Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

Separation of Gating Properties from Permeation and Block in mslo Large Conductance Ca-activated K+ Channels

In this and the following paper we have examined the kinetic and steady-state properties of macroscopic mslo Ca-activated K⁺ currents in order to interpret these currents in terms of the gating behavior of the mslo channel. To do so, however, it was necessary to first find conditions by which we could separate the effects that changes in Ca²⁺ concentration or membrane voltage have on channel permeation from the effects these stimuli have on channel gating. In this study we investigate three phenomena which are unrelated to gating but are manifest in macroscopic current records: a saturation of single channel current at high voltage, a rapid voltage-dependent Ca²⁺ block, and a slow voltage-dependent Ba²⁺ block. Where possible methods are described by which these phenomena can be separated from the effects that changes in Ca²⁺ concentration and membrane voltage have on channel gating. Where this is not possible, some assessment of the impact these effects have on gating parameters determined from macroscopic current measurements is provided. We have also found that without considering the effects of Ca²⁺ and voltage on channel permeation and block, macroscopic current measurements suggest that mslo channels do not reach the same maximum open probability at all Ca²⁺ concentrations. Taking into account permeation and blocking effects, however, we find that this is not the case. The maximum open probability of the mslo channel is the same or very similar over a Ca²⁺ concentration range spanning three orders of magnitude indicating that over this range the internal Ca²⁺ concentration does not limit the ability of the channel to be activated by voltage.     

Gating characteristics of a steeply voltage-dependent gap junction channel in rat Schwann cells

The gating properties of macroscopic and microscopic gap junctional currents were compared by applying the dual whole cell patch clamp technique to pairs of neonatal rat Schwann cells. In response to transjunctional voltage pulses (Vj), macroscopic gap junctional currents decayed exponentially with time constants ranging from < 1 to < 10 s before reaching steady-state levels. The relationship between normalized steady-state junctional conductance (Gss) and (Vj) was well described by a Boltzmann relationship with e-fold decay per 10.4 mV, representing an equivalent gating charge of 2.4. At Vj > 60 mV, Gss was virtually zero, a property that is unique among the gap junctions characterized to date. Determination of opening and closing rate constants for this process indicated that the voltage dependence of macroscopic conductance was governed predominantly by the closing rate constant. In 78% of the experiments, a single population of unitary junctional currents was detected corresponding to an unitary channel conductance of approximately 40 pS. The presence of only a limited number of junctional channels with identical unitary conductances made it possible to analyze their kinetics at the single channel level. Gating at the single channel level was further studied using a stochastic model to determine the open probability (Po) of individual channels in a multiple channel preparation. Po decreased with increasing Vj following a Boltzmann relationship similar to that describing the macroscopic Gss voltage dependence. These results indicate that, for Vj of a single polarity, the gating of the 40 pS gap junction channels expressed by Schwann cells can be described by a first order kinetic model of channel transitions between open and closed states.     

Inactivation viewed through single sodium channels

Recordings of the sodium current in tissue-cultured GH3 cells show that the rate of inactivation in whole cell and averaged single channel records is voltage dependent: tau h varied e-fold/approximately 26 mV. The source of this voltage dependence was investigated by examining the voltage dependence of individual rate constants, estimated by maximum likelihood analysis of single channel records, in a five-state kinetic model. The rate constant for inactivating from the open state, rather than closing, increased with depolarization, as did the probability that an open channel inactivates. The rate constant for closing from the open state had the opposite voltage dependence. Both rate constants contributed to the mean open time, which was not very voltage dependent. Both open time and burst duration were less than tau h for voltages up to -20 mV. The slowest time constant of activation, tau m, was measured from whole cell records, by fitting a single exponential either to tail currents or to activating currents in trypsin-treated cells, in which the inactivation was abolished. tau m was a bell-shaped function of voltage and had a voltage dependence similar to tau h at voltages more positive than -35 mV, but was smaller than tau h. At potentials more negative than about -10 mV, individual channels may open and close several times before inactivating. Therefore, averaged single channel records, which correspond with macroscopic current elicited by a depolarization, are best described by a convolution of the first latency density with the autocorrelation function rather than with 1 - (channel open time distribution). The voltage dependence of inactivation from the open state, in addition to that of the activation process, is a significant factor in determining the voltage dependence of macroscopic inactivation. Although the rates of activation and inactivation overlapped greatly, independent and coupled inactivation could not be statistically distinguished for two models examined. Although rates of activation affect the observed rate of inactivation at intermediate voltages, extrapolation of our estimates of rate constants suggests that at very depolarized voltages the activation process is so fast that it is an insignificant factor in the time course of inactivation. Prediction of gating currents shows that an inherently voltage-dependent inactivation process need not produce a conspicuous component in the gating current.     

The gating of single calcium-dependent potassium channels is described by an activation/blockade mechanism

Single calcium dependent potassium channels from cultured rat myoballs have been studied with the patch clamp technique, and current records subjected to statistical analysis. From the dependence of the mean open state probability on the internal calcium concentration, two calcium ions are required to open the channel. The open state and closed state lifetime distributions reveal that the usual activation model is not applicable to these channels. They are consistent with a two step gating mechanism that involves both activation by calcium and blockade by a calcium-sensitive gate.     

Effects of stochastic channel gating and distribution on the cardiac action potential

Ion channels exhibit stochastic conformational changes determining their gating behavior. In addition, the process of protein turnover leads to a natural variability of the number of membrane and gap junctional channels. Nevertheless, in computational models, these two aspects are scarcely considered and their impacts are largely unknown. We investigated the effects of stochastic current fluctuations and channel distributions on action potential duration (APD), intercellular conduction delays (ICDs) and conduction blocks using a modified ventricular cell model (Rudy et al.) with Markovian formulations of the principal ion currents (to simulate their stochastic open-close gating behavior) and with channel counts drawn from Poisson distributions (to simulate their natural variability). In single cells, APD variability (coefficient of variation: 1.6% at BCL=1000 ms) was essentially caused by stochastic channel gating of I_Ks, persistent I_Na and I_Ca,L. In cell strands, ICD variability induced by stochastic channel gating and Poissonian channel distributions was low under normal conditions. Nonetheless, at low intercellular coupling levels, Poissonian gap junctional channel distribution resulted in a large ICD variability (coefficient of variation >20%), highly heterogeneous conduction patterns and conduction blocks. Therefore, the stochastic behavior of current fluctuations and channel distributions can contribute to the heterogeneity of conduction patterns and to conduction block, as observed previously in experiments in cardiac tissue with altered intercellular coupling.     

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

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

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

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

Accurate and fast simulation of channel noise in conductance-based model neurons by diffusion approximation

Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here.     

Statistical analysis of channel current from a membrane patch. II. A stochastic theory of a multi-channel system in the steady-state

A general stochastic theory is presented for analysis of current records of a patch containing an arbitrary number (N) of independent homologous channels in the steady-state. We give the "basic theorem" that at the instant of any open (or shut) transition of a channel, the other N-1 channels are located in each state with a probability equal to those in the steady-state, if enough transitions are observed. Using the "basic theorem", we derived: (a) the time-dependent open and shut frequencies after a definite type of transition, and (b) the probability density functions (pdf) of the duration of any period between two successive transitions. Briefly, the main results obtained were: (1) The time-dependent open (or shut) transition frequency after every shut (or open) transition at t = 0 in an N-channel patch, fJSh,Op(t)(N) (or fJOp,Sh(t)(N)), is the same as that of a one-channel patch except for the value of the constant. (2) In the all-shut (or all-open) period of a patch, the average duration of the period is 1/N, and the slowest exponential decay constant contained in the pdf is N times those of a single channel patch, respectively. (3) An example calculation for small N showed that the stochastic properties of a single channel can be obtained even when N is uncertain, if the channel open probability is small and exponential decay constants are separated. (4) When the channels are in equilibrium, the pdf of duration of every type of period in the patch is described by a sum of exponential terms with positive coefficients. This also holds for fJSh,Op(t)(N) and fJOp,Sh(t)(N).     

Cardiac sodium channel Markov model with temperature dependence and recovery from inactivation

A Markov model of the cardiac sodium channel is presented. The model is similar to the CA1 hippocampal neuron sodium channel model developed by Kuo and Bean (1994. Neuron. 12:819-829) with the following modifications: 1) an additional open state is added; 2) open-inactivated transitions are made voltage-dependent; and 3) channel rate constants are exponential functions of enthalpy, entropy, and voltage and have explicit temperature dependence. Model parameters are determined using a simulated annealing algorithm to minimize the error between model responses and various experimental data sets. The model reproduces a wide range of experimental data including ionic currents, gating currents, tail currents, steady-state inactivation, recovery from inactivation, and open time distributions over a temperature range of 10 degrees C to 25 degrees C. The model also predicts measures of single channel activity such as first latency, probability of a null sweep, and probability of reopening.     

pH-regulated Slo3 K+ channels: properties of unitary currents

Here we have examined the voltage and pH dependence of unitary Slo3 channels and used analysis of current variance to define Slo3 unitary current properties over a broader range of voltages. Despite complexity in Slo3 channel openings that precludes simple definition of the unitary conductance, average current through single Slo3 channels varies linearly with voltage at positive activation potentials. Furthermore, the average Slo3 unitary current at a given activation potential does not change with pH. Consistent with macroscopic conductance estimates, the apparent open probability of Slo3 channel exhibits a pH-dependent maximum, with limiting values around 0.3 at the most elevated pH and voltage. Estimates of Slo3 conductance at negative potentials support a weaker intrinsic voltage dependence of gating than is observed for Slo1. For the pH-regulated Slo3 K(+) channel, the dependence of macroscopic conductance on pH suggests that the pH-sensitive mechanism regulates gating in an allosteric manner qualitatively similar to regulation of Slo1 by Ca(2+). Together, the results support the view that the regulation of macroscopic Slo3 currents by pH reflects regulation of gating equilibria, and not a direct effect of pH on ion permeation. Specifically, both voltage and pH regulate a closed-open conformational change in a largely independent fashion.     

Gating of Shaker K+ channels: I. Ionic and gating currents.

Ionic and gating currents from noninactivating Shaker B K+ channels were studied with the cut-open oocyte voltage clamp technique and compared with the macropatch clamp technique. The performance of the cut-open oocyte voltage clamp technique was evaluated from the electrical properties of the clamped upper domus membrane, K+ tail current measurements, and the time course of K+ currents after partial blockade. It was concluded that membrane currents less than 20 microA were spatially clamped with a time resolution of at least 50 microseconds. Subtracted, unsubtracted gating currents with the cut-open oocyte voltage clamp technique and gating currents recorded in cell attached macropatches had similar properties and time course, and the charge movement properties directly obtained from capacity measurements agreed with measurements of charge movement from subtracted records. An accurate estimate of the normalized open probability Po(V) was obtained from tail current measurements as a function of the prepulse V in high external K+. The Po(V) was zero at potentials more negative than -40 mV and increased sharply at this potential, then increased continuously until -20 mV, and finally slowly increased with voltages more positive than 0 mV. Deactivation tail currents decayed with two time constants and external potassium slowed down the faster component without affecting the slower component that is probably associated with the return between two of the closed states near the open state. In correlating gating currents and channel opening, Cole-Moore type experiments showed that charge moving in the negative region of voltage (-100 to -40 mV) is involved in the delay of the conductance activation but not in channel opening. The charge moving in the more positive voltage range (-40 to -10 mV) has a similar voltage dependence to the open probability of the channel, but it does not show the gradual increase with voltage seen in the Po(V).     

Structural determinants for the differences in voltage gating of chicken Cx56 and Cx45.6 gap-junctional hemichannels

The voltage- and calcium-dependent gating properties of two lens gap-junctional hemichannels were compared at the macroscopic and single channel level. In solutions containing zero added calcium and 1 mM Mg, chicken Cx56 hemichannels were mostly closed at negative potentials and application of depolarizing voltage clamp steps elicited a slowly activating outward current. In contrast, chicken Cx45.6 hemichannels were predominantly open at negative potentials and rapidly closed in response to application of large depolarizing potentials. Another difference was that macroscopic Cx45.6 currents were much smaller in size than the hemichannel currents induced by oocytes with similar amounts of cRNA for Cx56. The aim of this study was to identify which regions of the connexins were responsible for the differences in voltage-dependent gating and macroscopic current amplitude by constructing a series of chimeric Cx45.6-Cx56 channels. Our results show that two charged amino acids that are specific for the alpha3-group connexins (R9 in the N-terminus and E43 in the first extracellular loop) are important determinants for the difference in voltage-dependent gating between Cx45.6 and Cx56 hemichannels; the first transmembrane-spanning domain, M1, is an important determinant of macroscopic current magnitude; R9 and E43 are also determinants of single channel conductance and rectification.     

Effects of ultraviolet modification on the gating energetics of cyclic nucleotide-gated channels

Middendorf et al. (Middendorf, T.R., R.W. Aldrich, and D.A. Baylor. 2000. J. Gen. Physiol. 116:227-252) showed that ultraviolet light decreases the current through cloned cyclic nucleotide-gated channels from bovine retina activated by high concentrations of cGMP. Here we probe the mechanism of the current reduction. The channels' open probability before irradiation, P(o)(0), determined the sign of the change in current amplitude that occurred upon irradiation. UV always decreased the current through channels with high initial open probabilities [P(o)(0) > 0.3]. Manipulations that promoted channel opening antagonized the current reduction by UV. In contrast, UV always increased the current through channels with low initial open probabilities [P(o)(0) < or = 0.02], and the magnitude of the current increase varied inversely with P(o)(0). The dual effects of UV on channel currents and the correlation of both effects with P(o)(0) suggest that the channels contain two distinct classes of UV target residues whose photochemical modification exerts opposing effects on channel gating. We present a simple model based on this idea that accounts quantitatively for the UV effects on the currents and provides estimates for the photochemical quantum yields and free energy costs of modifying the UV targets. Simulations indicate that UV modification may be used to produce and quantify large changes in channel gating energetics in regimes where the associated changes in open probability are not measurable by existing techniques.     

Voltage-dependent gating of the cystic fibrosis transmembrane conductance regulator Cl- channel

When excised inside-out membrane patches are bathed in symmetrical Cl--rich solutions, the current-voltage (I-V) relationship of macroscopic cystic fibrosis transmembrane conductance regulator (CFTR) Cl- currents inwardly rectifies at large positive voltages. To investigate the mechanism of inward rectification, we studied CFTR Cl- channels in excised inside-out membrane patches from cells expressing wild-type human and murine CFTR using voltage-ramp and -step protocols. Using a voltage-ramp protocol, the magnitude of human CFTR Cl- current at +100 mV was 74 +/- 2% (n = 10) of that at -100 mV. This rectification of macroscopic CFTR Cl- current was reproduced in full by ensemble currents generated by averaging single-channel currents elicited by an identical voltage-ramp protocol. However, using a voltage-step protocol the single-channel current amplitude (i) of human CFTR at +100 mV was 88 +/- 2% (n = 10) of that at -100 mV. Based on these data, we hypothesized that voltage might alter the gating behavior of human CFTR. Using linear three-state kinetic schemes, we demonstrated that voltage has marked effects on channel gating. Membrane depolarization decreased both the duration of bursts and the interburst interval, but increased the duration of gaps within bursts. However, because the voltage dependencies of the different rate constants were in opposite directions, voltage was without large effect on the open probability (Po) of human CFTR. In contrast, the Po of murine CFTR was decreased markedly at positive voltages, suggesting that the rectification of murine CFTR is stronger than that of human CFTR. We conclude that inward rectification of CFTR is caused by a reduction in i and changes in gating kinetics. We suggest that inward rectification is an intrinsic property of the CFTR Cl- channel and not the result of pore block.     

一个改进的马尔科夫模型

离子通道是细胞膜里的大分子孔道,是跨越细胞膜里的蛋白质大分子,是神经、肌肉等细胞膜兴奋性的基础。人体细胞均具有完成特殊功能的离子通道,构建离子通道,尤其其门控行为的动力学模型,对于研究离子通道的相关课题具有重要意义。离子通道的开关反映了蛋白质构象变化的动力学过程。本文介绍了细胞膜离子通道的基本内容和几种常用模型,并根据Markov链对离子通道门控行为的一个二态（闭、开）模型的Markov过程进行了改进,得到了包含失活状态的离子通道门控行为的Markov模型。     

Exploring voltage-dependent ion channels in silico by hysteretic conductance

Kinetic models of voltage-dependent ion channels are normally inferred from time records of macroscopic current relaxation or microscopic single channel data. A complementary explorative approach is outlined. Hysteretic conductance refers to conductance delays in response to voltage changes, delays at either macroscopic or microscopic levels of observation. It enables complementary assessments of model assumptions and gating schemes of voltage-dependent channels, e.g. independent versus cooperative gating, and multiple gating modes. Under the Hodgkin-Huxley condition of independent gating, and under ideal measurement conditions, hysteretic conductance makes it also possible to estimate voltage-dependent rate functions. The argument is mainly theoretical, based on experimental observations, and illustrated by simulations of Markov kinetic models.     

Modification of potassium channel kinetics by amino group reagents.

We have examined the actions of several amino group reagents on delayed rectifier potassium channels in squid giant axons. Three general classes of reagents were used: (1) those that preserved the positive charge of amino groups; (2) those that neutralize the charge; and (3) those that replace the positive with a negative charge. All three types of reagents produced qualitatively similar effects on K channel properties. Trinitrobenzene sulfonic acid (TNBS) neutralizes the peptide terminal amino groups and the epsilon-amino group of lysine groups. TNBS (a) slowed the kinetics of macroscopic ionic currents; (b) increased the size of ionic currents at large positive voltages; (c) shifted the voltage-dependent probability of channel opening to more positive potentials but had no effect on the voltage sensitivity; and (d) altered several properties of K channel gating currents. The actions of TNBS on gating currents suggest the presence of multiple gating current components. These effects are not all coupled, suggesting that several amino groups on the external surface of K channels are important for channel gating. A simple kinetic model that considers the channel to be composed of independent heterologous subunits is consistent with most of the modifications produced by amino group reagents.