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Scheme S1.
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Figure 1. Filtering, noise and HMM convergence. (A) Portion of a leak-subtracted recording centered on the transition (arrow) between the +60 mV activating pulse and the â120 mV deactivating pulse. Due to amplifier saturation the current recording in the vicinity of the voltage jump is zero. The thick trace shows the original 30-kHz bandwidth; the thin trace is the result of inverse-filtering to 80 kHz. (B) Power spectrum (points) computed from 37-ms quiescent portions selected from 100 sweeps, and plotted on a linear frequency scale. Above 60 kHz, the spectral density falls due to the sharp-cutoff filter, which had a half-amplitude (1/4 power) frequency of 80 kHz. Small spectral peaks due to interference from the computer (65 kHz) and the patch-clamp amplifier's power supply (95 kHz) are visible. Superimposed on the data are the spectral densities computed from the AR noise model parameters of the HMM fit of this data set. It is seen that the shape of the spectrum is described better as the number of AR coefficients is increased from three to four and five. (C) Convergence of a hidden Markov model. The log likelihood and the value of the S2âS1 rate constant are plotted as a function of iteration number for the fitting of Model III (see Fig. 3) to the same data set obtained at â120 mV. The inset shows the rapid increase in log-likelihood value during the first 15 iterations.
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Figure 2. Single SNÎ channel deactivation. (A) Single-channel current recorded from a SNÎ channel with symmetrical 140 mM potassium solution, shown at 5-kHz bandwidth. Pulse protocol is shown below the current trace. The vicinity of the last closing transition, marked by a box and shown in B at expanded time scale, is subjected to HMM analysis. (B) Examples of the closing transitions shown at 15 kHz bandwidth. A simulated current step after Gaussian filtering to 15 kHz is overlaid on top of the current traces in the top two panels. (C) Sequence comparison between ShÎ and SNÎ. The proposed S6 transmembrane domain is marked. Dashes represent identical residues.
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Figure 3. Examples of the models tested. Each model has one open state (O), one to four substates (S and T), and two to three closed states (C), in which the leftmost closed state (C0) is an absorbing state. Deactivation always starts at the open state. Conductances and transition rates for each model are listed in Table .
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Figure 4. Comparison of the relative log-likelihood values generated by models with increasing complexity. Model numbers in parentheses are the same as in Fig. 3. Those not labeled are models having the same states as Model XI but with more transition pathways linking those states, including pathways between the S states and the T states. Dotted lines link results from the same model; open symbols represent fits starting from different initial values that ended at a different maxima. Dashed lines are drawn at slopes of 15 log-likelihood units per free parameter.
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Figure 5. Sublevels in the SNÎ channel have voltage-dependent lifetimes. (A) From the HMM analysis, estimates of the single-channel current are plotted as a function of membrane potential for the fully-open level and two sublevels. (B) Voltage dependence of the mean lifetime of each sublevel. Open symbols are results from a series of recordings from the same SNÎ channel; closed symbols at â120 mV are results from 765 closing events in a recording from an N-typeâinactivating SN channel. The number of sweeps analyzed at each voltage are as follows: â80 mV, 1,700; â100 mV, 1,286; â120 mV, 622 (SNÎ), 765 (SN); and â140 mV, 1,588. The superimposed dotted curves are the state lifetimes predicted by the model shown in D. (C) Voltage dependences of three rate constants and fits to an exponential function, from which partial charges are obtained. (D) Model III is shown with values for the rate constants (units of sâ1) given for V = â120 mV. The estimated errors are two-unit likelihood intervals as determined in Fig. 6. The partial valence associated with each rate constant is given in parentheses. The percentage of transitions taking the OâS2 path was obtained by adding an extra closing path from the open state and comparing the rate constants leading to S2 versus this other path. In most cases, this percent was higher than 90%. The absolute value of the rate of this transition could not be obtained because the dwell in O state was truncated by our data selection procedure. (E) Simulated single-channel data at â120 mV using the model shown in D and assuming a value of 1,500 sâ1 for the OâS2 rate constant.
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Figure 6. Likelihood intervals for rate constant estimates of Model III obtained from one of the data sets of deactivations at â120 mV. Solid curves are fit of the function ÎL = â(x â x0)2/s2, in which ÎL is the difference in the log likelihood from its maximum value, x is the value of the given rate constant, x0 is the optimal value for that rate, and s is the fitted one-unit likelihood interval for the estimated rate constant. The confidence intervals given in Fig. 5 B represent ±2s.
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Figure 7. Wild-type ShÎ channels close through sublevels. (A) Single-channel current recorded from a ShÎ channel with symmetrical potassium solution, shown at 5 kHz bandwidth. Pulse protocol is shown below the current trace. (B) Examples of the closing transitions shown at 15 kHz bandwidth. (C) Comparison of the relative log-likelihood values for ShÎ channels. Model numbers in parentheses indicate the same models as in Fig. 3; unlabeled symbols represent extensions of Model II having additional transitions. Dotted lines link results obtained from the same model with different initial parameter values. Open symbols represent fitting results that contain rate estimates exceeding 100,000 sâ1. Dashed lines are drawn with slopes of 15 log-likelihood units per free parameter.
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Figure 8. Recovery from inactivation to the fully-open level does not traverse sublevels. (A) Partial gating scheme illustrating multistep activation and a single-step inactivation process. (B1) Representative current sweeps of a SN channel that has N-type inactivation intact, shown at 5-kHz bandwidth. The channel activates and subsequently inactivates upon depolarization to +60 mV. An event representing recovery from inactivation at â120 mV is marked by a box and shown in detail in B2, where it is shown at 15 kHz with enlarged scales. Data segments for recovery from inactivation (a) and deactivation (b) were analyzed separately. (C) Comparison of HMM analysis of the recovery from inactivation (La) and deactivation (Lb) from 765 sweeps. A state enclosed by a box has a fixed current amplitude whose value is shown. N, number of free parameters in each model; La and Lb, relative log-likelihood values for a and b, respectively. Results were obtained with four AR coefficients.
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Figure 9. A mutation at the ion selectivity filter dramatically affects sublevels. (A) Molecular model of the KcsA potassium channel highlighting the position of T75, which is equivalent to T442 in Shaker B. (B) A representative single-channel current trace recorded from a T442S mutant channel, SNSÎ, with the pulse protocol shown underneath. Arrows point to dwells at sublevels during deactivation. (C) Lifetimes in sublevels of the SNSÎ channel, measured by threshold analysis (open symbols), are compared with those of SNÎ channels. (D) A cartoon showing three possible gates in the Shaker channel: (1) the S6 gate; (2) the pore gate; and (3) the N-type inactivation gate.
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