On the stochastic properties of single ion channels

It is desirable to be able to predict, from a specified mechanism, the appearance of currents that flow through single ion channels (a) to enable interpretation of experiments in which single channel currents are observed, and (b) to allow physical meaning to be attached to the results observed in kinetic (noise and relaxation) experiments in which the aggregate of many single channel currents is observed. With this object, distributions (and the means) are derived for the length of the sojourn in any specified subset of states (e.g. all shut states). In general these are found to depend not only on the state in which the sojourn starts, but also on the state that immediately follows the sojourn. The methods described allow derivation of the distribution of, for example, (a) the number of openings, and total length of the burst of openings, that may occur during a single occupancy, and (b) the apparent gap between such bursts. The methods are illustrated by their application to two simple theories of agonist action. The Castillo-Katz (non-cooperative) mechanism predicts, for example, that the number of openings per occupancy, and the apparent burst length, are independent of agonist concentration whereas a simple cooperative mechanism predicts that both will increase with agonist concentration.     

Discrete-state theory in nerve impulse modelling.

Hodgkin-Huxley models have been the standard for describing ionic current kinetics. However, many single channel behaviors cannot be described using traditional Hodgkin-Huxley models; they can be described by expanding the Hodgkin-Huxley models to have multiple resting and inactivated states. The model, based on charge translocation between a finite number of discrete Markovian states, is a biophysical kinetic model, according to current generalizations of channel structure, capable of reproducing channel behavior. The elaboration of the model is based on the Markov process. This type of model assumes that each channel has a discrete number of states that are connected by a kinetic diagram that defines the allowable transitions between these states and the rates at which these transitions occur. The application of the model presented here leads to results in accordance with the experimental data regarding the shape and characteristics of the nerve impulse registered along the nerve fibre. Unlike the traditional Hodgkin-Huxley models, the model based on the Markov processes has the advantage of removing the empirical equations, simplifying the computation of the membrane potential and revealing the single-channel variables. The average behavior is obtained by the repetition in one channel of the same stimulus, a number of times equal to the number of channels, which means that the macroscopic variables are predictable by the repetition, a certain number of times, of the same observations in a single channel.     

A nonequilibrium binary elements-based kinetic model for benzodiazepine regulation of GABAA receptors

Ion channels, like many other proteins, are composed of multiple structural domains. A stimulus that impinges on one domain, such as binding of a ligand to its recognition site, can influence the activity of another domain, such as a transmembrane channel gate, through interdomain interactions. Kinetic schemes that describe the function of interacting domains typically incorporate a minimal number of states and transitions, and do not explicitly model interactions between domains. Here, we develop a kinetic model of the GABA_A receptor, a ligand-gated ion channel modulated by numerous compounds including benzodiazepines, a class of drugs used clinically as sedatives and anxiolytics. Our model explicitly treats both the kinetics of distinct functional domains within the receptor and the interactions between these domains. The model describes not only how benzodiazepines that potentiate GABA_A receptor activity, such as diazepam, affect peak current dose–response relationships in the presence of desensitization, but also their effect on the detailed kinetics of current activation, desensitization, and deactivation in response to various stimulation protocols. Finally, our model explains positive modulation by benzodiazepines of receptor currents elicited by either full or partial agonists, and can resolve conflicting observations arguing for benzodiazepine modulation of agonist binding versus channel gating.     

Characterizing Single-Channel Behavior of GluA3 Receptors

AMPA receptors play a major role in excitatory neurotransmission in the CNS and are involved in numerous neurological disorders. Agonists bind to each of four bilobed LBDs of this tetrameric receptor, and upon binding, the lobes close to envelope the agonist, leading to channel activation. However, AMPA receptors exhibit complex activation kinetics, the mechanism of which has not yet been determined. We report here single-channel studies of a homomeric AMPA receptor (GluA3) activated by the full agonist, glutamate, and a partial agonist, fluorowillardiine. Both agonists activate the channel to the same three open conductance levels but with different open probabilities in each level. The closed probability (P_c) varied within records, particularly at low agonist concentrations. By sorting discrete segments of the record according to P_c using the X-means algorithm, we defined five modes of activity. The kinetic behavior could then be analyzed for both agonists over a range of agonist concentrations with a relatively simple model (three closed states and two open states for each open conductance level). The structural mechanism underlying the modal behavior is not clear; however, it occurs on a timescale consistent with hydrogen bonding across the lobe interface in the LBD.     

Single-Channel Kinetic Analysis for Activation and Desensitization of Homomeric 5-HT3A Receptors

The 5-HT₃A receptor is a member of the Cys-loop family of ligand-gated ion channels. To perform kinetic analysis, we mutated the 5-HT3A subunit to obtain a high-conductance form so that single-channel currents can be detected. At all 5-HT concentrations (>0.1 μM), channel activity appears as openings in quick succession that form bursts, which coalesce into clusters. By combining single-channel and macroscopic data, we generated a kinetic model that perfectly describes activation, deactivation, and desensitization. The model shows that full activation arises from receptors with three molecules of agonist bound. It reveals an earlier conformational change of the fully liganded receptor that occurs while the channel is still closed. From this pre-open closed state, the receptor enters into an open-closed cycle involving three open states, which form the cluster whose duration parallels the time constant of desensitization. A similar model lacking the pre-open closed state can describe the data only if the opening rates are fixed to account for the slow activation rate. The application of the model to M4 mutant receptors shows that position 10′ contributes to channel opening and closing rates. Thus, our kinetic model provides a foundation for understanding structural bases of activation and drug action.     

Opening mechanism of a cyclic nucleotide-gated channel based on analysis of single channels locked in each liganded state.

Cyclic nucleotide-gated channels contain four subunits, each with a binding site for cGMP or cAMP in the cytoplasmic COOH-terminal domain. Previous studies of the kinetic mechanism of activation have been hampered by the complication that ligands are continuously binding and unbinding at each of these sites. Thus, even at the single channel level, it has been difficult to distinguish changes in behavior that arise from a channel with a fixed number of ligands bound from those that occur upon the binding and unbinding of ligands. For example, it is often assumed that complex behaviors like multiple conductance levels and bursting occur only as a consequence of changes in the number of bound ligands. We have overcome these ambiguities by covalently tethering one ligand at a time to single rod cyclic nucleotide-gated channels (Ruiz, ML., and J.W. Karpen. 1997. Nature. 389:389-392). We find that with a fixed number of ligands locked in place the channel freely moves between three conductance states and undergoes bursting behavior. Furthermore, a thorough kinetic analysis of channels locked in doubly, triply, and fully liganded states reveals more than one kinetically distinguishable state at each conductance level. Thus, even when the channel contains a fixed number of bound ligands, it can assume at least nine distinct states. Such complex behavior is inconsistent with simple concerted or sequential allosteric models. The data at each level of liganding can be successfully described by the same connected state model (with different rate constants), suggesting that the channel undergoes the same set of conformational changes regardless of the number of bound ligands. A general allosteric model, which postulates one conformational change per subunit in both the absence and presence of ligand, comes close to providing enough kinetically distinct states. We propose an extension of this model, in which more than one conformational change per subunit can occur during the process of channel activation.     

On identification of Na+ channel gating schemes using moving-average filtered hidden Markov models

 Transitions between distinct kinetic states of an ion channel are described by a Markov process. Hidden Markov models (HMM) have been successfully applied in the analysis of single ion channel recordings with a small signal-to-noise ratio. However, we have recently shown that the anti-aliasing low-pass filter misleads parameter estimation. Here, we show for the case of a Na⁺ channel recording that the standard HMM do neither allow parameter estimation nor a correct identification of the gating scheme. In particular, the number of closed and open states is determined incorrectly, whereas a modified HMM considering the anti-aliasing filter (moving-average filtered HMM) is able to reproduce the characteristic properties of the time series and to perform gating scheme identification. Received: 11 February 1999 / Revised version: 18 June 1999 / Accepted: 21 June 1999     

Activation of a nicotinic acetylcholine receptor.

We studied activation of the nicotinic acetylcholine (ACh) receptor on cells of a mouse clonal muscle cell line (BC3H1). We analyzed single-channel currents through outside-out patches elicited with various concentrations of acetylcholine (ACh), carbamylcholine (Carb) and suberyldicholine (Sub). Our goal is to determine a likely reaction scheme for receptor activation by agonist and to determine values of rate constants for transitions in that scheme. Over a wide range of agonist concentrations the open-time duration histograms are not described by single exponential functions, but are well-described by the sum of two exponentials, a brief-duration and a long-duration component. At high concentration, channel openings occur in groups and these groups contain an excess number of brief openings. We conclude that there are two open states of the ACh receptor with different mean open times and that a single receptor may open to either open state. The concentration dependence of the numbers of brief and long openings indicates that brief openings do not result from the opening of channels of receptors which have only one agonist molecule bound to them. Closed-time duration histograms exhibit a major brief component at low concentrations. We have used the method proposed by Colquhoun and Sakmann (1981) to analyze these brief closings and to extract estimates for the rates of channel opening (beta) and agonist dissociation (k-2). We find that this estimate of beta does not predict our closed-time histograms at high agonist concentration (ACh: 30-300 microM; Carb: 300-1,000 microM). We conclude that brief closings at low agonist concentrations do not result solely from transitions between the doubly-liganded open and the doubly-liganded closed states. Instead, we postulate the existence of a second closed-channel state coupled to the open state.     

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.     

Channel properties of the purified acetylcholine receptor from Torpedo californica reconstituted in planar lipid bilayer membranes

The electrophysiological properties of the cation channel of the purified nicotinic acetylcholine receptor (AChR) reconstituted in planar lipid bilayers were characterized. Single-channel currents were activated by acetylcholine, carbamylcholine and suberyldicholine. The single channel conductance (28 pS in 0.3 M NaCl) was ohmic and independent of the agonist. Single channel currents increased with Na+ concentration to a maximum conductance of 95 pS and showed a half-saturation point of 395 mM. The apparent ion selectivity sequence, derived from single-channel current recordings, is: NH+4 greater than Cs+ greater than Rb+ greater than or equal to Na+ Cl-, F-, SO2-(4). The distribution of channel open times was fit by a sum of two exponentials, reflecting the existence of at least two distinct open states. The time constants depend on the choice of agonist, being consistently longer for suberyldicholine than for carbamylcholine. Similar channel properties were recorded in bilayers formed from monolayers at the tip of patch pipets . Single-channel currents occur in paroxysms of channel activity followed by quiescent periods. This pattern is more pronounced as the agonist concentration increases, and is reflected in histograms of channel-opening frequencies. Computer simulations with a three-state model, consisting of two closed (unliganded and liganded) and one open state, do not resemble the recorded pattern of channel activity, especially at high agonist concentration. Inclusion of a desensitized liganded state reproduces the qualitative features of channel recordings. The occurrence of paroxysms of channel activity thus seems to result from the transit of AChR through its active conformation, from which it can open several times before desensitizing.     

Acetylcholine receptor in planar lipid bilayers. Characterization of the channel properties of the purified nicotinic acetylcholine receptor from Torpedo californica reconstituted in planar lipid bilayers

The properties of the channel of the purified acetylcholine receptor (AChR) were investigated after reconstitution in planar lipid bilayers. The time course of the agonist-induced conductance exhibits a transient peak that relaxes to a steady state value. The macroscopic steady state membrane conductance increases with agonist concentration, reaching saturation at 10(-5) M for carbamylcholine (CCh). The agonist-induced membrane conductance was inhibited by d-tubocurarine (50% inhibition, IC50, at approximately 10(-6) M) and hexamethonium (IC50 approximately 10(-5) M). The single channel conductance, gamma, is ohmic and independent of the agonist. At 0.3 M monovalent salt concentrations, gamma = 28 pS for Na+, 30 pS for Rb+, 38 pS for Cs+, and 50 pS for NH+4. The distribution of channel open times was fit by a sum of two exponentials, reflecting the existence of two distinct open states. tau o1 and tau o2, the fast and slow components of the distribution of open times, are independent of the agonist concentration: for CCh this was verified in the range of 10(-6) M less than C less than 10(-3)M. tau 01 and tau o2 are approximately three times longer for suberyldicholine ( SubCh ) than for CCh. tau o1 and tau o2 are moderately voltage dependent, increasing as the applied voltage in the compartment containing agonist is made more positive with respect to the other. At desensitizing concentrations of agonist, the AChR channel openings occurred in a characteristic pattern of sudden paroxysms of channel activity followed by quiescent periods. A local anesthetic derivative of lidocaine ( QX -222) reduced both tau o1 and tau o2. This effect was dependent on both the concentration of QX -222 and the applied voltage. Thus, the AChR purified from Torpedo electric organ and reconstituted in planar lipid bilayers exhibits ion conduction and kinetic and pharmacological properties similar to AChR in intact muscle postsynaptic membranes.     

Modeling Facilitative Sugar Transporters: Transitions Between Single and Double Ligand Occupancy of Multiconformational Channel Models Explain Anomalous Kinetics

The four-state simple carrier model (SCM) is employed to describe ligand translocation by diverse passive membrane transporters. However, its application to systems like facilitative sugar transporters (GLUTs) is controversial: unidirectional fluxes under zero-trans and equilibrium-exchange experimental conditions fit a SCM, but flux data from infinite-cis and infinite-trans experiments appear not to fit the same SCM. More complex kinetic models have been proposed to explain this ``anomalous' behavior of GLUTs, but none of them accounts for all the experimental findings. We propose an alternative model in which GLUTs are channels subject to conformational transitions, and further assume that the results from zero-trans and equilibrium-exchange experiments as well as trans-effects corresponds to a single-occupancy channel regime, whereas the results from the infinite-cis and infinite-trans experiments correspond to a regime including higher channel occupancies. We test the plausibility of this hypothesis by studying a kinetic model of a two-site channel with two conformational states. In each state, the channel can bind the ligand from only one of the compartments. Under single-occupancy, for conditions corresponding to zero-trans and equilibrium-exchange experiments, the model behaves as a SCM capable of exhibiting trans-stimulations. For a regime including higher degrees of occupancy and infinite-cis and infinite-trans conditions, the same channel model can exhibit a behavior qualitatively similar to a SCM, albeit with kinetic parameters different from those for the single-occupancy regime. Numerical results obtained with our model are consistent with available experimental data on facilitative glucose transport across erythrocyte membranes. Hence, if GLUTs are multiconformational channels, their particular kinetic properties can result from transitions between single and double channel occupancies. Received: 12 April 1995/Revised: 28 August 1995     

Exploring the Mechanism of General Anesthesia: Kinetic Analysis of GABAA Receptor Electrophysiology

A kinetic model of the effect of agonist and anesthetics on ligand-gated ion channels, developed in earlier work, is further refined and used to predict traces observed in fast-perfusion electrophysiological studies on recombinant GABA_A receptors under a wide range of agonist and/or anesthetic concentrations. The model incorporates only three conformational states (resting, open, and desensitized) but allows for the modulation of the conformational free energy landscape connecting these states resulting from adsorption of agonist and/or anesthetic to the bilayer in which the protein is embedded. The model is shown to reproduce the diverse and complex features of experimental traces remarkably well, including both anesthetic-induced and agonist-induced traces, as well as the modulation of agonist-induced traces by anesthetic, either coapplied or continuously present. The solutions to the kinetic equations, which give the time-dependence of each of the nine protein states (three ligation states for each of the three conformations), describe the flow of probability among these states and thus reveal the kinetic underpinnings of the traces. Many of the parameters in the model, such as the desorption rate constants of anesthetic and agonist, are directly related to model-independent experimental measurements and thus can serve as a definitive test of its validity.     

Microscopic model for selective permeation in ion channels.

Ionic permeation in the selectivity filter of ion channels is analyzed by a microscopic model based on molecular kinetic theory. The energy and flux equations are derived by assuming that: (a) the selectivity filter is formed by a symmetrical array of carbonyl groups; (b) ion movement is near the axis of the channel; (c) a fraction of water molecules is separated from the ion while it moves across the selectivity filter; (d) the applied voltage drops linearly across the selectivity filter; (e) ions move independently. Energy profiles, single channel conductances, and the degree of hydration of K+ in a hypothetical K+ channel are examined by varying the following microscopic parameters: ion radius and mass, channel radius, number of effective water dipoles, and number of carbonyl groups. The i-V curve is linear up to +/- 170 mV. If the positions of energy maxima and minima are fixed, this linear range is reduced to +/- 50 mV. Channel radius and ion-water interactions are found to be two major channel structural determinants for selectivity sequences. Both radius and mass of an ion are important in selectivity mediated by these interactions. The theory predicts a total of 15 possible kinetic selectivity sequences for alkali cations in ion channels with a single selectivity filter.     

Stoichiometry of Kir channels with phosphatidylinositol bisphosphate

Phosphatidylinositol bisphosphate (PIP2) is the most abundant phosphoinositide in the plasma membranes of cells and its interaction with many ion channel proteins has proven to be a critical factor enabling ion channel gating. All members of the inwardly rectifying potassium (Kir) channel family depend on PIP2 for their activity, displaying distinct affinities and stereospecificities of interaction with the phosphoinositide. Here, we explored the stoichiometry of Kir channels with PIP2. We first showed that PIP2 regulated the activity of Kir3.4 channels mainly by altering their bursting behavior. Detailed burst analysis indicates that the channels assumed up to four open states and a connectivity of four between open and closed states depending on the available PIP2 levels. Moreover, by controlling the number of PIP2-sensitive subunits in the stoichiometry of a tetrameric Kir2.1 channel, we showed that characteristic channel activity was obtained when at least two wild-type subunits were present. Our studies support a kinetic model for gating of Kir channels by PIP2, where each of the four open states corresponds to the channel activated by one to four PIP2 molecules.     

Estimating kinetic constants from single channel data.

The process underlying the opening and closing of ionic channels in biological or artificial lipid membranes can be modeled kinetically as a time-homogeneous Markov chain. The elements of the chain are kinetic states that can be either open or closed. A maximum likelihood procedure is described for estimating the transition rates between these states from single channel data. The method has been implemented for linear kinetic schemes of fewer than six states, and is suitable for nonstationary data in which one or more independent channels are functioning simultaneously. It also provides standard errors for all estimates of rate constants and permits testing of smoothly parameterized subhypotheses of a general model. We have illustrated our approach by analysis of single channel data simulated on a computer and have described a procedure for analysis of experimental data.     

Effect of protein kinase A-induced phosphorylation on the gating mechanism of the brain Na+ channel: model fitting to whole-cell current traces.

The activity of the voltage-gated Na+ channel is subjected to modulation through covalent modifications. It has been previously shown that brain Na+ currents are reduced following the activation of the protein kinase A (PKA) pathway, but the effect of the phosphorylation on the gating mechanism of the channel has not been demonstrated so far. In this study, we analyze the whole-cell Na+ current recorded in the absence or presence of forskolin, which stimulates the PKA pathway. A minimal molecular model of the gating mechanism of the Na+ channel is defined to fit the experimental data: it consists of three closed states, one open state, and two inactivated states. We experimentally demonstrate that the kinetics of inactivation from the closed states are not affected by phosphorylation. The results obtained by computer fitting indicate that, among all the kinetic parameters describing the transitions between states, only one parameter is significantly modified in the presence of forskolin, and corresponds to the acceleration of the inactivation from the open state. This conclusion is supported by the analysis of current traces obtained from cells in the presence of a phosphatase inhibitor or loaded with the PKA catalytic unit, and is in agreement with previously reported single channel records.     

Stoichiometry of Kir channels with phosphatidylinositol bisphosphate

Phosphatidylinositol bisphosphate (PIP(2)) is the most abundant phosphoinositide in the plasma membrane of cells and its interaction with many ion channel proteins has proven to be a critical factor enabling ion channel gating. All members of the inwardly rectifying potassium (Kir) channel family depend on PIP(2) for their activity, displaying distinct affinities and stereospecificities of interaction with the phosphoinositide. Here, we explored the stoichiometry of Kir channels with PIP(2). We first showed that PIP(2) regulated the activity of Kir3.4 channels mainly by altering their bursting behavior. Detailed burst analysis indicates that the channels assumed up to four open states and a connectivity of four between open and closed states depending on the available PIP(2) levels. Moreover, by controlling the number of PIP(2)-sensitive subunits in the stoichiometry of a tetrameric Kir2.1 channel, we showed that characteristic channel activity was obtained when at least two wild-type subunits were present. Our studies support a kinetic model for gating of Kir channels by PIP(2), where each of the four open states corresponds to the channel activated by one to four PIP(2) molecules.