel current amplitudes or maximum rates of rise. (B) Maximum slope line analysis. Maximum slope lines fitted to 30 mV leak-subtracted currents with/without stretch from A. The fit quantifies characteristic properties of current activation: SMax − maximum slope, and td − delay.
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Figure 2. . The effect of stretch on 5aa currents elicited by a depolarizing step from −90 mV. Gray, current with stretch (−45 mm Hg suction); black, current without stretch (before and after). 20 μM Gd3+ in pipette. The most dramatic stretch effect for Shaker channels, including 5aa, occurs at the foot of the g(V), where stretch enables a voltage step to elicit current where there had been none without stretch. In these two examples (A and B; different patches), −40 mV was just at (A) or below (B) the threshold for this effect, whereas −20 mV was above it. Macroscopic currents were absent without stretch, but at −20 mV with stretch a large time-dependent current developed. At larger depolarizations (0 and 40 mV in A), stretch accelerated both current rise and decline.
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Figure 3. . Trying to find a ΔV substitute for stretch. (A) 5aa currents evoked by stepping from −90 to 50 mV with −45 mm Hg suction (gray trace) and from −90 to 55–95 mV in 10-mV increments without suction (black traces). (B) The rising phases of the currents at 50 mV with stretch (gray) and 85 mV without stretch (black) are scaleable (inset, scaled currents on an expanded time scale). Downscaling was necessary to compensate for the increased driving force, but activation kinetics were otherwise identical: in its effect on channel activation, the 35-mV voltage increment was equivalent to the particular membrane stretch produced by −45 mmHg. Note, however, that compared with the accelerating effect of the stretch stimulus (at 50 mV), acceleration of inactivation by the depolarizing stimulus was less pronounced.
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Figure 4. . Voltage dependence of delay td and maximum slope SMax for a family of 5aa currents evoked by stepping from a holding potential of −90 mV in 10-mV increments from −10 to 100 mV. (A) Sample currents (−10, 10, 30, 50, 70, 90 mV), with fitted maximum slope lines. (B) td (squares) and SMax (circles) versus voltage. td(V) can be fitted by a single exponential like Eq. 4, an expression from Scheme I. From the fit with Eq. 4, zα = 0.68 and α0= 12 s−1 follow. The corresponding expression for SMaxP in Scheme I is Eq. 3. To describe SMax(V), the voltage-dependent driving force has been included by multiplying Eq. 3 with (V + 90 mV) and a constant. This, with the gating charge from the td(V) fit, yields a satisfying description of SMax(V). (C) Fitting current rising phases with Eq. 4. Sample currents at −10, 30, and 90 mV from the current family in A. PO(t) from Eq. 2, with α0 and zα from the td(V) fit in B, has been scaled to match the rising phases—the irreversible four-step model describes the current rise well. The remainder of the capacitive transient at 90 mV is unlikely to obstruct determination of channel current properties (see Fig. 1).
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SCHEME I.
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Figure 5. . The four-step model describes 5aa current properties with and without stretch. (A) Delay and maximum slope from 0 to 90 mV, with (gray symbols) and without −45 mm Hg suction (open symbols); voltage was stepped to the indicated levels from −90 mV. td(V) without stretch is fitted with Eq. 4, α0 = 1.4 s−1 and zα = 0.85. With stretch, the same zα, and α0 = 2.3 s−1 provide a good description. SMax(V) with and without stretch is well described by Eq. 3, multiplied by a linear driving force, and with the zα value from the delay fit. (B) Sample currents with (gray) and without stretch, before and after (black). PO(t) from Eq. 2 with zα and α0 from the td(V) fits in A has been scaled to match the current rise.
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SCHEME II.
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Figure 6. . 5aa slow inactivation. (A) In a typical experiment (15 μM Gd3+ in the pipette) currents were evoked by stepping from a holding potential of −90 mV in 10-mV increments from −10 to 80 mV, and recorded on a long time scale. (B) Currents from A. Amplitudes are normalized and the time scaled for each voltage so the rising phases overlap. Activation cannot be rate limiting for inactivation, because the scaled decline slows progressively with increasing voltage, with the sole exception of the response at −10 mV (the noisy, lighter colored trace). (C) The current falling phases can be fitted by Eq. 1. The time constant τ of the monophasic exponential decay to a constant plateau is given on each panel. (D) τ(V) declines with increasing voltage.
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Figure 7. . Voltage dependence of the time constant τ from fits of Eq. 1 to current falling phases, with and without stretch, pooled from a number of experiments. Most data were obtained with 10–20 μM Gd3+ in the pipette, but some data without Gd3+ are also shown, as indicated. Stretch reduced τ at all voltages. Gd3+ had no discernible effect, but since it right shifts the g(V) relation (Gu et al. 2001), time constants could be obtained at −10 and −20 mV without Gd3+ but not with it.
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Figure 8. . 5aa current decline and stretch. (A) Sample currents with and without −45 mm Hg suction. The falling phases have been fitted with Eq. 1. Stretch decreases τ. (B) τ (V) with (gray circles) and without stretch (black squares). (C and D) The sample currents with and without stretch from A have been normalized, their times scaled to match all rising phases. The scaled falling phase decelerates with increasing voltage. Above 10 mV, where the falling phase is not rate limited by activation, the inactivation rate dominates falling phase dynamics. (E) Normalized currents with and without stretch. For each voltage, the time scale of the current with stretch has been expanded to match the rising phase of the current without stretch. Note that the entire scaled time courses are virtually identical. C and D showed that activation is not rate limiting for the current decline at these voltages and thus stretch, in contrast to voltage, seems to affect activation and inactivation in exactly the same way.
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Figure 9. . Influence of inactivation on our analysis. (A and B) Simulated kinetic data for a model with four irreversible activation steps plus inactivation (Scheme II). α0= 1, zα = 0.85, inactivation rate i = 0.5 (arbitrary units). (A) Simulated td(V) and SMax(V) relations, from fits of maximum slope lines to simulated PO(t) (symbols). For comparability with the experimental results, simulated SMaxP in Figs. 9–13 have been multiplied by (V + 90 mV) to mimic a driving force. Thick lines, predicted delay and maximum slope without inactivation (i = 0). This line (delay), fit of td(V) with Eq. 3, α0= 1.15 and zα = 0.76. Thin line (slope), fit of the maximum slope with Eq. 2 and zα from the delay fit. (B) Simulated PO(t) (thick lines) and fits with Eq. 2, α0 and zα from the delay fit (thin lines). The fits look reasonable at low voltages. At 90 mV, α(V) predicted from the fit of td(V) is too slow. To fit the simulated rise, Eq. 2 must be upscaled beyond the peak PO amplitude. (B, insets) Sample currents from an experiment. Currents and their fits with Eq. 2 and the apparent parameters from the delay fit (α0= 15 s−1, zα = 0.58, not depicted) behave like the simulated data in that they show the same failure to reproduce PO saturation at large voltages.
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Figure 10. . Effect of back reactions with zβ < zα on td(V) and SMax(V). Simulated data. α0= 1, β0= 0.2, zα = 0.85, zβ = 1.7 (arbitrary units). From Eqs. A3 and A4, \documentclass[10pt]{article}
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\begin{equation*}\overline{{\mathit{V}}}\end{equation*}\end{document}≈−9 mV and V1/2 ≈ 0 mV. Symbols, simulated td(V) and SMax(V). Solid lines, predicted delay or maximum slope without deactivation (Eqs. 3 and 4). Simulated SMax(V) is indistinguishable from its counterpart for the irreversible model; simulated td(V) is well described at large voltages, but bends characteristically below 0 mV. PO(t) rise, and amplitudes (except at the lowest voltages) are well fitted by Eq. 2 (not depicted). (Inset) One of two observations of a bend in td(V). Experimental td(V) and SMax(V) relations with (gray symbols) and without stretch (open symbols). Delay fits (from which the bend region has been excluded) by Eq. 4, apparent α0= 1.9 s−1 and α0= 3.3 s−1 without and with stretch, respectively, zα = 1 for both. Maximum slope fitted with Eq. 3 (multiplied by linear driving force) and the same zα. As in the simulated example, current rise and amplitudes were well fitted by Eq. 2 with the parameters from the delay fit, except at the lowest voltages, where amplitudes were smaller (not depicted).
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Figure 11. . Effect of back reactions with zβ < zα. α0= 1, β0= 1, zα = 0.85, zβ = 0.17 (arbitrary units). From Eqs. A3 and A4, \documentclass[10pt]{article}
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\begin{equation*}\overline{{\mathit{V}}}\end{equation*}\end{document}≈−40 mV and V1/2 ≈ 42 mV. (A) Symbols, simulated td(V) and SMax(V). Thick lines, predicted delay or maximum slope without deactivation. Deactivation shortens the delay and reduces the maximum slope for all but the highest voltages. Thin line (delay), fit of td(V) with the four-step model (Eq. 4), apparent α0= 1.7 and zα = 0.66. Thin line (slope), fit of Smax(V) with Eq. 3 and zα forced to the value from the delay fit. The resulting fit is not very good, Smax(V) is somewhat steeper. (Inset) Experimental example. td(V) and SMax(V) relations (delay fit, α0= 6.7 s−1, zα = 0.57; slope fit, zα forced to 0.57). (B) Fits of simulated PO(t) rise (thick lines) with Eq. 2 and the apparent parameters from the td(V) fit (thin lines). At 0 mV, the fit amplitude is larger, and at 30 mV it is smaller than the peak PO. At 90 mV, Eq. 2 needs to be strongly upscaled to fit the early rising phase of PO. (Inset) Experimental example, as in A. When fitting the recorded current rise with Eq. 2 and the parameters from the delay fit, the fit amplitudes at 0 and 30 mV are smaller, and at 90 mV larger than the peak current.
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Figure 12. . Simulated kinetic data for four irreversible subunit activation steps and one final concerted transition with rate \documentclass[10pt]{article}
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\begin{equation*}c=c_{0}e^{z_{c}0.039V}\end{equation*}\end{document}. α0= 1, zα = 0.85, γ0= 2, zγ = 0.42 (arbitrary units). (A) Symbols, simulated td(V) and SMax(V). Thick lines, predicted delay or maximum slope without (i.e., with infinitely fast) last step. Naturally, the additional step prolongs the delay and slows the current rise. Thin line (delay), fit of td(V) with Eq. 4, apparent α0= 0.65 and zα = 0.9. Thin line (slope), fit of SMax(V) with Eq. 3, zα forced to the value from the delay fit. SMax(V) is really somewhat biphasic, but can be described satisfyingly with zα = 0.9. (B) Simulated PO(t) (thick lines) and fits with Eq. 2, α0 and zα from the delay fit (thin lines). The fits look reasonable at all voltages. Upscaling of the fit expression is required, but if we encountered this effect in our experiments, we couldn't easily differentiate it from that of inactivation.
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Figure 13. . Simulated kinetic data for two irreversible subunit activation steps. (A and B) Similar rates at all voltages. α10= 1, α20= 1.2, zα1 = 0.85, zα2 = 0.85 (arbitrary units). (A) Symbols, simulated td(V) and Smax(V). Thin line (delay), fit with the four-step model (Eq. 4), apparent α0= 0.42 and zα = 0.85. Thin line (slope), fit with Eq. 3 and zα from the delay fit. (B) Simulated PO(t) (thick lines) and fits with Eq. 2, α0 and zα from the delay fit (thin lines). Upscaling of the fit expression is required for all voltages to reproduce the simulated maximum slope. The fitted curves from the four-step model are less sigmoidal. (C and D) Different rates (one is rate-limiting at high voltages). α10= 1, α20= 1, zα1 = 0.85, zα2 = 1.35 (arbitrary units). (C) Symbols, simulated td(V) and SMax(V). Thin line (delay), fit of td(V) with Eq. 4, apparent α0= 0.4 and zα = 1.2. Thick line (delay), Eq. 4 with α0= 1 and zα = 0.85, the parameters that fit SMax(V). Thin line (slope), trying to describe SMax with Eq. 3 and zα from the delay fit. Thick line (slope), Eq. 3 with α0= 1 and zα = 0.85 describes SMax(V) well. (B) Simulated PO(t) (thick lines) and fits with Eq. 2, α0 and zα from the delay fit (thin lines). At small voltages, the maximum slope can be reproduced by upscaling Eq. 2, but the eight-step PO(t) is always more sigmoidal. At large voltages (e.g., 90 mV), PO(t) rises more slowly than Eq. 2 as the step with α10= 1 and zα1 = 0.85 becomes rate limiting: at 90 mV, it is threefold slower than the rate predicted from the delay fit.
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