Identifying kinetic gating mechanisms for ion channels by using two-dimensional distributions of simulated dwell times.

Ion channels are integral membrane proteins that regulate ionic flux through cell membranes by opening and closing (or gating) their pores. The gating can be monitored by observing step changes in the current flowing through single channels. Analysis of the durations of the open and closed intervals and of the correlations among the interval durations can give insight into the gating mechanism. Although it is well known that the correlation information can be essential to distinguish among possible gating mechanisms, it has been difficult to use this information because it has not been possible to correct the predicted correlations for the distortion of the single-channel data because of filtering and noise. To overcome this limitation we present a method based on a comparison of simulated and experimental two-dimensional dwell-time distributions constructed by analysing simulated and experimental single-channel currents in an identical manner. The simulated currents incorporate the true effects of filtering and noise, the two-dimensional distributions retain the correlation information, and the identical analysis allows direct maximum-likelihood comparison of the simulated and experimental two-dimensional distributions. We show that the two-dimensional simulation method has a greatly increased ability to distinguish among models, compared with methods that use one-dimensional distributions.     

Testing for microscopic reversibility in the gating of maxi K+ channels using two-dimensional dwell-time distributions.

An assumption usually made when developing kinetic models for the gating of ion channels is that the transitions among the various states involved in the gating obey microscopic reversibility. If this assumption is incorrect, then the models and estimated rate constants made with the assumption would be in error. This paper examines whether the gating of a large conductance Ca-activated K+ channel in skeletal muscle is consistent with microscopic reversibility. If microscopic reversibility is obeyed, then the number of forward and backward transitions per unit time for each individual reaction step will, on average, be identical and, consequently, the gating must show time reversibility. To look for time reversibility, two-dimensional dwell-time distributions of the durations of open and closed intervals were obtained from single-channel current records analyzed in the forward and in the backward directions. Two-dimensional dwell-time distributions of pairs of open intervals and of pairs of closed intervals were also analyzed to extend the resolution of the method to special circumstances in which intervals from different closed (or open) states might have similar durations. No significant differences were observed between the forward and backward analysis of the two-dimensional dwell-time distributions, suggesting time reversibility. Thus, we find no evidence to indicate that the gating of the maxi K+ channel violates microscopic reversibility.     

Two-dimensional kinetic analysis suggests nonsequential gating of mechanosensitive channels in Xenopus oocytes

Xenopus oocytes express mechanosensitive (MS(XO)) channels that can be studied in excised patches of membrane with the patch-clamp technique. This study examines the steady-state kinetic gating properties of MS(XO) channels using detailed single-channel analysis. The open and closed one-dimensional dwell-time distributions were described by the sums of 2-3 open and 5-7 closed exponential components, respectively, indicating that the channels enter at least 2-3 open and 5-7 closed kinetic states during gating. Dependency plots revealed that the durations of adjacent open and closed intervals were correlated, indicating two or more gateway states in the gating mechanism for MS channels. Maximum likelihood fitting of two-dimensional dwell-time distributions to both generic and specific models was used to examine gating mechanism and rank models. A kinetic scheme with five closed and five open states, in which each closed state could make a direct transition to an open state (two-tiered model) could account for the major features of the single-channel data. Two-tiered models that allowed direct transitions to subconductance open states in addition to the fully open state were also consistent with multiple gateway states. Thus, the gating mechanism of MS(XO) channels differs from the sequential (linear) gating mechanisms considered for MS channels in bacteria, chick skeletal muscle, and Necturus proximal tubule.     

Rapid kinetic analysis of multichannel records by a simultaneous fit to all dwell-time histograms

A method is presented for rapidly extracting single-channel transition rate constants from patch-clamp recordings containing signals from several channels. The procedure is based on a simultaneous fit of the observed dwell-time distributions for all conductance levels, using a maximum likelihood approach. This algorithm allows estimation of single-channel rate constants in cases where more advanced methods may be impractical because of their extremely long computational time. A correction is included for the limited time resolution of the recording system, according to theory developed by Roux and Sauvé (Biophys. J. 48:149-158, 1985), by accounting for the impact of undetected transitions on the dwell-time distributions, and by introducing an improved practical implementation of a fixed dead time for the case of more than one channel. This feature allows application of the method to noisy data, after filtering. A computer program implementing the method is tested successfully on a variety of simulated multichannel current traces.     

Ca2+-dependent gating mechanisms for dSlo, a large-conductance Ca2+-activated K+ (BK) channel.

The Ca2+-dependent gating mechanism of cloned BK channels from Drosophila (dSlo) was studied. Both a natural variant (A1/C2/E1/G3/IO) and a mutant (S942A) were expressed in Xenopus oocytes, and single-channel currents were recorded from excised patches of membrane. Stability plots were used to define stable segments of data. Unlike native BK channels from rat skeletal muscle in which increasing internal Ca2+ concentration (Cai2+) in the range of 5 to 30 microM increases mean open time, increasing Cai2+ in this range for dSlo had little effect on mean open time. However, further increases in Cai2+ to 300 or 3000 microM then typically increased dSlo mean open time. Kinetic schemes for the observed Ca2+-dependent gating kinetics of dSlo were evaluated by fitting two-dimensional dwell-time distributions using maximum likelihood techniques and by comparing observed dependency plots with those predicted by the models. Previously described kinetic schemes that largely account for the Ca2+-dependent kinetics of native BK channels from rat skeletal muscle did not adequately describe the Ca2+ dependence of dSlo. An expanded version of these schemes which, in addition to the Ca2+-activation steps, permitted a Ca2+-facilitated transition from each open state to a closed state, could approximate the Ca2+-dependent kinetics of dSlo, suggesting that Ca2+ may exert dual effects on gating.     

Two-dimensional components and hidden dependencies provide insight into ion channel gating mechanisms

Correlations between the durations of adjacent open and shut intervals recorded from ion channels contain information about the underlying gating mechanism. This study presents an additional approach to extracting the correlation information. Detailed correlation information is obtained directly from single-channel data and quantified in a manner that can provide insight into the connections among the states underlying the gating. The information is obtained independently of any specific kinetic scheme, except for the general assumption of Markov gating. The durations of adjacent open and shut intervals are binned into two-dimensional (2-D) dwell-time distributions. The 2-D (joint) distributions are fitted with sums of 2-D exponential components to determine the number of 2-D components, their volumes, and their open and closed time constants. The dependency of each 2-D component is calculated by comparing its observed volume to the volume that would be expected if open and shut intervals paired independently. The estimated component dependencies are then used to suggest gating mechanisms and to provide a powerful means of examining whether proposed gating mechanisms have the correct connections among states. The sensitivity of the 2-D method can identify hidden components and dependencies that can go undetected by previous correlation methods.     

The power of two-dimensional dwell-time analysis for model discrimination, temporal resolution, multichannel analysis and level detection

Two-dimensional (2D) dwell-time analysis of time series of single-channel patch-clamp current was improved by employing a Hinkley detector for jump detection, introducing a genetic fit algorithm, replacing maximum likelihood by a least square criterion, averaging over a field of 9 or 25 bins in the 2D plane and normalizing per measuring time, not per events. Using simulated time series for the generation of the "theoretical" 2D histograms from assumed Markov models enabled the incorporation of the measured filter response and noise. The effects of these improvements were tested with respect to the temporal resolution, accuracy of the determination of the rate constants of the Markov model, sensitivity to noise and requirement of open time and length of the time series. The 2D fit was better than the classical hidden Markov model (HMM) fit in all tested fields. The temporal resolution of the two most efficient algorithms, the 2D fit and the subsequent HMM/beta fit, enabled the determination of rate constants 10 times faster than the corner frequency of the low-pass filter. The 2D fit was much less sensitive to noise. The requirement of computing time is a problem of the 2D fit (100 times that of the HMM fit) but can now be handled by personal computers. The studies revealed a fringe benefit of 2D analysis: it can reveal the "true" single-channel current when the filter has reduced the apparent current level by averaging over undetected fast gating.     

Open channel noise. IV. Estimation of rapid kinetics of formamide block in gramicidin A channels.

Blocking events in currents through biological ion channels occur over a wide range of characteristic times. The interruptions in single-channel currents from blocking events may be characterized by the direct measurement of gap durations or by analyzing open-channel current histograms, provided that the events are not much shorter than the time resolution of single-channel recordings (approximately 10 microseconds). Here we present a method for the characterization of channel block on a much faster time scale by combining open-channel noise measurements with subsequent model fits according to a theoretical approach (Frehland, E. 1978. Biophysical Chemistry. 8:255-265). Although the bandwidth limitations in open-channel noise experiments are the same as in conventional single-channel experiments, from the dependence of the mean current and the spectral density of the noise on the concentration of the blocking agent, kinetics of very brief blocking events can be estimated. As an example we have analyzed the open-channel noise of K+ currents through the gramicidin A channel in the presence of various concentrations of formamide, a weak blocker, at neutral pH. We estimate the blocking and unblocking rates to be approximately 10(7)s-1 at 1 M formamide and discuss possible mechanisms for the blocking process.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

The Power of Two-Dimensional Dwell-Time Analysis for Model Discrimination, Temporal Resolution, Multichannel Analysis and Level Detection

Two-dimensional (2D) dwell-time analysis of time series of single-channel patch-clamp current was improved by employing a Hinkley detector for jump detection, introducing a genetic fit algorithm, replacing maximum likelihood by a least square criterion, averaging over a field of 9 or 25 bins in the 2D plane and normalizing per measuring time, not per events. Using simulated time series for the generation of the “theoretical” 2D histograms from assumed Markov models enabled the incorporation of the measured filter response and noise. The effects of these improvements were tested with respect to the temporal resolution, accuracy of the determination of the rate constants of the Markov model, sensitivity to noise and requirement of open time and length of the time series. The 2D fit was better than the classical hidden Markov model (HMM) fit in all tested fields. The temporal resolution of the two most efficient algorithms, the 2D fit and the subsequent HMM/beta fit, enabled the determination of rate constants 10 times faster than the corner frequency of the low-pass filter. The 2D fit was much less sensitive to noise. The requirement of computing time is a problem of the 2D fit (100 times that of the HMM fit) but can now be handled by personal computers. The studies revealed a fringe benefit of 2D analysis: it can reveal the “true” single-channel current when the filter has reduced the apparent current level by averaging over undetected fast gating.     

Dependency plots suggest the kinetic structure of ion channels.

Ion channels are integral membrane proteins that regulate ionic flux through cell membranes by opening and closing (gating) their pores. The gating can be monitored by observing step changes in the current flowing through single channels, and analysis of the observed open and closed interval durations has provided a window to develop kinetic models for the gating process. One difficulty in developing such models has been to determine the connections (transition pathways) among the various kinetic states involved in the gating. To help overcome this difficulty we present a transform (dependency plot) of the single-channel data that can give immediate insight into the connections. A dependency plot is derived by calculating a contingency table from a two-dimensional (joint density) dwell-time distribution of adjacent open and closed intervals by assuming that the two classified criteria are the open and closed durations of each pair of adjacent intervals. A three-dimensional surface plot of the fractional difference between the numbers of observed interval pairs and the numbers expected if the durations of adjacent intervals are independent then gives the dependency plot. An excess of interval pairs in the dependency plot suggests that the open and closed states (or compound states) that give rise to the interval pairs in excess are directly connected. A deficit of interval pairs suggests that the open and closed states (or compound states) that give rise to the interval pairs in deficit are either not directly connected or that there are additional open-closed transition pathways arising from the directly connected states.     

On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory

Several approaches to ion-channel gating modelling have been proposed. Although many models describe the dwell-time distributions correctly, they are incapable of predicting and explaining the long-term correlations between the lengths of adjacent openings and closings of a channel. In this paper we propose two simple random-walk models of the gating dynamics of voltage and Ca(2+)-activated potassium channels which qualitatively reproduce the dwell-time distributions, and describe the experimentally observed long-term memory quite well. Biological interpretation of both models is presented. In particular, the origin of the correlations is associated with fluctuations of channel mass density. The long-term memory effect, as measured by Hurst R/S analysis of experimental single-channel patch-clamp recordings, is close to the behaviour predicted by our models. The flexibility of the models enables their use as templates for other types of ion channel.     

Single-channel recording of ligand-gated ion channels

Electrophysiological recording of single-channel currents is the most direct method available for obtaining detailed and precise information about the kinetic behavior of ion channels. A wide variety of cell types can be used for single-channel recording, but to obtain the highest resolution of the briefest channel opening and closing events, low-noise recordings, coupled with a minimal filtering frequency, are required. Here, we present a protocol designed to help those with some electrophysiological expertise who wish to explore the properties of native and recombinant single ligand-gated ion channels. We have focused on the practical aspects of recording single GABA channels from cell-attached and outside-out patches and also introduced some of the preliminary considerations that are necessary for the analysis of single-channel data, including an introduction to single-channel analysis software.     

Yet another approach to the dwell-time omission problem of single-channel analysis.

Distortion of the open-time or closed-time distributions of single channel currents, due to limited time resolution of the recording system, has been addressed by many authors. The calculation of the modified distributions generally involves the numerical inversion of a Laplace transform and is difficult to apply in fitting multistate kinetic schemes to data. Our approach is to introduce "virtual states" into the kinetic scheme, as suggested by Blatz and Magleby (1986. Biophys. J. 49:967-980) to account for missed events. To simplify the assignment of rate constants in multistate schemes we make use of Kienker's (1989. Proc. R. Soc. Lond. 236:296-309) theory to first transform schemes to uncoupled form. Our approach provides a good approximation to the exact solution, while allowing the observable dwell-time distributions, and also the second-order probability density functions, to be computed by standard matrix techniques.     

Gating Kinetics of Single Large-Conductance Ca2+-Activated K+ Channels in High Ca2+ Suggest a Two-Tiered Allosteric Gating Mechanism?

The Ca²⁺-dependent gating mechanism of large-conductance calcium-activated K⁺ (BK) channels from cultured rat skeletal muscle was examined from low (4 μM) to high (1,024 μM) intracellular concentrations of calcium (Ca²⁺ _i) using single-channel recording. Open probability (P _o) increased with increasing Ca²⁺ _i (K _0.5 11.2 ± 0.3 μM at +30 mV, Hill coefficient of 3.5 ± 0.3), reaching a maximum of ∼0.97 for Ca²⁺ _i ∼ 100 μM. Increasing Ca²⁺ _i further to 1,024 μM had little additional effect on either P _o or the single-channel kinetics. The channels gated among at least three to four open and four to five closed states at high levels of Ca²⁺ _i (>100 μM), compared with three to four open and five to seven closed states at lower Ca²⁺ _i. The ability of kinetic schemes to account for the single-channel kinetics was examined with simultaneous maximum likelihood fitting of two-dimensional (2-D) dwell-time distributions obtained from low to high Ca²⁺ _i. Kinetic schemes drawn from the 10-state Monod-Wyman-Changeux model could not describe the dwell-time distributions from low to high Ca²⁺ _i. Kinetic schemes drawn from Eigen''s general model for a ligand-activated tetrameric protein could approximate the dwell-time distributions but not the dependency (correlations) between adjacent intervals at high Ca²⁺ _i. However, models drawn from a general 50 state two-tiered scheme, in which there were 25 closed states on the upper tier and 25 open states on the lower tier, could approximate both the dwell-time distributions and the dependency from low to high Ca²⁺ _i. In the two-tiered model, the BK channel can open directly from each closed state, and a minimum of five open and five closed states are available for gating at any given Ca²⁺ _i. A model that assumed that the apparent Ca²⁺-binding steps can reach a maximum rate at high Ca²⁺ _i could also approximate the gating from low to high Ca²⁺ _i. The considered models can serve as working hypotheses for the gating of BK channels.     

Optical single-channel recording: imaging Ca2+ flux through individual N-type voltage-gated channels expressed in Xenopus oocytes

Functional studies of single membrane ion channels were made possible by the introduction of the patch-clamp technique, which allows single-channel currents to be measured with unprecedented resolution. Nevertheless, patch clamping has some limitations: including the need for physical access of the patch pipette, possible disruption of local cellular architecture, inability to monitor multiple channels, and lack of spatial information. Here, we demonstrate the use of confocal fluorescence microscopy as a non-invasive technique to optically monitor the gating of individual Ca2+ channels. Near-membrane fluorescence signals track the gating of N-type Ca2+ channels with a kinetic resolution of about 10ms, provide a simultaneous and independent readout from several channels, and allow their locations to be mapped with sub-micrometer spatial resolution. Optical single-channel recording should be applicable to diverse voltage- and ligand-gated Ca2+-permeable channels, and has the potential for high-throughput functional analysis of single channels.     

Single-channel properties of a volume-sensitive anion conductance. Current activation occurs by abrupt switching of closed channels to an open state

Swelling-induced loss of organic osmolytes from cells is mediated by an outwardly rectified, volume-sensitive anion channel termed VSOAC (Volume-Sensitive Organic osmolyte/Anion Channel). Similar swelling- activated anion channels have been described in numerous cell types. The unitary conductance and gating kinetics of VSOAC have been uncertain, however. Stationary noise analysis and single-channel measurements have produced estimates for the unitary conductance of swelling-activated, outwardly rectified anion channels that vary by > 15-fold. We used a combination of stationary and nonstationary noise analyses and single-channel measurements to estimate the unitary properties of VSOAC. Current noise was analyzed initially by assuming that graded changes in macroscopic current were due to graded changes in channel open probability. Stationary noise analysis predicts that the unitary conductance of VSOAC is approximately 1 pS at 0 mV. In sharp contrast, nonstationary noise analysis demonstrates that VSOAC is a 40-50 pS channel at +120 mV (approximately 15 pS at 0 mV). Measurement of single-channel events in whole-cell currents and outside- out membrane patches confirmed the nonstationary noise analysis results. The discrepancy between stationary and nonstationary noise analyses and single-channel measurements indicates that swelling- induced current activation is not mediated by a graded increase in channel open probability as assumed initially. Instead, activation of VSOAC appears to involve an abrupt switching of single channels from an OFF state, where channel open probability is zero, to an ON state, where open probability is near unity.