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Figure 1. . Allosteric activation model for BK channel. (A) The allosteric activation by calcium originates a 10-state Monod-Wymann-Changeaux (MWC) activation model. For each calcium binding site occupied, the equilibrium constant for channel opening (L) is multiplied by the allosteric factor C. The same factor multiplies the calcium binding equilibrium constant (K) when the channel is open. (B) The allosteric activation by voltage also originates a 10-state MWC model. In this case, the allosteric factor is D and the equilibrium constant for voltage sensor activation is J. (C) The combination of A and B originates a two-tiered, 50-state model. (D) The complete 70-state model takes into account some interaction between voltage sensor activation and calcium binding (allosteric factor E). Note that when E = 1, the model reduces to 50-state as in C (modified from Horrigan and Aldrich, 2002).
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Figure 2. . Effects of β1 and β2IR subunits on BK channel steady-state activation parameters. (A) Macroscopic currents recorded in the inside-out configuration at 5 nM (left) and 2.8 μM (right) intracellular calcium. The respective voltage protocols are shown at the bottom. (B) Averaged P(O)/V curves at 2.8 μM (triangles) and 5 nM Ca2+ (circles). n = 4–9. Lines are the best fit to a Boltzmann distribution (Eq. 2). Fit parameters are: α, V0.5 = 209 mV, z = 0.89 (5 nM); V0.5 = 42 mV, z = 1.54 (2.8 μM). α+β1, V0.5 = 244 mV, z = 0.67 (5 nM); V0.5 = −5 mV, z = 1.15 (2.8 μM). α+β2IR, V0.5 = 190 mV, z = 0.96 (5 nM); V0.5 = −33 mV, z = 1.59 (2.8 μM) (C) Average of the obtained V0.5 values plotted against calcium concentration. (D) Average of the obtained z values plotted against calcium concentration. Error bars are SD, n = 4–7.
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Figure 3. . ΔG vs. [Ca2+] relationships for BK channel. ΔG was calculated for each individual experiment as −zFV0.5. Plotted values are mean ± SD (n = 4–9). Lines represent the best fit to a sigmoid concentration-effect curve (Eq. 6). Best-fit values are listed in Table I.
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Figure 4. . Unitary events quantification at very low open probabilities. Inside-out patches containing hundreds of α (A), α+β1 (C), or α+β2IR (E) channels were exposed to the indicated membrane potentials in the virtual absence of intracellular calcium (5 nM). All-points histograms were constructed from 20–45-s recordings for α (B), α+β1 (D), and α+β2IR (F) channel containing patches. Amplitudes are expressed as conductance in pS.
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Figure 5. . Limiting voltage dependence for α, α+β1, and α+β2IR channels. Semilogarithmic P(O)/V plots for α (A), α+β1 (B), and α+β2IR (C) channels, obtained from unitary events quantification (V < 60 mV) or macroscopic recordings (V > 30 mV). Values shown are mean ± SD of 3–9 points, obtained from 8 (α and α+β2IR) or 9 (α+β1) different patches. In some cases, the SD bars fall within the symbols. Linear intervals were fitted to Eq. 9 and the corresponding z values are shown beside the plots. The inset in B shows a smoothed differential of the lnP(O)/V data, expressed as electronic equivalents (scaled by R×T/F). The straight dotted line marks the Y = 0.3 position while the dotted curve is a simple exponential fit of the data added as visual reference. (D) Simultaneous fit of the P(O)/V data to Eq. 11 restricting a shared zL value for all datasets (lines). Parameters for the best fit are listed in Table II. Symbols are the same experimental data shown in A–C.
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Figure 6. . The maximum slope of the ln(P(O))/V relationship is highly affected by the zJ parameter. (A) Semilogarithmic P(O)/V plots for α, α+β1, and α+β2IR channels. This is the same data as in Fig. 5 D, showing the P(O) > 10−5 range for a better appreciation of the maximum slope. Duplicate points correspond to data obtained by two different methods (unitary events quantification and macroscopic recordings). Lines represent the fit of the maximum slope found in the 0–100-mV range. (B) P(O)/V curves (dotted lines) simulated with Eq. 11 are plotted in the P(O) > 3 × 10−5 range. In each plot, the thicker dotted line is the same, corresponding to the following parameters: Vh(J) = 140 mV, zJ = 0.61, L0 = 4.7 × 10−6, zL = 0.28, and D = 14.4. Thinner dotted lines were constructed varying zJ (top left), D (top right), zL (bottom left), L0 (bottom right, gray), or Vh(J) (bottom right, black) in the indicated ranges. Continuous straight lines represent the maximum d(lnP(O))/dV value of each simulated curve.
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Figure 7. . The β subunits and the voltage dependence of the deactivation kinetics in the absence of calcium. (A) Tail current traces evoked by a −60-mV pulse after opening the channels with a 200-mV activation pulse. Current magnitude of each trace was normalized by the mean current at the 200-mV steady state. Over the traces, the best fit to an exponential decay is shown. (B) Deactivation time constant (τdeact) plotted against voltage. Symbols represent mean ± SD. n = 3–4. For the α subunit, most of the SD bars fall within the symbols. (C) The range from −220 to −190 mV of each plot was fitted to Eq. 15 (dotted lines). The fit shown is with a shared value of zγ (0.13). The range from 20 to 100 mV (α), −50 to 100 mV (α+β1), and 0 to 75 mV (α+β2IR) was fit to the same equation (continuous lines). Obtained z values were 0.46, 0.23, and 0.47, respectively.
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Figure 8. . Proposed kinetic scheme for BK channel in the absence of calcium. (A) Kinetic scheme proposed by Horrigan et al. (1999). α = α(0)exp(zαFV/RT), β = β(0)exp(−zβFV/RT), δn = δn(0)exp(zδFV/RT), and γn = γn(0)exp(−zγFV/RT). The correspondence with the scheme in Fig. 1 B is verified by the following equalities: J = α/β, L0 = δ0/γ0, zJ = zα + zβ, zL = zδ + zγ, and D = (δn+1/γn+1)/(δn/γn) = f2. (B) Abbreviated kinetic scheme that considers the movement of the voltage sensors as in instantaneous equilibrium.
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Figure 9. . The β subunits and the voltage dependence of the activation kinetics in the absence of calcium. (A) Current traces evoked by a 200-mV pulse after a −80-mV prepulse. Current magnitude of each trace was normalized by the mean current at steady state. Over the traces, the best fit to an exponential raise is shown. For α and α+β1, the fit was extrapolated (segmented lines) to show that the traces are effectively normalized to the same maximum. (B) Activation time constant (τact) plotted against activation voltage. Symbols represent mean ± SD. n = 3–5. (C) The lineal range of each plot was fit to Eq. 15 with a negative z.
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Figure 10. . A change in zJ can account for the increase of the apparent calcium sensitivity in the presence of the β1 subunit. (A) Averaged experimental P(O)/V curves (symbols) and predictions of the best fit to Eq. 1 (lines) for α channels. Within the plot, the parameters of the best fit are shown. See the text for the restrictions applied. (B) Averaged experimental P(O)/V curves (symbols) and predictions of the best fit to Eq. 1 (lines) for α+β1 channels. In the left plot, the fit was done with the same parameters obtained in A (best fit parameters for α) and varying only zJ. In the right plot, L0 was fixed to 3 × 10−7 and zJ, C, and D were varied. (C and D) Families of predicted P(O)/V curves were fitted to a Boltzmann distribution (Eq. 2) and the obtained V0.5 (C) and z (D) values are plotted against calcium concentration (lines). Symbols represent the experimental values (these are the same values plotted in Fig. 2, C and D).
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Figure 11. . Fit of the P(O)/V curves for α+β2IR to Eq. 1. (A) Averaged experimental P(O)/V curves (symbols) and predictions of the best fit to Eq. 1 (lines) for α channels. Fit 1 was performed using the following restrictions: 140 < Vh(J) < 160, 0.25 < zL < 0.4, 0.25 < zJ < 0.6. Fit 2 was performed using the following restrictions: 140 < Vh(J) < 160, zL = 0.3, 0.45 < zJ < 0.6, E > 0. In each case, the best-fit parameters obtained are listed to the right of the plot. (B and C) Families of predicted P(O)/V curves were fitted to a Boltzmann distribution (Eq. 2) and the obtained V0.5 (B) and z (C) values are plotted against calcium concentration (lines). Symbols represent the experimental values for α+β2IR (these are the same values plotted in Fig. 2, C and D).
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Figure 12. . Effects of a reduction of zJ in the allosteric activation model. (A) A reduction of zJ in the allosteric model affects the ΔG/[Ca2+] relationships in the same way as the β1 subunit. Symbols represent ΔG values predicted by the allosteric activation model. For each calcium concentration, V0.5 and z values were calculated by fitting predicted P(O)/V curves to Eq. 2 and ΔG was calculated as −zFV0.5. The parameters used are the same as in Fig. 10. Lines represent a fit of the data to Eq. 6. For α, the best-fit logEC50 value is 0.54 and EC50 = 3.4 μM. For α+β1 (zJ = 0.3), logEC50 = 0.28 and EC50 = 1.9 μM. For α+β1 (var. zJ, L0, C, and D), logEC50 = 0.30 and EC50 = 2.0 μM. (B) A reduction of zJ implies a higher J for any V < Vh(J). J values were calculated as exp[zJF(V − Vh(J))/RT] with Vh(J) = 140 mV and the indicated zJ values. Dotted lines indicate that J = 1 when V = Vh(J).
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