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Figure 1. Ramp protocol for recording Kir2.1 currents. (A) Voltage protocol and current trace (experiment R1). The voltage protocol (red line) consists of a 100-ms prepulse at −200 mV, a 400-ms positive ramp (Pos) from −200 to +200 mV, a 100-ms interramp plateau phase at +200 mV, and a 400-ms negative ramp (Neg) from +200 to −200 mV. The ionic current (colored blue for Pos segment; orange for Neg segment; otherwise black) was recorded from an inside-out patch with symmetrical 20 mM [K+]. (B) Scaled G-V curves obtained from I-V relationships determined from the double ramp protocol using the formula G(V) = scale ⋅ I(V)/(V − Vos) − Gmin (Table 1). Ramp segments are color-coded the same as in A. (C) Conductance Hill plot (WH[g]) constructed from the G-V curves in A. Red arrows demarcate the outer linear asymptotic regions used for fitting.
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Figure 2. Hill plots of nominally blocker-free experiments. Hill-transformed decimated conductance data for seven experiments (R1–R7); average of Pos and Neg traces. The dashed lines are linear regression fits to the positive-voltage and negative-voltage asymptotic regions.
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Figure 3. Gating schemes. (A) The C-O model. C, closed; O, open. (B) The COSB model. Small circles represent partial (green) and complete (red) blocking sites. (C) Allosteric model with four voltage-sensing J gating particles (orange half-circles). R, resting; A, activated. Intermediate configuration states are not depicted. (D) Two-populations model. The two pore species (1 and 2) have different conductances and gating charges.
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Figure 4. Hill analysis of ramp data with varying spermine concentration. Positive ramps from −200 to 200 mV with varying internal spermine concentrations from different experiments. The control experiment is nominally free of spermine. The Hill-transformed conductances were fitted by eye to the COSB model in the intermediate region around V = 0 mV. The Gmin error factor b was set to zero. Adjusted variables are: [B] = 1.0 nM; ΔqL = 0.13 eo, VL = 46 mV, Δq1 = 3.8 eo, V1 = 8 mV, Δq2 = 1.5 eo, and V2 = 33 mV.
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Figure 5. Fits of blocker-free gating models to the Hill plot. Experiment R2. The experimental Hill plot is in gray, whereas fitted lines are solid red. Linear asymptotic regions are designated by black (shallow outer segments) or green (steeper intermediate region) dashed lines, accompanied by the model-specific formula that describes them. (A) The COSB model. The value of φ was set to 0.2 as previously established (Liu et al., 2012). Fitted parameters were: ΔqL = 0.16 eo; VL = −41 mV; ΔqK1 = 3.49 eo; VK1 = 1.58 mV; ΔqK2 = 1.28 eo; VK2 = 20.3 mV; b = 0.0074. The COSB model predicts that the blocking transition S→B should be represented by a steep positive asymptote with limiting slope ΔqK2 (dashed green line). However, to more closely fit the data, the positive-voltage asymptote was clipped at the value of kTlnb (lower dashed black line). (B) Allosteric scheme (n = 1). The shallow upper and lower extreme asymptotes are functions of the pore sensor L and the strength of its interaction with J, whereas the steeper intermediate slope (Hill coefficient) is a lower limit for the total gating charge dominated by the J particle. The very small difference in slope between the shallow asymptotes reflects a tiny coupling charge (ΔqD = 0.003 eo). ΔqL = 0.16 eo; VL = −39 mV; ΔqJ = 1.65 eo; VJ = 60 mV; WD = −91 meV (−2.1 kcal/mol). (C) The two-population model. The common slope of the two shallow asymptotes reflects gating in the channel with the smaller gating charge (channel 1). The steeper intermediate slope is the weighted mean of particle potentials from both channels. ΔqL1 = 0.33 eo, VL1 = −205 mV, ΔqL2 = 1.81 eo, VL2 = 4 mV; f1 = 0.50. (D–F) Residuals from least-squares fit. Linear regression lines for the residuals at the outer asymptotic regions are in red. Residuals points were normally distributed around their regression line (P value range: 0.49–0.99, Shapiro–Wilk test). The slopes of the outer residuals in the allosteric model, and in the Neg residual of the COSB model, are statistically equal to zero (P = 0.99, F test). However, nonzero slopes in the residuals are found in the Pos asymptotes of the COSB and two-populations model (P < 0.001) and in the Neg asymptote of the two-populations model (P = 0.02). (G–I) χ2 values used in the fit statistic, with inverse weights 0.148 meV2 (Neg) and 2.12 meV2 (Pos), equal to the variances calculated from residuals in the nondecimated data.
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Figure 6. Fit of two-population model assuming Δq1 = ΔqL. Data same as in Fig. 4. Fitted variables are Δq1 = 0.16 eo, V1 = −328 mV, Δq2 = 1.47 eo, V2 = −8 mV; f1 = 0.16.
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Figure 7. AIC analysis. Histogram of bootstrap AIC scores for three candidate models in experiments R1–R7. The AIC scores obtained from decimated tracings are indicated by thin vertical lines whose values are shown in parentheses.
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Figure 8. Confidence intervals for allosteric model parameters in a single experiment (R3). (A) χ2 values plotted against the variable of interest (x) are obtained by fitting the remaining variables to the decimated data set while keeping x fixed The critical χ2 value (horizontal orange line) for 95% confidence intervals was obtained from the F distribution: χ2crit = χ2min {1 + [m/(n − m)]F(0.05,m,n − m)}, where χ2min is obtained from fitting all variables, n is the number of data points, and m is the number of adjustable parameters (Kemmer and Keller, 2010). In the particular case of the allosteric model shown here, m = 4 (ΔqL, VL, WD, ΔqD). The variables that determine the steep intermediate portion of the Hill plot (ΔqJ = 1.67 eo; VJ = 61.9 mV) were kept constant. The mean and lower and upper confidence limits are indicated by the vertical gray lines. (B) Distributions of parameter values from 200 bootstrap samples. Mean values μ and 95% confidence intervals are indicated by vertical black lines. Confidence intervals were determined from μ ± 1.96σ, where σ is the sample standard deviation.
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Figure 9. Mean parameter values and confidence intervals for all experiments. Same procedure described in Fig. 8, applied to all experiments and gating schemes. Blue columns, decimated data; orange columns, bootstrap sampled data. (A) Allosteric model. (B) COSB model. (C) Two-populations model. Error bars indicate 95% confidence intervals. Columns labeled "mean" represent the overall mean and 95% confidence intervals from individual means in experiments R1–R7. Colored horizontal lines indicate which experiments were performed on oocytes from the same frog.
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Figure 10. Equivalent circuit and kinetic model. (A) Patch-clamp elements. Vc, command potential; Vos, voltage offset; I, membrane current; Ios, current offset; Rs, series resistance; Vm, membrane potential; Vrev, reversal potential (assigned zero for symmetric solutions); Cm, membrane capacitance; Rm, leak resistance. The governing equation is I′ = (V′ − Vm)/Rs = Cm(dVm/dt) + G(Vm − Vrev), solved numerically for Vm with initial conditions Vm(0) = V′(0)/(1 + Rs/Rm). The membrane conductance G is given by Gmin + Po(Gmax – Gmin), where Po is open probability, Gmin = 1/Rm, and Gmax – Gmin = NgL (product of channel number and unit conductance). (B) Kinetic scheme for the n = 1 allosteric model of Kir. The four states are labeled according to their contribution to the partition function. Expressions for rate constants are shown next to transition arrows. Ratios of forward to backward rate constants simplify to one of these transition equilibrium constants: L, J, LD, or JD, which also equal to the ratio of product to reactant terms in the partition function. Prefactors νL and νJ are transition frequencies for the L and J particles, respectively. The linear free energy variables xL and xJ range from 0 to 1. State probabilities were calculated by numerically integrating the rate differential equations governing the model using Vm as input, with initial conditions determined by the starting equilibrium distribution. (C) Newly formed inside-out patch without leak, capacity, or series resistance compensation (experiment R1). A fit (black line) of the ionic current (blue) to the linear elements of the equivalent circuit yielded Rs = 10.9 Mohm and Cm = 7.33 pF. The “leak” current contains contributions from both Rm and channel openings. The inset shows an expanded view of the inward capacity transient. The voltage protocol is shown at the bottom (red, Vc; black, Vm). Vos = −0.04 mV; Ios = 0.0019 pA. (D) Same patch as in C, but after capacity and 70% series resistance compensation. Equivalent circuit parameters were changed to Rs = 3.23 Mohm, Cm = 1.0 pF. Same patch subject to double ramp protocol. Experimental recordings of membrane current I and command voltage Vc are in blue and red, respectively. Dashed horizontal lines indicate zero values for current and voltage tracings. The inset is an enlargement of the current at very positive voltages. The black lines represent calculations of I and Vm using simultaneously equivalent circuit and gating models in response to Vc with adjustable variables derived from the Kir kinetic model and simultaneously fitted to I and the Hill plot. Ios = −0.00074 pA; Vos = 0 mV. Rs and Cm, same as in D. (F) Hill plots derived from Pos (blue) and Neg (orange) ramp currents, showing mild hysteresis. The black lines are the dynamic fits. The center green line is the calculated equilibrium Hill energy. Model variables: NgL = 30 nS; ΔqL = 0.17 eo; VL = −44 mV; νL = 2.0 kHz; xL = 0.2; ΔqJ = 1.69 eo; VJ = 55 mV; νJ = 0.15 kHz; xJ = 0.65; WD = −90 meV (−2.1 kcal/mol).
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