Estimating kinetic parameters for single channels with simulation. A general method that resolves the missed event problem and accounts for noise.

Analysis of currents recorded from single channels is complicated by the limited time resolution (filtering) of the data which can prevent the detection of brief intervals. Although a number of approaches have been used to correct for the undetected intervals (missed events) when identifying kinetic models and estimating parameters, none of them provide a general method which takes into account the true effects of noise and limited time resolution. This paper presents such a method. The approach is to use simulated single-channel currents to incorporate the true effects of filtering and noise on missed events and interval durations. The simulated currents are then analyzed in a manner identical to that used to analyze the experimental currents. An iterative search process using likelihood comparison of two-dimensional dwell-time distributions obtained from the simulated and experimental single-channel currents then allows the most likely rate constants to be determined. The large errors and false solutions that can result from the more typically applied assumptions of no noise and an absolute dead time (idealized filtering) are excluded by the iterative simulation method, and the correlation information contained in the two-dimensional distributions should increase the ability to distinguish among different gating mechanisms. The iterative simulation method is generally applicable to channels which typically open to a single conductance level. For these channels the method places no restrictions on the proposed gating mechanism or the form of the predicted dwell-time distributions.     

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.     

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.     

Gating of single non-Shaker A-type potassium channels in larval Drosophila neurons

The voltage-dependent gating of transient A2-type potassium channels from primary cultures of larval Drosophila central nervous system neurons was studied using whole-cell and single-channel voltage clamp. A2 channels are genetically distinct from the Shaker A1 channels observed in Drosophila muscle, and differ in single-channel conductance, voltage dependence, and gating kinetics. Single A2 channels were recorded and analyzed at -30, -10, +10, and +30 mV. The channels opened in bursts in response to depolarizing steps, with three to four openings per burst and two to three bursts per 480-ms pulse (2.8-ms burst criterion). Mean open durations were in a range of 2-4 ms and mean burst durations in a range of 9-17 ms. With the exception of the first latency distributions, none of the means of the distributions measured showed a consistent trend with voltage. Macroscopic inactivation of both whole-cell A currents and ensemble average currents of single A2 channels was well fitted by a sum of two exponentials. The fast time constants in different cells were in a range of 9-25 ms, and the slow time constants in a range of 60-140 ms. A six-state kinetic model (three closed, one open, two inactivated states) was tested at four command voltages by fitting frequency histograms of open durations, burst durations, burst closed durations, number of openings per burst, and number of bursts per trace. The model provided good fits to these data, as well as to the ensemble averages. With the exception of the rates leading to initial opening, the transitions in the model were largely independent of voltage.     

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.     

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.     

Recruiting RyRs to Open in a Ca2+ Release Unit: Single-RyR Gating Properties Make RyR Group Dynamics

In cardiac myocytes, clusters of type-2 ryanodine receptors (RyR2s) release Ca²⁺ from the sarcoplasmic reticulum (SR) via a positive feedback mechanism in which fluxed Ca²⁺ activates nearby RyRs. Although the general principles of this are understood, less is known about how single-RyR gating properties define the RyR group dynamics in an array of many channels. Here, we examine this using simulations with three models of RyR gating that have identical open probabilities: the commonly used two-state Markov gating model, one that utilizes multiple exponentials to fit single-channel open time (OT) and closed time (CT) distributions, and an extension of this multiexponential model that also includes experimentally measured correlations between single-channel OTs and CTs. The simulations of RyR clusters that utilize the multiexponential gating model produce infrequent Ca²⁺ release events with relatively few open RyRs. Ca²⁺ release events become even smaller when OT/CT correlations are included. This occurs because the correlations produce a small but consistent bias against recruiting more RyRs to open during the middle of a Ca²⁺ release event, between the initiation and termination phases (which are unaltered compared to the uncorrelated simulations). In comparison, the two-state model produces frequent, large, and long Ca²⁺ release events because it had a recruitment bias in favor of opening more RyRs. This difference stems from the two-state model’s single-RyR OT and CT distributions being qualitatively different from the experimental ones. Thus, the details of single-RyR gating can profoundly affect SR Ca²⁺ release even if open probability and mean OTs and CTs are identical. We also show that Ca²⁺ release events can terminate spontaneously without any reduction in SR [Ca²⁺], luminal regulation, Ca²⁺-dependent inactivation, or physical coupling between RyRs when Ca²⁺ flux is below a threshold value. This supports and extends the pernicious attrition/induction decay hypothesis that SR Ca²⁺ release events terminate below a threshold Ca²⁺ flux.     

Shaker potassium channel gating. III: Evaluation of kinetic models for activation

Predictions of different classes of gating models involving identical conformational changes in each of four subunits were compared to the gating behavior of Shaker potassium channels without N-type inactivation. Each model was tested to see if it could simulate the voltage dependence of the steady state open probability, and the kinetics of the single-channel currents, macroscopic ionic currents and macroscopic gating currents using a single set of parameters. Activation schemes based upon four identical single-step activation processes were found to be incompatible with the experimental results, as were those involving a concerted, opening transition. A model where the opening of the channel requires two conformational changes in each of the four subunits can adequately account for the steady state and kinetic behavior of the channel. In this model, the gating in each subunit is independent except for a stabilization of the open state when all four subunits are activated, and an unstable closed conformation that the channel enters after opening. A small amount of negative cooperativity between the subunits must be added to account quantitatively for the dependence of the activation time course on holding voltage.     

Gating and flickery block differentially affected by rubidium in homomeric KCNQ1 and heteromeric KCNQ1/KCNE1 potassium channels

The voltage-gated potassium channel KCNQ1 associates with the small KCNE1 subunit to form the cardiac IKs delayed rectifier potassium current and mutations in both genes can lead to the long QT syndrome. KCNQ1 can form functional homotetrameric channels, however with drastically different biophysical properties compared to heteromeric KCNQ1/KCNE1 channels. We analyzed gating and conductance of these channels expressed in Xenopus oocytes using the two-electrode voltage-clamp and the patch-clamp technique and high extracellular potassium (K) and rubidium (Rb) solutions. Inward tail currents of homomeric KCNQ1 channels are increased about threefold upon substitution of 100 mM potassium with 100 mM rubidium despite a smaller rubidium permeability, suggesting an effect of rubidium on gating. However, the kinetics of tail currents and the steady-state activation curve are only slightly changed in rubidium. Single-channel amplitude at negative voltages was estimated by nonstationary noise analysis, and it was found that rubidium has only a small effect on homomeric channels (1.2-fold increase) when measured at a 5-kHz bandwidth. The apparent single-channel conductance was decreased after filtering the data at lower cutoff frequencies indicative of a relatively fast "flickery/block" process. The relative conductance in rubidium compared to potassium increased at lower cutoff frequencies (about twofold at 10 Hz), suggesting that the main effect of rubidium is to decrease the probability of channel blockage leading to an increase of inward currents without large changes in gating properties. Macroscopic inward tail currents of heteromeric KCNQ1/KCNE1 channels in rubidium are reduced by about twofold and show a pronounced sigmoidal time course that develops with a delay similar to the inactivation process of homomeric KCNQ1, and is indicative of the presence of several open states. The single channel amplitude of heteromers is about twofold smaller in rubidium than in potassium at a bandwidth of 5 kHz. Filtering at lower cutoff frequencies reduces the apparent single-channel conductance, the ratio of the conductance in rubidium versus potassium is, however, independent of the cutoff frequency. Our results suggest the presence of a relatively rapid process (flicker) that can occur almost independently of the gating state. Occupancy by rubidium at negative voltages favors the flicker-open state and slows the flickering rate in homomeric channels, whereas rubidium does not affect the flickering in heteromeric channels. The effects of KCNE1 on the conduction properties are consistent with an interaction of KCNE1 in the outer vestibule of the channel.     

A plant Shaker-like K+ channel switches between two distinct gating modes resulting in either inward-rectifying or "leak" current

Among the Shaker-like plant potassium channels, AKT2 is remarkable because it mediates both instantaneous "leak-like" and time-dependent hyperpolarisation-activated currents. This unique gating behaviour has been analysed in Xenopus oocytes and in COS and Chinese hamster ovary cells. Whole-cell and single-channel data show that (i) AKT2 channels display two distinct gating modes, (ii) the gating of a given AKT2 channel can change from one mode to the other and (iii) this conversion is under the control of post-translational factor(s). This behaviour is strongly reminiscent of that of the KCNK2 channel, recently reported to be controlled by its phosphorylation state.     

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.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

Sodium ionic and gating currents in mammalian cells

Ionic and gating currents from voltage-gated sodium channels were recorded in mouse neuroblastoma cells using the path-clamp technique. Displacement currents were measured from whole-cell recordings. The gating charge displaced during step depolarizations increased with the applied membrane potential and reached saturating levels above 20 mV Prolonged large depolarizations produced partial immobilization of the gating charge, and only about one third of the displaced charge was quickly reversed upon return to negative holding potentials. The activation and inactivation properties of macroscopic sodium currents were characterized by voltage-clamp analysis of large outside-out patches and the single-channel conductance was estimated from nonstationary noise analysis. The general properties of the sodium channels in mouse neuroblastoma cells are very similar to those previously reported for various preparations of invertebrate and vertebrate nerve cells. Offprint requests to: O. Moran     

A model of the gating of ion channels

The gating of ion channels in biological membranes has usually been described in terms of Markov transitions between a few discrete open or closed states. Such models predict that the distributions of open and closed durations decay as a sum of exponential terms. Recent experimental data have indicated that certain channels are not easily described by these models. We show that distributions of open and closed times similar to those seen experimentally are predicted by a model that involves only one open and closed state but that assumes the activation energy of the gating process to be stochastic. This model involves only a few parameters and these have direct physical interpretations. Measurements of the correlation between the durations of successive open or closed events is shown to provide an experimental method for distinguishing between this and other models.     

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, macroscopic, and gating currents from sodium channels in the squid giant axon.

Single-channel, macroscopic ionic, and macroscopic gating currents were recorded from the voltage-dependent sodium channel using patch-clamp techniques on the cut-open squid giant axon. To obtain a complete set of physiological measurements of sodium channel gating under identical conditions, and to facilitate comparison with previous work, comparison was made between currents recorded in the absence of extracellular divalent cations and in the presence of physiological concentrations of extracellular Ca2+ (10 mM) and Mg2+ (50 mM). The single-channel currents were well resolved when divalent cations were not included in the extracellular solution, but were decreased in amplitude in the presence of Ca2+ and Mg2+ ions. The instantaneous current-voltage relationship obtained from macroscopic tail current measurements similarly was depressed by divalents, and showed a negative slope-conductance region for inward current at negative potentials. Voltage dependent parameters of channel gating were shifted 9-13 mV towards depolarized potentials by external divalent cations, including the peak fraction of channels open versus voltage, the time constant of tail current decline, the prepulse inactivation versus voltage relationship, and the charge-voltage relationship for gating currents. The effects of divalent cations are consistent with open channel block by Ca2+ and Mg2+ together with divalent screening of membrane charges.     

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.