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.     

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.     

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.     

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.     

Single-channel currents from diethylpyrocarbonate-modified NMDA receptors in cultured rat brain cortical neurons

The role of histidine residues in the function of N-methyl-D-aspartate (NMDA)-activated channels was tested with the histidine-modifying reagent diethylpyrocarbonate (DEP) applied to cells and membrane patches from rat brain cortical neurons in culture. Channels in excised outside-out patches that were treated with 3 mM DEP for 15-30 s (pH 6.5) showed an average 3.4-fold potentiation in steady state open probability when exposed to NMDA and glycine. Analysis of the underlying alterations in channel gating revealed no changes in the numbers of kinetic states: distributions of open intervals were fitted with three exponential components, and four components described the shut intervals, in both control and DEP-modified channels. However, the distribution of shut intervals was obviously different after DEP treatment, consistent with the single-channel current record. After modification, the proportion of long shut states was decreased while the time constants were largely unaffected. Burst kinetics reflected these effects with an increase in the average number of openings/burst from 1.5 (control) to 2.2 (DEP), and a decrease in the average interburst interval from 54.1 to 38.2 ms. These effects were most likely due to histidine modification because other reagents (n- acetylimidazole and 2,4,6-trinitrobenzene 1-sulfonic acid) that are specific for residues other than histidine failed to reproduce the effects of DEP, whereas hydroxylamine could restore channel open probability to control levels. In contrast to these effects on channel gating, DEP had no effect on average single-channel conductance or reversal potential under bi-ionic (Na+:Cs+) conditions. Inhibition by zinc was also unaffected by DEP. We propose a channel gating model in which transitions between single- and multi-opening burst modes give rise to the channel activity observed under steady state conditions. When adjusted to account for the effects of DEP, this model suggests that one or more extracellular histidine residues involved in channel gating are associated with a single kinetic state.     

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.     

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.     

A note on correlations in single ion channel records

General expressions are derived for the correlation coefficients between the length of an opening and that of the nth subsequent opening for a single ion channel. Analogous results are given for the correlation between shut times, and between an open time and subsequent shut times. An alternative derivation of the results of Fredkin et al. (in Proc. Berkeley Conf. in honor of Neyman & Kiefer, vol. 1, pp. 269-289 (1985] is given, and their results are extended to the case where openings occur in bursts. Expressions are given for the correlation between the first and nth opening in a burst, between the lengths of bursts, and between the number of openings per burst. Each of these sorts of correlation can give information about the connections that exist between the various states of the system; interpretations of the correlations are discussed. Expressions are derived for the distributions of the nth open time, shut time, burst length, etc. following the application of a perturbation (e.g. a voltage jump or a concentration jump). It is shown that these distributions will all be the same (namely the equilibrium distribution) only in the case where the openings, burst lengths, etc. are not correlated. Certain reaction schemes predict a component in the distribution of the number of openings per burst that has a unit mean (i.e. a component of isolated single openings). For some schemes this component is predicted to have zero amplitude, in principle, whereas in others it may be quite prominent. The presence or absence of this component can give information about the way in which the various states of the system are connected. The interpretation in terms of mechanism is discussed.     

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.     

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.     

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.     

Cholinergic excitation of smooth muscles: Multiple signaling pathways linking M2 and M3 muscarinic receptors to cationic channels

Acetylcholine, the main neurotransmitter of the parasympathetic nervous system, depolarizes various smooth muscles and initiates their contraction via activating muscarinic cholinergic receptors. In most visceral smooth muscle tissues, such as the gastrointestinal tract, airways, and the urinary system, muscarinic receptors are comprised of predominant M₂ (about 80%)and minor M₃ (about 20%) subtypes. Cholinergic excitation is generally mediated by the opening of ion channels selective for monovalent cations (under physiological conditions, Na⁺ and K⁺); among them the cationic channel of an about 60 pS unitary conductance has been recently identified as the main target for acetylcholine action. The signal transduction leading to channel opening is very complex and involves activation of G_o protein (an M₂ effect), activation of phospholipase C (an M₃ effect), and [Ca²⁺]_i and voltage dependence of channel opening. These multiple signaling pathways were difficult to reconcile with the channel gating mechanisms since only a simplified two-state channel mechanism (e.g., one open and one shut state) was until recently available. However, our recent studies of channel gating in isolated outside-out membrane patches revealed a greater complexity. Thus, this cationic channel shows transitions between at least eight states, four open and four shut, with strong connections between adjacent shut and open states. Therefore, four pairs of connected states have been identified, which showed voltage-dependent transitions in each pair of shut/open states. Since the membrane potential did not affect the relative proportions between the pairs, we have assumed that these effects are controlled by ligands that bind to the channel and, thus, stabilize its various open conformations. In this work, direct tests of the above hypothesis have been performed, and their results showed that spontaneous brief channel gating exists in the absence of receptor or G-protein activation, which is strongly voltage-dependent (increasing at depolarized potentials). Furthermore, this activity was potentiated at a low agonist concentration, while channel openings generally remained brief. An increasing receptor occupancy by the agonist produced long channel openings, indicating a shift of gating towards a long open/brief shut pair of the channel states. These findings are interpreted in the context of the established signal transduction pathways;certain predictions for the whole-cell current are also examined.Neirofiziologiya/Neurophysiology, Vol. 36, Nos. 5/6, pp. 446–454, September–December, 2004.This revised version was published online in April 2005 with a corrected cover date and copyright year.     

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.     

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.     

Mechanism of Maxi-K Channel Activation by Dehydrosoyasaponin-I

Dehydrosoyasaponin-I (DHS-I) is a potent activator of high-conductance, calcium-activated potassium (maxi-K) channels. Interaction of DHS-I with maxi-K channels from bovine aortic smooth muscle was studied after incorporating single channels into planar lipid bilayers. Nanomolar amounts of intracellular DHS-I caused the appearance of discrete episodes of high channel open probability interrupted by periods of apparently normal activity. Statistical analysis of these periods revealed two clearly separable gating modes that likely reflect binding and unbinding of DHS-I. Kinetic analysis of durations of DHS-I-modified modes suggested DHS-I activates maxi-K channels through a high-order reaction. Average durations of DHS-I-modified modes increased with DHS-I concentration, and distributions of these mode durations contained two or more exponential components. In addition, dose-dependent increases in channel open probability from low initial values were high order with average Hill slopes of 2.4–2.9 under different conditions, suggesting at least three to four DHS-I molecules bind to maximally activate the channel. Changes in membrane potential over a 60-mV range appeared to have little effect on DHS-I binding. DHS-I modified calcium- and voltage-dependent channel gating. 100 nM DHS-I caused a threefold decrease in concentration of calcium required to half maximally open channels. DHS-I shifted the midpoint voltage for channel opening to more hyperpolarized potentials with a maximum shift of −105 mV. 100 nM DHS-I had a larger effect on voltage-dependent compared with calcium-dependent channel gating, suggesting DHS-I may differentiate these gating mechanisms. A model specifying four identical, noninteracting binding sites, where DHS-I binds to open conformations with 10–20-fold higher affinity than to closed conformations, explained changes in voltage-dependent gating and DHS-I-induced modes. This model of channel activation by DHS-I may provide a framework for understanding protein structures underlying maxi-K channel gating, and may provide a basis for understanding ligand activation of other ion channels.     

Markovian models of low and high activity levels of cardiac ryanodine receptors

The modal gating behavior of single sheep cardiac sarcoplasmic reticulum (SR) Ca2+-release/ryanodine receptor (RyR) channels was assessed. We find that the gating of RyR channels spontaneously shifts between high (H) and low (L) levels of activity and inactive periods where no channel openings are detected (I). Moreover, we find that there is evidence for multiple gating modes within H activity, which we term H1 and H2 mode. Our results demonstrate that the underlying mechanisms regulating gating are similar in native and purified channels. Dwell-time distributions of L activity were best fitted by three open and five closed significant exponential components whereas dwell-time distributions of H1 activity were best fitted by two to three open and four closed significant exponential components. Increases in cytosolic [Ca2+] cause an increase in open probability (Po) within L activity and an increase in the probability of occurrence of H activity. Open lifetime distributions within L activity were Ca2+ independent whereas open lifetime distributions within H activity were Ca2+ dependent. This study is the first attempt to estimate RyR single-channel kinetic parameters from sequences of idealized dwell-times and to develop kinetic models of RyR gating using the criterion of maximum likelihood. We propose distinct kinetic schemes for L, H1, and H2 activity that describe the major features of sheep cardiac RyR channel gating at these levels of activity.     

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- and cold-dependent gating of single TRPM8 ion channels

Transient receptor potential (TRP) channels play critical roles in cell signaling by coupling various environmental factors to changes in membrane potential that modulate calcium influx. TRP channels are typically activated in a polymodal manner, thus integrating multiple stimuli. Although much progress has been made, the underlying mechanisms of TRP channel activation are largely unknown. The TRPM8 cation channel has been extensively investigated as a major neuronal cold sensor but is also activated by voltage, calcium store depletion, and some lipids as well as by compounds that produce cooling sensations, such as menthol or icilin. Several models of TRPM8 activation have been proposed to explain the interaction between these diverse stimuli. However, a kinetic scheme is not yet available that can describe the detailed single-channel kinetics to gain further insight into the underlying gating mechanism. To work toward this goal, we investigated voltage-dependent single-channel gating in cell-attached patches at two different temperatures (20 and 30 °C) using HEK293 cells stably expressing TRPM8. Both membrane depolarization and cooling increased channel open probability (P(o)) mainly by decreasing the duration of closed intervals, with a smaller increase in the duration of open intervals. Maximum likelihood analysis of dwell times at both temperatures indicated gating in a minimum of five closed and two open states, and global fitting over a wide range of voltages identified a seven-state model that described the voltage dependence of P(o), the single-channel kinetics, and the response of whole-cell currents to voltage ramps and steps. The major action of depolarization and cooling was to accelerate forward transitions between the same two sets of adjacent closed states. The seven-state model provides a general mechanism to account for TRPM8 activation by membrane depolarization at two temperatures and can serve as a starting point for further investigations of multimodal TRP activation.