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

Microscopic mechanism of PIEZO1 activation by pressure-induced membrane stretch

Mechanosensitive PIEZO1 ion channels open in response to membrane stretch. Yet, the underlying microscopic mechanism of this activation remains unknown. To probe this mechanism, we used cell-attached pressure-clamp recordings to measure single channel currents at different steady-state negative pipette pressures, spanning the full range of the channel's pressure sensitivity. Pressure-dependent activation occurs through a sharp reduction of the mean shut duration and through a moderate increase of the mean open duration. Across all tested pressures, the distribution of open and shut dwell times best follows sums of two and three exponential components, respectively. As the magnitude of the pressure stimulus increases, the time constants of most of these exponential components gradually change, in opposite directions for open and shut dwell times, and to a similar extent. In addition, while the relative amplitudes of fast and slow components remain unchanged for open intervals, they fully reverse for shut intervals, further reducing the mean shut duration. Using two-dimensional dwell time analysis, Markov-chain modeling, and simulations, we identified a minimal five-states model which recapitulates essential characteristics of single channel data, including microscopic reversibility, correlations between adjacent open and shut intervals, and asymmetric modulation of dwell times by pressure. This study identifies a microscopic mechanism for the activation of PIEZO1 channels by pressure-induced membrane stretch and deepens our fundamental understanding of mechanotransduction by a vertebrate member of the PIEZO channel family.

Hypothesized diprotomeric enzyme complex supported by stochastic modelling of palytoxin-induced Na/K pump channels

The sodium-potassium pump (Na⁺/K⁺ pump) is crucial for cell physiology. Despite great advances in the understanding of this ionic pumping system, its mechanism is not completely understood. We propose the use of a statistical model checker to investigate palytoxin (PTX)-induced Na⁺/K⁺ pump channels. We modelled a system of reactions representing transitions between the conformational substates of the channel with parameters, concentrations of the substates and reaction rates extracted from simulations reported in the literature, based on electrophysiological recordings in a whole-cell configuration. The model was implemented using the UPPAAL-SMC platform. Comparing simulations and probabilistic queries from stochastic system semantics with experimental data, it was possible to propose additional reactions to reproduce the single-channel dynamic. The probabilistic analyses and simulations suggest that the PTX-induced Na⁺/K⁺ pump channel functions as a diprotomeric complex in which protein-protein interactions increase the affinity of the Na⁺/K⁺ pump for PTX.

Estimating kinetic mechanisms with prior knowledge I: Linear parameter constraints

New mathematical tools are needed to incorporate existing knowledge into kinetic models of ion channels and other proteins. Salari et al. describe an algebraic transformation that can enforce linearly interdependent parameters into kinetic models in order to test new hypotheses.

To understand how ion channels and other proteins function at the molecular and cellular levels, one must decrypt their kinetic mechanisms. Sophisticated algorithms have been developed that can be used to extract kinetic parameters from a variety of experimental data types. However, formulating models that not only explain new data, but are also consistent with existing knowledge, remains a challenge. Here, we present a two-part study describing a mathematical and computational formalism that can be used to enforce prior knowledge into the model using constraints. In this first part, we focus on constraints that enforce explicit linear relationships involving rate constants or other model parameters. We develop a simple, linear algebra–based transformation that can be applied to enforce many types of model properties and assumptions, such as microscopic reversibility, allosteric gating, and equality and inequality parameter relationships. This transformation converts the set of linearly interdependent model parameters into a reduced set of independent parameters, which can be passed to an automated search engine for model optimization. In the companion article, we introduce a complementary method that can be used to enforce arbitrary parameter relationships and any constraints that quantify the behavior of the model under certain conditions. The procedures described in this study can, in principle, be coupled to any of the existing methods for solving molecular kinetics for ion channels or other proteins. These concepts can be used not only to enforce existing knowledge but also to formulate and test new hypotheses.

Elucidation of molecular kinetic schemes from macroscopic traces using system identification

Overall cellular responses to biologically-relevant stimuli are mediated by networks of simpler lower-level processes. Although information about some of these processes can now be obtained by visualizing and recording events at the molecular level, this is still possible only in especially favorable cases. Therefore the development of methods to extract the dynamics and relationships between the different lower-level (microscopic) processes from the overall (macroscopic) response remains a crucial challenge in the understanding of many aspects of physiology. Here we have devised a hybrid computational-analytical method to accomplish this task, the SYStems-based MOLecular kinetic scheme Extractor (SYSMOLE). SYSMOLE utilizes system-identification input-output analysis to obtain a transfer function between the stimulus and the overall cellular response in the Laplace-transformed domain. It then derives a Markov-chain state molecular kinetic scheme uniquely associated with the transfer function by means of a classification procedure and an analytical step that imposes general biological constraints. We first tested SYSMOLE with synthetic data and evaluated its performance in terms of its rate of convergence to the correct molecular kinetic scheme and its robustness to noise. We then examined its performance on real experimental traces by analyzing macroscopic calcium-current traces elicited by membrane depolarization. SYSMOLE derived the correct, previously known molecular kinetic scheme describing the activation and inactivation of the underlying calcium channels and correctly identified the accepted mechanism of action of nifedipine, a calcium-channel blocker clinically used in patients with cardiovascular disease. Finally, we applied SYSMOLE to study the pharmacology of a new class of glutamate antipsychotic drugs and their crosstalk mechanism through a heteromeric complex of G protein-coupled receptors. Our results indicate that our methodology can be successfully applied to accurately derive molecular kinetic schemes from experimental macroscopic traces, and we anticipate that it may be useful in the study of a wide variety of biological systems.

Unraveling the lower-level (microscopic) processes underlying the overall (macroscopic) cell response to a given stimulus is a challenging problem in cell physiology. This has been a classic problem in biophysics, where the ability to record the activity of single ion channels that generate a macroscopic ion current has allowed a measure of direct access to the underlying microscopic processes. These classic studies have demonstrated that very different groupings of the microscopic processes can yield extremely similar macroscopic responses. Biologists in fields other than biophysics are frequently confronted with the same macroscopic-to-microscopic problem, usually, however, without any direct access to the microscopic processes. Thus, the development of computational methods to deduce from the available macroscopic measurements the nature of the underlying microscopic processes can be expected to substantially advance the study of many areas of cell physiology. Toward that aim, here we have derived and tested a hybrid computational-analytical method to extract information about the microscopic processes that is hidden in macroscopic experimental traces. Our method is independent of the particular system under study, and thus can be applied to new as well as previously-recorded macroscopic traces obtained in a wide variety of biological systems.

Quantifying the cooperative subunit action in a multimeric membrane receptor

In multimeric membrane receptors the cooperative action of the subunits prevents exact knowledge about the operation and the interaction of the individual subunits. We propose a method that permits quantification of ligand binding to and activation effects of the individual binding sites in a multimeric membrane receptor. The power of this method is demonstrated by gaining detailed insight into the subunit action in olfactory cyclic nucleotide-gated CNGA2 ion channels.

Single-channel kinetics of BK (Slo1) channels

Single-channel kinetics has proven a powerful tool to reveal information about the gating mechanisms that control the opening and closing of ion channels. This introductory review focuses on the gating of large conductance Ca²⁺- and voltage-activated K⁺ (BK or Slo1) channels at the single-channel level. It starts with single-channel current records and progresses to presentation and analysis of single-channel data and the development of gating mechanisms in terms of discrete state Markov (DSM) models. The DSM models are formulated in terms of the tetrameric modular structure of BK channels, consisting of a central transmembrane pore-gate domain (PGD) attached to four surrounding transmembrane voltage sensing domains (VSD) and a large intracellular cytosolic domain (CTD), also referred to as the gating ring. The modular structure and data analysis shows that the Ca²⁺ and voltage dependent gating considered separately can each be approximated by 10-state two-tiered models with five closed states on the upper tier and five open states on the lower tier. The modular structure and joint Ca²⁺ and voltage dependent gating are consistent with a 50 state two-tiered model with 25 closed states on the upper tier and 25 open states on the lower tier. Adding an additional tier of brief closed (flicker states) to the 10-state or 50-state models improved the description of the gating. For fixed experimental conditions a channel would gate in only a subset of the potential number of states. The detected number of states and the correlations between adjacent interval durations are consistent with the tiered models. The examined models can account for the single-channel kinetics and the bursting behavior of gating. Ca²⁺ and voltage activate BK channels by predominantly increasing the effective opening rate of the channel with a smaller decrease in the effective closing rate. Ca²⁺ and depolarization thus activate by mainly destabilizing the closed states.

SOLVING ION CHANNEL KINETICS WITH THE QuB SOFTWARE

The ability to record the currents from single ion channels led to the need to extract the underlying kinetic model from such data. This inverse hidden Markov problem is difficult but led to the creation of a software suite called QuB utilizing likelihood optimization. This review presents the software. The software is open source and, in addition to solving kinetic models, has many generic database operations including report generation with publishable graphics, function fitting and scripting for new and repeated processing and AD/DA I/O. The core algorithms allow for constraints such as fixed rates or maintaining detailed balance in the model. All rate constants can be driven by a stimulus and the system can analyze nonstationary data. QuB also can analyze the kinetics of multichannel data where individual events cannot be discriminated, but the fitting algorithms utilize the signal variance as well as the mean to fit models. QuB can be applied to any data appropriately modeled with Markov kinetics and has been utilized to solve ion channels but also the movement of motor proteins, the sleep cycles in mice, and physics processes.

[Formula: see text]Special Issue Comment: This is a review about the software QuB that can extract a model from the trajectory. It is connected with the review about treatments when solving single molecules,60 and the reviews about enzymes.61,62

MATHEMATICAL TREATMENTS THAT SOLVE SINGLE MOLECULES

In this article, we talk about the ways that scientists can solve single molecule trajectories. Solving single molecules, that is, finding the model from the data, is complicated at least as much as measuring single molecules. We must filter the noise and take care of every step in the analysis when constructing the most accurate model from the data. Here, we present valuable solutions. Ways that solve clean discrete data are first presented. We review here our reduced dimensions forms (RDFs): unique models that are canonical forms of discrete data, and the statistical and numerical toolbox that builds a RDF from finite, clean, two-state data. We then review our most recent filter that "tackles" the noise when measuring two state noisy photon trajectories. The filter is a numerical algorithm with various special statistical treatments that is based on a general likelihood function that we have developed recently. We show the strengths of the filter (also over other approaches) and talk about its various new variants. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet, extensions are needed in "tackling" the noise from other data, e.g. continuous data. Only the combined procedures enable creating the most accurate model from noisy discrete trajectories from single molecules. These concepts and methods (with adjustments) are valuable also when solving continuous trajectories and fluorescence resonance energy transfer trajectories. We also present a set of simple methods that can help any scientist with treating the trajectory perhaps encouraging applying the involved methods. The involved methods will appear in software that we are developing now, helping therefore the experimentalists utilizing these methods on real data. Comparisons with other known methods in this field are made.

[Formula: see text]Special Issue Comment: This article about mathematical treatments when solving single molecules is related to the reviews in this Special Issue about measuring enzymes67 and about FRET experiments2 and about the software QUB.6

Transduction of voltage and Ca2+ signals by Slo1 BK channels

Large-conductance Ca2+ -and voltage-gated K+ channels are activated by an increase in intracellular Ca2+ concentration and/or depolarization. The channel activation mechanism is well described by an allosteric model encompassing the gate, voltage sensors, and Ca2+ sensors, and the model is an excellent framework to understand the influences of auxiliary β and γ subunits and regulatory factors such as Mg2+. Recent advances permit elucidation of structural correlates of the biophysical mechanism.

Coupling and cooperativity in voltage activation of a limited-state BK channel gating in saturating Ca2+

Voltage-dependent gating mechanisms of large conductance Ca²⁺ and voltage-activated (BK) channels were investigated using two-dimensional maximum likelihood analysis of single-channel open and closed intervals. To obtain sufficient data at negative as well as positive voltages, single-channel currents were recorded at saturating Ca²⁺ from BK channels mutated to remove the RCK1 Ca²⁺ and Mg²⁺ sensors. The saturating Ca²⁺ acting on the Ca²⁺ bowl sensors of the resulting BK_B channels increased channel activity while driving the gating into a reduced number of states, simplifying the model. Five highly constrained idealized gating mechanisms based on extensions of the Monod-Wyman-Changeux model for allosteric proteins were examined. A 10-state model without coupling between the voltage sensors and the opening/closing transitions partially described the voltage dependence of Po but not the single-channel kinetics. With allowed coupling, the model gave improved descriptions of Po and approximated the single-channel kinetics; each activated voltage sensor increased the opening rate approximately an additional 23-fold while having little effect on the closing rate. Allowing cooperativity among voltage sensors further improved the description of the data: each activated voltage sensor increased the activation rate of the remaining voltage sensors approximately fourfold, with little effect on the deactivation rate. The coupling factor was decreased in models with cooperativity from ∼23 to ∼18. Whether the apparent cooperativity among voltage sensors arises from imposing highly idealized models or from actual cooperativity will require additional studies to resolve. For both cooperative and noncooperative models, allowing transitions to five additional brief (flicker) closed states further improved the description of the data. These observations show that the voltage-dependent single-channel kinetics of BK_B channels can be approximated by highly idealized allosteric models in which voltage sensor movement increases Po mainly through an increase in channel opening rates, with limited effects on closing rates.

Intracellular Mg2+ influences both open and closed times of a native Ca2+-activated BK channel in cultured human renal proximal tubule cells

Effects of intracellular Mg2+ on a native Ca(2+)-and voltage-sensitive large-conductance K+ channel in cultured human renal proximal tubule cells were examined with the patch-clamp technique in the inside-out mode. At an intracellular concentration of Ca2+ ([Ca2+](i)) of 10(-5)-10(-4) M, addition of 1-10 mM: Mg2+ increased the open probability (P(o)) of the channel, which shifted the P(o) -membrane potential (V(m)) relationship to the negative voltage direction without causing an appreciable change in the gating charge (Boltzmann constant). However, the Mg(2+)-induced increase in P(o) was suppressed at a relatively low [Ca2+](i) (10(-5.5)-10(-6) M). Dwell-time histograms have revealed that addition of Mg2+ mainly increased P(o) by extending open times at 10(-5) M Ca2+ and extending both open and closed times simultaneously at 10(-5.5) M Ca2+. Since our data showed that raising the [Ca2+](i) from 10(-5) to 10(-4) M increased P(o) mainly by shortening the closed time, extension of the closed time at 10(-5.5) M Ca(2+) would result from the Mg(2+)-inhibited Ca(2+)-dependent activation. At a constant V(m), adding Mg2+ enhanced the sigmoidicity of the P(o)-[Ca2+](i) relationship with an increase in the Hill coefficient. These results suggest that the major action of Mg2+ on this channel is to elevate P(o) by lengthening the open time, while extension of the closed time at a relatively low [Ca2+](i) results from a lowering of the sensitivity to Ca2+ of the channel by Mg2+, which causes the increase in the Hill coefficient.

Toolbox for analyzing finite two-state trajectories

In many experiments, the aim is to deduce an underlying multisubstate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory. However, a two-state trajectory that is generated from an on-off KS contains only partial information about the KS, and so, in many cases, more than one KS can be associated with the data. We recently showed that the optimal way to solve this problem is to use canonical forms of reduced dimensions (RDs). RD forms are on-off networks with connections only between substates of different states, where the connections can have nonexponential waiting time probability density functions (WT-PDFs). In theory, only a single RD form can be associated with the data. To utilize RD forms in the analysis of the data, a RD form should be associated with the data. Here, we give a toolbox for building a RD form from a finite time, noiseless, two-state trajectory. The methods in the toolbox are based on known statistical methods in data analysis, combined with statistical methods and numerical algorithms designed specifically for the current problem. Our toolbox is self-contained-it builds a mechanism based only on the information it extracts from the data, and its implementation is fast (analyzing a 10;{6}cycle trajectory from a 30-parameter mechanism takes a couple of hours on a PC with a 2.66GHz processor). The toolbox is automated and is freely available for academic research upon electronic request.

Testing for violations of microscopic reversibility in ATP-sensitive potassium channel gating

In pancreatic beta cells, insulin secretion is tightly controlled by the cells' metabolic state via the ATP-sensitive potassium (KATP) channel. ATP is a key mediator in this signaling process, where its role as an inhibitor of KATP channels has been extensively studied. Since the channel contains an ATPase as an accessory subunit, the possibility that ATP hydrolysis mediates KATP channel opening has also been proposed. However, a rigorous test of coupling between ATP hydrolysis and channel gating has not previously been performed. In the present work, we examine whether KATP channel gating obeys detailed balance in order to determine whether ATP hydrolysis is strongly coupled to the gating of the KATP channel. Single-channel records were obtained from inside-out patches of transiently transfected HEK-293 cells. Channel activity in membrane patches with exactly one channel shows no violations of microscopic reversibility. Although KATP channel gating shows long closed times on the time scale where ATP hydrolysis takes place, the time symmetry of channel gating indicates that it is not tightly coupled to ATP hydrolysis. This lack of coupling suggests that channel gating operates close to equilibrium; although detailed balance is not expected to hold for ATP hydrolysis, it still does so in channel gating. On the basis of these results, the function of the ATPase active site in channel gating may be to sense nucleotides by differential binding of ATP and ADP, rather than to drive a thermodynamically unfavorable conformational change.

Testing for Renewal and Detailed Balance Violations in Single-Molecule Blinking Processes

This paper examines methods to test one- and two-dimensional histograms for several features including the renewal properties, detailed balance violations, and experimental condition dependences. The tests are simple to implement and allow rigorous statistical determination of the existence of these kinetic features. The tests are used to determine the lower bound on the number of measurements necessary to differentiate underlying kinetic models.

Utilizing the information content in two-state trajectories

The signal from many single-molecule experiments monitoring molecular processes, such as enzyme turnover by means of fluorescence and opening and closing of ion channel through the flux of ions, consists of a time series of stochastic "on" and "off" (or open and closed) periods, termed a two-state trajectory. This signal reflects the dynamics in the underlying multisubstate on-off kinetic scheme (KS) of the process. The determination of the underlying KS is difficult and sometimes even impossible because of the loss of information in the mapping of the multidimensional KS onto two dimensions. Here we introduce a previously undescribed procedure that efficiently and optimally relates the signal to all equivalent underlying KS. This procedure partitions the space of KS into canonical (unique) forms that can handle any KS and obtains the topology and other details of the canonical form from the data without the need for fitting. Also established are relationships between the data and the topology of the canonical form to the on-off connectivity of a KS. The suggested canonical forms constitute a powerful tool in discriminating between KS. Based on our approach, the upper bound on the information content in two-state trajectories is determined.

Gating reaction mechanisms for NMDA receptor channels

NMDA receptors (NMDARs) mediate the slow component of excitatory transmission in the CNS and play key roles in synaptic plasticity and excitotoxicity. We investigated the gating reaction mechanism of fully liganded NR1/NR2A recombinant NMDARs (expressed in Xenopus oocytes) by fitting all possible three-closed/two-open-state, noncyclic kinetic schemes to currents elicited by saturating concentrations of glutamate plus glycine. The adequacy of each scheme was assessed by maximum likelihood values and autocorrelation coefficients of single-channel currents, as well as by the predicted time courses of transient macroscopic currents. Two schemes provided the best description for NMDAR gating at both the single-channel and macroscopic levels. These two schemes had coupled open states, only one gateway between the closed and open aggregates, and at least two preopening closed states. These two models could be condensed into a cyclic reaction mechanism. Using a linear reaction scheme, the overall "gating" rates (from the initial stable closed state to the final stable open state) are 177 and 4.4 s(-1).

On the relationships between kinetic schemes and two-state single molecule trajectories

Trajectories of a signal that fluctuates between two states which originate from single molecule activities have become ubiquitous. Common examples are trajectories of ionic flux through individual membrane channels and of photon counts collected from diffusion, activity, and conformational changes of biopolymers. By analyzing the trajectory, one wishes to deduce the underlying mechanism, which is usually described by a multisubstate kinetic scheme. In previous works [O. Flomenborn, J. Klafter, and A. Szabo, Biophys. J. 88, 3780 (2005); O. Flomenbom and J. Klafter, Acta Phys. Pol. B 36, 1527 (2005)], we divided kinetic schemes that generate two-state trajectories into two types: reducible schemes and irreducible schemes. A full characterization of the reducible ones was given. We showed that all the information in trajectories generated from reducible schemes is contained in the waiting time probability density functions (PDFs) of the two states. It follows that reducible schemes with the same waiting time PDFs are not distinguishable; namely, such schemes lead to identical two-state trajectories in the statistical sense. In this work, we further characterize the topologies of kinetic schemes, now of irreducible schemes, and further study two-state trajectories from the two types of scheme. We suggest various methods for extracting information about the underlying kinetic scheme from the trajectory (e.g., calculate the binned successive waiting times PDFs and analyze the ordered waiting time trajectories), and point out the advantages and disadvantages of each. We show that the binned successive waiting times PDFs are not only more robust than other functions when analyzing finite trajectories, but contain, in most cases, more information about the underlying kinetic scheme than other functions in the limit of infinitely long trajectories. For some cases, however, analyzing the ordered waiting times trajectory may supply unique information about the underlying kinetic scheme.

Using independent open-to-closed transitions to simplify aggregated Markov models of ion channel gating kinetics

Deducing plausible reaction schemes from single-channel current traces is time-consuming and difficult. The goal is to find the simplest scheme that fits the data, but there are many ways to connect even a small number of states (>2 million schemes with four open and four closed states). Many schemes make identical predictions. An exhaustive search over model space does not address the many equivalent schemes that will result. We have found a canonical form that can express all reaction schemes for binary channels. This form has the minimal number of rate constants for any rank (number of independent open-closed transitions), unlike other canonical forms such as the well established "uncoupled" scheme. Because all of the interconductance transitions in the new form are independent, we refer to it as the manifest interconductance rank (MIR) form. In the case of four open and four closed states, there are four MIR form schemes, corresponding to ranks 1-4. For many models proposed in the literature for specific ion channels, the equivalent MIR form has dramatically fewer links than the uncoupled form. By using the MIR form we prove that all rank 1 topologies with a given number of open and closed states make identical predictions in steady state, thus narrowing the search space for simple models. Moreover, we prove that fitting to canonical form preserves detailed balance. We also propose an efficient hierarchical algorithm for searching for the simplest possible model consistent with a given data set.

Maximum likelihood estimation of ion channel kinetics from macroscopic currents

We describe a maximum likelihood method for direct estimation of rate constants from macroscopic ion channel data for kinetic models of arbitrary size and topology. The number of channels in the preparation, and the mean and standard deviation of the unitary current can be estimated, and a priori constraints can be imposed on rate constants. The method allows for arbitrary stimulation protocols, including stimuli with finite rise time, trains of ligand or voltage steps, and global fitting across different experimental conditions. The initial state occupancies can be optimized from the fit kinetics. Utilizing arbitrary stimulation protocols and using the mean and the variance of the current reduce or eliminate problems of model identifiability (Kienker, 1989). The algorithm is faster than a recent method that uses the full autocovariance matrix (Celentano and Hawkes, 2004), in part due to the analytical calculation of the likelihood gradients. We tested the method with simulated data and with real macroscopic currents from acetylcholine receptors, elicited in response to brief pulses of carbachol. Given appropriate stimulation protocols, our method chose a reasonable model size and topology.

How to impose microscopic reversibility in complex reaction mechanisms

Most, but not all, ion channels appear to obey the law of microscopic reversibility (or detailed balance). During the fitting of reaction mechanisms it is therefore often required that cycles in the mechanism should obey microscopic reversibility at all times. In complex reaction mechanisms, especially those that contain cubic arrangements of states, it may not be obvious how to achieve this. Three general methods for imposing microscopic reversibility are described. The first method works by setting the 'obvious' four-state cycles in the correct order. The second method, based on the idea of a spanning tree, works by finding independent cycles (which will often have more than four states) such that the order in which they are set does not matter. The third method uses linear algebra to solve for constrained rates.

Calcium regulation of single ryanodine receptor channel gating analyzed using HMM/MCMC statistical methods

Type-II ryanodine receptor channels (RYRs) play a fundamental role in intracellular Ca²⁺ dynamics in heart. The processes of activation, inactivation, and regulation of these channels have been the subject of intensive research and the focus of recent debates. Typically, approaches to understand these processes involve statistical analysis of single RYRs, involving signal restoration, model estimation, and selection. These tasks are usually performed by following rather phenomenological criteria that turn models into self-fulfilling prophecies. Here, a thorough statistical treatment is applied by modeling single RYRs using aggregated hidden Markov models. Inferences are made using Bayesian statistics and stochastic search methods known as Markov chain Monte Carlo. These methods allow extension of the temporal resolution of the analysis far beyond the limits of previous approaches and provide a direct measure of the uncertainties associated with every estimation step, together with a direct assessment of why and where a particular model fails. Analyses of single RYRs at several Ca²⁺ concentrations are made by considering 16 models, some of them previously reported in the literature. Results clearly show that single RYRs have Ca²⁺-dependent gating modes. Moreover, our results demonstrate that single RYRs responding to a sudden change in Ca²⁺ display adaptation kinetics. Interestingly, best ranked models predict microscopic reversibility when monovalent cations are used as the main permeating species. Finally, the extended bandwidth revealed the existence of novel fast buzz-mode at low Ca²⁺ concentrations.

Electrodiffusion near an ion channel and the effect of mobile buffer

A numerical method was set up to calculate the dynamic concentration behavior of charged particles in the vicinity of an ion channel. It takes into account the electric potential due to the charge of the transported ions. Additionally, the finite on- and off-kinetics of mobile ion-buffers such as EGTA can be added to the simulations. The calculations were carried out using a modified Crank-Nicolson algorithm to solve the partial differential equations describing the problem. It was found that the electrostatic effect on the concentration of permeating ions is negligible in the presence of physiological salt concentrations. Nevertheless, there are electrostatic effects on other ion species near the channel mouth. Studies on the effect of a Ba2+ -current through a Ca2+ -channel onto the Ca2+ -concentration in the bath, and on the amplification of the Ca2+ -effect on the BK-channel due to the K+ -flux are presented. Additionally, the effect of mobile buffers was simulated and the numerical results are compared with some common analytical approximations.

Oxidative regulation of large conductance calcium-activated potassium channels

Reactive oxygen/nitrogen species are readily generated in vivo, playing roles in many physiological and pathological conditions, such as Alzheimer's disease and Parkinson's disease, by oxidatively modifying various proteins. Previous studies indicate that large conductance Ca(2+)-activated K(+) channels (BK(Ca) or Slo) are subject to redox regulation. However, conflicting results exist whether oxidation increases or decreases the channel activity. We used chloramine-T, which preferentially oxidizes methionine, to examine the functional consequences of methionine oxidation in the cloned human Slo (hSlo) channel expressed in mammalian cells. In the virtual absence of Ca(2+), the oxidant shifted the steady-state macroscopic conductance to a more negative direction and slowed deactivation. The results obtained suggest that oxidation enhances specific voltage-dependent opening transitions and slows the rate-limiting closing transition. Enhancement of the hSlo activity was partially reversed by the enzyme peptide methionine sulfoxide reductase, suggesting that the upregulation is mediated by methionine oxidation. In contrast, hydrogen peroxide and cysteine-specific reagents, DTNB, MTSEA, and PCMB, decreased the channel activity. Chloramine-T was much less effective when concurrently applied with the K(+) channel blocker TEA, which is consistent with the possibility that the target methionine lies within the channel pore. Regulation of the Slo channel by methionine oxidation may represent an important link between cellular electrical excitability and metabolism.

Model selection in non-nested hidden Markov models for ion channel gating

An important task in the application of Markov models to the analysis of ion channel data is the determination of the correct gating scheme of the ion channel under investigation. Some prior knowledge from other experiments can reduce significantly the number of possible models. If these models are standard statistical procedures nested like likelihood ratio testing, provide reliable selection methods. In the case of non-nested models, information criteria like AIC, BIC, etc., are used. However, it is not known if any of these criteria provide a reliable selection method and which is the best one in the context of ion channel gating. We provide an alternative approach to model selection in the case of non-nested models with an equal number of open and closed states. The models to choose from are embedded in a properly defined general model. Therefore, we circumvent the problems of model selection in the non-nested case and can apply model selection procedures for nested models.

The effects of non-identifiability on testing for detailed balance in aggregated Markov models for ion-channel gating

Aggregated Markov models are a widely used tool to model patch clamp data measured from single ion channels. These channels must obey the principle of detailed balance in thermodynamic equilibrium; otherwise, the channel is driven by an external source of energy. We investigate the power of a likelihood ratio test for detailed balance for a number of data points which is in the order of magnitude of patch clamp experiments. We show that for certain models with nearly equal dwell times, a test for detailed balance suffers from a loss of power to detect violations of detailed balance which is due to the non-identifiability of the transition rates for models with equal dwell times.

Voltage and Ca2+ activation of single large-conductance Ca2+-activated K+ channels described by a two-tiered allosteric gating mechanism

The voltage- and Ca2+-dependent gating mechanism of large-conductance Ca2+-activated K+ (BK) channels from cultured rat skeletal muscle was studied using single-channel analysis. Channel open probability (Po) increased with depolarization, as determined by limiting slope measurements (11 mV per e-fold change in Po; effective gating charge, q(eff), of 2.3 +/- 0.6 e(o)). Estimates of q(eff) were little changed for intracellular Ca2+ (Ca2+(i)) ranging from 0.0003 to 1,024 microM. Increasing Ca2+(i) from 0.03 to 1,024 microM shifted the voltage for half maximal activation (V(1/2)) 175 mV in the hyperpolarizing direction. V(1/2) was independent of Ca2+(i) for Ca2+(i) < or = 0.03 microM, indicating that the channel can be activated in the absence of Ca2+(i). Open and closed dwell-time distributions for data obtained at different Ca2+(i) and voltage, but at the same Po, were different, indicating that the major action of voltage is not through concentrating Ca2+ at the binding sites. The voltage dependence of Po arose from a decrease in the mean closing rate with depolarization (q(eff) = -0.5 e(o)) and an increase in the mean opening rate (q(eff) = 1.8 e(o)), consistent with voltage-dependent steps in both the activation and deactivation pathways. A 50-state two-tiered model with separate voltage- and Ca2+-dependent steps was consistent with the major features of the voltage and Ca2+ dependence of the single-channel kinetics over wide ranges of Ca2+(i) (approximately 0 through 1,024 microM), voltage (+80 to -80 mV), and Po (10(-4) to 0.96). In the model, the voltage dependence of the gating arises mainly from voltage-dependent transitions between closed (C-C) and open (O-O) states, with less voltage dependence for transitions between open and closed states (C-O), and with no voltage dependence for Ca2+-binding and unbinding. The two-tiered model can serve as a working hypothesis for the Ca2+- and voltage-dependent gating of the BK channel.

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 microM) to high (1,024 microM) intracellular concentrations of calcium (Ca2+i) using single-channel recording. Open probability (Po) increased with increasing Ca2+i (K0. 5 11.2 +/- 0.3 microM at +30 mV, Hill coefficient of 3.5 +/- 0.3), reaching a maximum of approximately 0.97 for Ca2+i approximately 100 microM. Increasing Ca2+i further to 1,024 microM had little additional effect on either Po 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 microM), 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.

Kinetic structure of large-conductance Ca2+-activated K+ channels suggests that the gating includes transitions through intermediate or secondary states. A mechanism for flickers

Mechanisms for the Ca2+-dependent gating of single large-conductance Ca2+-activated K+ channels from cultured rat skeletal muscle were developed using two-dimensional analysis of single-channel currents recorded with the patch clamp technique. To extract and display the essential kinetic information, the kinetic structure, from the single channel currents, adjacent open and closed intervals were binned as pairs and plotted as two-dimensional dwell-time distributions, and the excesses and deficits of the interval pairs over that expected for independent pairing were plotted as dependency plots. The basic features of the kinetic structure were generally the same among single large-conductance Ca2+-activated K+ channels, but channel-specific differences were readily apparent, suggesting heterogeneities in the gating. Simple gating schemes drawn from the Monod- Wyman-Changeux (MWC) model for allosteric proteins could approximate the basic features of the Ca2+ dependence of the kinetic structure. However, consistent differences between the observed and predicted dependency plots suggested that additional brief lifetime closed states not included in MWC-type models were involved in the gating. Adding these additional brief closed states to the MWC-type models, either beyond the activation pathway (secondary closed states) or within the activation pathway (intermediate closed states), improved the description of the Ca2+ dependence of the kinetic structure. Secondary closed states are consistent with the closing of secondary gates or channel block. Intermediate closed states are consistent with mechanisms in which the channel activates by passing through a series of intermediate conformations between the more stable open and closed states. It is the added secondary or intermediate closed states that give rise to the majority of the brief closings (flickers) in the gating.

Time-irreversible subconductance gating associated with Ba2+ block of large conductance Ca2+-activated K+ channels

Ba2+ block of large conductance Ca2+-activated K+ channels was studied in patches of membrane excised from cultures of rat skeletal muscle using the patch clamp technique. Under conditions in which a blocking Ba2+ ion would dissociate to the external solution (150 mM N-methyl-D-glucamine+o, 500 mM K+i, 10 microM Ba2+i, +30 mV, and 100 microM Ca2+i to fully activate the channel), Ba2+ blocks with a mean duration of approximately 2 s occurred, on average, once every approximately 100 ms of channel open time. Of these Ba2+ blocks, 78% terminated with a single step in the current to the fully open level and 22% terminated with a transition to a subconductance level at approximately 0.26 of the fully open level (preopening) before stepping to the fully open level. Only one apparent preclosing was observed in approximately 10,000 Ba2+ blocks. Thus, the preopenings represent Ba2+-induced time-irreversible subconductance gating. The fraction of Ba2+ blocks terminating with a preopening and the duration of preopenings (exponentially distributed, mean = 0.75 ms) appeared independent of changes in [Ba2+]i or membrane potential. The fractional conductance of the preopenings increased from 0.24 at +10 mV to 0.39 at +90 mV. In contrast, the average subconductance level during normal gating in the absence of Ba2+ was independent of membrane potential, suggesting different mechanisms for preopenings and normal subconductance levels. Preopenings were also observed with 10 mM Ba2+o and no added Ba2+i. Adding K+, Rb+, or Na+ to the external solution decreased the fraction of Ba2+ blocks with preopenings, with K+ and Rb+ being more effective than Na+. These results are consistent with models in which the blocking Ba2+ ion either induces a preopening gate, and then dissociates to the external solution, or moves to a site located on the external side of the Ba2+ blocking site and acts directly as the preopening gate.

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.

External barium influences the gating charge movement of Shaker potassium channels

External Ba2+ speeds the OFF gating currents (IgOFF) of Shaker K+ channels but only upon repolarization from potentials that are expected to open the channel pore. To study this effect we used a nonconducting and noninactivating mutant of the Shaker K+ channel, ShH4-IR (W434F). External Ba2+ slightly decreases the quantity of ON gating charge (QON) upon depolarization to potentials near -30 mV but has little effect on the quantity of charge upon stepping to more hyperpolarized or depolarized potentials. More strikingly, Ba2+ significantly increases the decay rate of IgOFF upon repolarization to -90 mV from potentials positive to approximately -55 mV. For Ba2+ to have this effect, the depolarizing command must be maintained for a duration that is dependent on the depolarizing potential (> 4 ms at -30 mV and > 1 ms at 0 mV). The actions of Ba2+ on the gating current are dose-dependent (EC50 approximately 0.2 mM) and are not produced by either Ca2+ or Mg2+ (2 mM). The results suggest that Ba2+ binds to a specific site on the Shaker K+ channel that destabilizes the open conformation and thus facilitates the return of gating charge upon repolarization.

High Ca2+ concentrations induce a low activity mode and reveal Ca2(+)-independent long shut intervals in BK channels from rat muscle

1. Large-conductance calcium-activated K+ channels (BK channels) often display long closed intervals at higher levels of Ca2+. To gain further insight into possible mechanisms for these intervals, currents were recorded from single BK channels, using the patch clamp technique, from patches of membrane excised from primary cultures of rat skeletal muscle. 2. High intracellular calcium concentrations ([Ca2+]i; 10-1000 microM) induced a low activity mode and revealed isolated long shut intervals. Neither of these phenomena were due to the Ba2+ that typically contaminates reagent grade salts. 3. The low activity mode was characterized by typically single brief open intervals with mean durations of 0.1 ms, separated by long shut intervals with mean durations of 100 ms. The very low open probability of about 0.001 during the low activity mode would make a sojourn to this mode functionally equivalent to a sojourn to an inactive state. The durations of sojourns in the low activity mode were exponentially distributed, with the mean durations ranging from about 1 s in 10 microM Ca(i)2+, to 4.5 s in 1000 microM Ca(i)2+. With increased filtering, the brief open intervals would escape detection so that a sojourn to the low activity mode would appear as a single shut interval. A typical channel spent less than 5% of its time in the low activity mode for [Ca2+]i < 10 microM. This increased to about 30% for [Ca2+]i > 100-1000 microM. A kinetic model with three closed states and two open states could approximate the gating of the low activity mode. 4. The isolated long shut intervals were not from the low activity mode, suggesting a different underlying mechanism. Their frequency of occurrence of about 0.3 s-1 did not increase with increasing [Ca2+]i, indicating that they did not arise from a slow Ca2+ block. Their durations were exponentially distributed, with a mean of 127 ms, which was independent of [Ca2+]i, suggesting that a single Ca(2+)-independent closed state or block underlies the isolated long shut intervals. At higher [Ca2+]i, up to 60% of the shut time could be spent in the isolated long shut intervals. 5. These observations suggest that activation of BK channels by high [Ca2+]i can be limited by sojourns to a low activity mode and also by isolated long shut intervals, two additional phenomena that will have to be accounted for in the gating of BK channels.