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.     

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.     

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.     

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.     

Single channel kinetics of a glutamate receptor.

The glutamate receptor-channel of locust muscle membrane was studied using the patch-clamp technique. Muscles were pretreated with concanavalin A to block receptor-channel desensitization, thus facilitating analysis of receptor-channel gating kinetics. Single channel kinetics were analyzed to aid in identification of the molecular basis of channel gating. Channel dwell-time distributions and dwell-time autocorrelation functions were calculated from single channel data recorded in the precence of 10-4M glutamate. Analysis of the dwell time distributions in terms of mixtures of exponential functions revealed there to be at least three open states of the receptor-channel and at least four closed states. Autocorrelation function analysis showed there to be at least three pathways linking the open states with the closed. This results in a minimal scheme for gating of the glutamate receptor-channel, which is suggestive of allosteric models of receptor-channel gating.     

Single Channel Kinetics of a Glutamate Receptor

The glutamate receptor-channel of locust muscle membrane was studied using the patch-clamp technique. Muscles were pretreated with concanavalin A to block receptor-channel desensitization, thus facilitating analysis of receptor-channel gating kinetics. Single channel kinetics were analyzed to aid in identification of the molecular basis of channel gating. Channel dwell-time distributions and dwell-time autocorrelation functions were calculated from single channel data recorded in the presence of 10^-4 M glutamate. Analysis of the dwell time distributions in terms of mixtures of exponential functions revealed there to be at least three open states of the receptor-channel and at least four closed states. Autocorrelation function analysis showed there to be at least three pathways linking the open states with the closed. This results in a minimal scheme for gating of the glutamate receptor-channel, which is suggestive of allosteric models of receptor-channel gating.     

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.     

Linking exponential components to kinetic states in Markov models for single-channel gating

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1-59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.     

Two distinct gating mechanisms in gap junction channels: CO2-sensitive and voltage-sensitive.

The chemical gating of single-gap junction channels was studied by the dual whole-cell voltage-clamp method in HeLa cells transfected with connexin43 (HeLa43) and in fibroblasts from sciatic nerves. Junctional current (Ij), single-channel conductance, and Ij kinetics were studied in cell pairs during CO2 uncoupling and recoupling at small transjunctional voltages (Vj < 35 mV: Vj gating absent) and at high Vj (Vj > 40 mV: Vj gating strongly activated). In the absence of Vj gating, CO2 exclusively caused Ij slow transitions from open to closed channel states (mean transition time: approximately 10 ms), corresponding to a single-channel conductance of approximately 120 pS. At Vj > 40 mV, Vj gating induced fast Ij flickering between open, gamma j(main state), and residual, gamma j(residual), states (transition time: approximately 2 ms). The ratio gamma j(main state)/gamma j(residual) was approximately 4-5. No obvious correlation between Ij fast flickering and CO2 treatment was noticed. At high Vj, in addition to slow Ij transitions between open and closed states, CO2 induced slow transitions between residual and closed states. During recoupling, each channel reopened by a slow transition (mean transition time: approximately 10 ms) from closed to open state (rarely from closed to residual state). Fast Ij flickering between open and residual states followed. The data are in agreement with the hypothesis that gap junction channels possess two gating mechanisms, and indicate that CO2 induces channel gating exclusively by the slow gating mechanism.     

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.     

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.     

Two-dimensional probability density analysis of single channel currents from reconstituted acetylcholine receptors and sodium channels

Two-dimensional probability density analysis of single channel current recordings was applied to two purified channel proteins reconstituted in planar lipid bilayers: Torpedo acetylcholine receptors and voltage-sensitive sodium channels from rat brain. The information contained in the dynamic history of the gating process, i.e., the time sequence of opening and closing events was extracted from two-dimensional distributions of transitions between identifiable states. This approach allows one to identify kinetic models consistent with the observables. Gating of acetylcholine receptors expresses "memory" of the transition history: the receptor has two channel open (O) states; the residence time in each of them strongly depends on both the preceding open time and the intervening closed interval. Correspondingly, the residence time in the closed (C) states depends on both the preceding open time and the preceding closed time. This result confirms the scheme that considers, at least, two transition pathways between the open and closed states and extends the details of the model in that it defines that the short-lived open state is primarily entered from long-lived closed states while the long-lived open state is accessed mainly through short-lived closed states. Since ligand binding to the acetylcholine-binding sites is a reaction with channel closed states, we infer that the longest closed state (approximately 19 ms) is unliganded, the intermediate closed state (approximately 2 ms) is singly liganded and makes transitions to the short open state (approximately 0.5 ms) and the shortest closed state (approximately 0.4 ms) is doubly liganded and isomerizes to long open states (approximately 5 ms). This is the simplest interpretation consistent with available data. In contrast, sodium channels modified with batrachotoxin to eliminate inactivation show no correlation in the sequence of channel opening and closing events, i.e., have no memory of the transition history. This result is, therefore, consistent with any kinetic scheme that considers a single transition pathway between open and closed states, and confirms the C-C-O model previously inferred from one-dimensional distribution analysis. The strategy described should be of general validity in the analysis of single channel events from channel proteins in both natural and reconstituted membranes.     

Gating kinetics of four classes of voltage-dependent K+ channels in pheochromocytoma cells

Clonal pheochromocytoma (PC-12) cells have four different types of voltage-dependent K+ channels whose activation does not require high concentrations of Ca++ on the cytoplasmic side of the membrane (Hoshi, T., and R. W. Aldrich, 1988, Journal of General Physiology, 91:73-106). The durations of open and closed events of these four different types of voltage-dependent K+ channels were measured using the excised configuration of the patch-clamp method. The open durations of a class of K+ channels termed the Kz channel, which activates rapidly and inactivates slowly in response to depolarizing pulses, had two exponential components. The closed durations of the Kz channel had at least four exponential components. The time constants of the fastest of the two exponential components in the closed durations were very similar to those of the two exponential components present in the first-latency distribution. The first latencies of the Kz channel decreased steeply with depolarization, contributing to the increased probability of the channel being open with depolarization. The Kz channel also had a very slow gating process that resulted in a clustering of blank sweeps. A gating scheme containing two open states and five closed states is consistent with the observations. The Ky channel had one exponential component in the open durations and three exponential components in the closed durations. The first latencies varied greatly depending on the prepulse voltage and duration. The results were consistent with a sequential model with a large number of closed states and one open state. The Kx channel, which required large hyperpolarizing prepulses to remove steady state inactivation and did not show inactivation with maintained depolarization, had two exponential components in the open durations and three exponential components in the closed durations. The burst behavior of the Kx channel involved many more than two states. The transient Kw channel had one exponential component in the open durations and the mean open time increased with depolarization. The first latencies of the Kw channel were steeply dependent on the voltage, decreasing with depolarization.     

Thermodynamically irreversible gating of ryanodine receptors in situ revealed by stereotyped duration of release in Ca(2+) sparks

For a single or a group of Markov channels gating reversibly, distributions of open and closed times should be the sum of positively weighted decaying exponentials. Violation of this microscopic reversibility has been demonstrated previously on a number of occasions at the single channel level, and has been attributed to possible channel coupling to external sources of free energy. Here we show that distribution of durations of Ca(2+) release underlying Ca(2+) sparks in intact cardiac myocytes exhibits a prominent mode at approximately 8 ms. Analysis of the cycle time for repetitive sparks at hyperactive sites revealed no intervals briefer than approximately 35 ms and a mode at approximately 90 ms. These results indicate that, regardless of whether Ca(2+) sparks are single-channel or multi-channel in origin, they are generated by thermodynamically irreversible stochastic processes. In contrast, data from planar lipid bilayer experiments were consistent with reversible gating of RyR under asymmetric cis (4 microM) and trans Ca(2+) (10 mM), suggesting that the irreversibility for Ca(2+) spark genesis may reside at a supramolecular level. Modeling suggests that Ca(2+)-induced Ca(2+) release among adjacent RyRs may couple the external energy derived from Ca(2+) gradients across the SR to RyR gating in situ, and drive the irreversible generation of Ca(2+) sparks.     

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.     

Voltage-dependent Gating of Single Wild-Type and S4 Mutant KAT1 Inward Rectifier Potassium Channels

The voltage-dependent gating mechanism of KAT1 inward rectifier potassium channels was studied using single channel current recordings from Xenopus oocytes injected with KAT1 mRNA. The inward rectification properties of KAT1 result from an intrinsic gating mechanism in the KAT1 channel protein, not from pore block by an extrinsic cation species. KAT1 channels activate with hyperpolarizing potentials from −110 through −190 mV with a slow voltage-dependent time course. Transitions before first opening are voltage dependent and account for much of the voltage dependence of activation, while transitions after first opening are only slightly voltage dependent. Using burst analysis, transitions near the open state were analyzed in detail. A kinetic model with multiple closed states before first opening, a single open state, a single closed state after first opening, and a closed-state inactivation pathway accurately describes the single channel and macroscopic data. Two mutations neutralizing charged residues in the S4 region (R177Q and R176L) were introduced, and their effects on single channel gating properties were examined. Both mutations resulted in depolarizing shifts in the steady state conductance–voltage relationship, shortened first latencies to opening, decreased probability of terminating bursts, and increased burst durations. These effects on gating were well described by changes in the rate constants in the kinetic model describing KAT1 channel gating. All transitions before the open state were affected by the mutations, while the transitions after the open state were unaffected, implying that the S4 region contributes to the early steps in gating for KAT1 channels.     

Glutamate receptor-channel gating. Maximum likelihood analysis of gigaohm seal recordings from locust muscle.

Gigaohm recordings have been made from glutamate receptor channels in excised, outside-out patches of collagenase-treated locust muscle membrane. The channels in the excised patches exhibit the kinetic state switching first seen in megaohm recordings from intact muscle fibers. Analysis of channel dwell time distributions reveals that the gating mechanism contains at least four open states and at least four closed states. Dwell time autocorrelation function analysis shows that there are at least three gateways linking the open states of the channel with the closed states. A maximum likelihood procedure has been used to fit six different gating models to the single channel data. Of these models, a cooperative model yields the best fit, and accurately predicts most features of the observed channel gating kinetics.     

A Modeling Study of T-Type Ca Channel Gating and Modulation by L-Cysteine in Rat Nociceptors

L-cysteine (L-cys) increases the amplitude of T-type Ca²⁺ currents in rat T-rich nociceptor-like dorsal root ganglia neurons. The modulation of T-type Ca²⁺ channel gating by L-cys was studied by fitting Markov state models to whole-cell currents recorded from T-rich neurons. The best fitting model tested included three resting states and inactivation from the second resting state and the open state. Inactivation and the final opening step were voltage-independent, whereas transitions between the resting states and deactivation were voltage-dependent. The transition rates between the first two resting states were an order of magnitude faster than those between the second and third resting states, and the voltage-dependency of forward transitions through resting states was two to three times greater than for analogous backward transitions. Analysis with the best fitting model suggested that L-cys increases current amplitude mainly by increasing the transition rate from resting to open and decreasing the transition rate from open to inactivated. An additional model was developed that could account for the bi-exponential time course of recovery from inactivation of the currents and the high frequency of blank sweeps in single channel recordings. This model detected basically the same effects of L-cys on channel gating as the best fitting model.     

Gating kinetics of potassium channel in rat dorsal root ganglion neurons analyzed with fractal model

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and kinetic rate constants connecting these states are constant. To study the gating kinetics of voltage-dependent K(+) channel in rat dorsal root ganglion neurons, K(+) channel current were recorded using cell-attached patch-clamp technique. The K(+) channel characteristic of kinetics were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimension D for the closed times and for the open times depend on the pipette potential. For the open and closed times of kinetic setpoint, it was found dependent on the applied pipette potential, which indicated that the ion channel gating kinetics had nonlinear kinetic properties. Thus, the open and closed durations, which had the voltage dependence of the gating of this ion channel, were well described by the fractal model.