Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

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.     

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.     

Properties of single voltage-gated proton channels in human eosinophils estimated by noise analysis and by direct measurement

Voltage-gated proton channels were studied under voltage clamp in excised, inside-out patches of human eosinophils, at various pHi with pHo 7.5 or 6.5 pipette solutions. H+ current fluctuations were observed consistently when the membrane was depolarized to voltages that activated H+ current. At pHi < or = 5.5 the variance increased nonmonotonically with depolarization to a maximum near the midpoint of the H+ conductance-voltage relationship, gH-V, and then decreased, supporting the idea that the noise is generated by H+ channel gating. Power spectral analysis indicated Lorentzian and 1/f components, both related to H+ currents. Unitary H+ current amplitude was estimated from stationary or quasi-stationary variance, sigmaH2. We analyze sigmaH2 data obtained at various voltages on a linearized plot that provides estimates of both unitary conductance and the number of channels in the patch, without requiring knowledge of open probability. The unitary conductance averaged 38 fS at pHi 6.5, and increased nearly fourfold to 140 fS at pHi 5.5, but was independent of pHo. In contrast, the macroscopic gH was only 1.8-fold larger at pHi 5.5 than at pHi 6.5. The maximum H+ channel open probability during large depolarizations was 0.75 at pHi 6.5 and 0.95 at pHi 5.5. Because the unitary conductance increases at lower pHi more than the macroscopic gH, the number of functional channels must decrease. Single H+ channel currents were too small to record directly at physiological pH, but at pHi < or = 5.5 near Vthreshold (the voltage at which gH turns on), single channel-like current events were observed with amplitudes 7-16 fA.     

Single sodium channels from the squid giant axon

Since the work of A. L. Hodgkin and A. F. Huxley (1952. J. Physiol. [Lond.].117:500-544) the squid giant axon has been considered the classical preparation for the study of voltage-dependent sodium and potassium channels. In this preparation much data have been gathered on macroscopic and gating currents but no single sodium channel data have been available. This paper reports patch clamp recording of single sodium channel events from the cut-open squid axon. It is shown that the single channel conductance in the absence of external divalent ions is approximately 14 pS, similar to sodium channels recorded from other preparations, and that their kinetic properties are consistent with previous results on gating and macroscopic currents obtained from the perfused squid axon preparation.     

Ionic conductances of squid giant fiber lobe neurons

The cell bodies of the neurons in the giant fiber lobe (GFL) of the squid stellate ganglion give rise to axons that fuse and thereby form the third-order giant axon, whose initial portion functions as the postsynaptic element of the squid giant synapse. We have developed a preparation of dissociated, cultured cells from this lobe and have studied the voltage-dependent conductances using patch-clamp techniques. This system offers a unique opportunity for comparing the properties and regional differentiation of ionic channels in somatic and axonal membranes within the same cell. Some of these cells contain a small inward Na current which resembles that found in axon with respect to tetrodotoxin sensitivity, voltage dependence, and inactivation. More prominent is a macroscopic inward current, carried by Ca2+, which is likely to be the result of at least two kinetically distinct types of channels. These Ca channels differ in their closing kinetics, voltage range and time course of activation, and the extent to which their conductance inactivates. The dominant current in these GFL neurons is outward and is carried by K+. It can be accounted for by a single type of voltage-dependent channel. This conductance resembles the K conductance of the axon, except that it partially inactivates during relatively short depolarizations. Ensemble fluctuation analysis of K currents obtained from excised outside-out patches is consistent with a single type of K channel and yields estimates for the single channel conductance of approximately 13 pS, independently of membrane potential. A preliminary analysis of single channel data supports the conclusion that there is a single type of voltage-dependent, inactivating K channel in the GFL neurons.     

Mechanism of blocking K+-channels with tetraethylammonium

Using the patch-voltage-clamp method action of tetraethylammonium on the fast (30 pS) and slow K+ channels was investigated. The slow K+ channels were presented by two types: with whole (30 pS) and decreased (20 pS) conductance. In all cases tetraethylammonium decreased the current magnitude and modified the channel kinetic parameters. Apparent blocking constants determined from the current decreasing are as 8-50 and 4-12 mM for the slow K+ channels with whole and decreased conductance, respectively, and 0.05-0.08 mM--for the fast K+ channel. The potential dependency of the blocking constants correlates with that of the channel conductance. Probability of the channel open state for the slow K+ channels decreases, and that for the fast K+ channel increases under application of tetraethylammonium. It is concluded that there are two sites of tetraethylammonium binding: the first site is into the channel pore, and the second one--into the regulatory centre responsible for the channel kinetic behaviour. Blocking of general conductance of the slow channels is accompanied by proportional decrease of the channel substate conductances without change of their number and cooperatively. Block of the fast K+ channel occurs without change of the channel elementary conductance but with decrease of the number of the channel substates and reversible violation of the channel transition cooperativity. The data are discussed from the point of the hypothesis on the channel clustery organization.     

Kinetic properties of a voltage-dependent junctional conductance

We have proposed that the gap junctions between amphibian blastomeres are comprised of voltage-sensitive channels. The kinetic properties of the junctional conductance are here studied under voltage clamp. When the transjunctional voltage is stepped to a new voltage of the same polarity, the junctional conductance changes as a single exponential to a steady-state level. The time constant of the conductance change is determined by the existing transjunctional voltage and is independent of the previous voltage. For each voltage polarity, the relations between voltage, time constant, and steady-state conductance are well modeled by a reversible two-state reaction scheme in which the calculated rate constants for the transitions between the states are exponential functions of voltage. The calculated rate constant for the transition to the low-conductance state is approximately twice as voltage dependent as that for the transition to the high-conductance state. When the transjunctional voltage polarity is reversed, the junctional conductance undergoes a transient recovery. The polarity reversal data are well modeled by a reaction scheme in which the junctional channel has two gates, each with opposite voltage sensitivity, and in which an open gate may close only if the gate in series with it is open. A simple explanation for this contingent gating is a mechanism in which each gate senses only the local voltage drop within the channel.     

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.     

Potassium conductance of the squid giant axon. Single-channel studies

The patch-clamp technique was implemented in the cut-open squid giant axon and used to record single K channels. We present evidence for the existence of three distinct types of channel activities. In patches that contained three to eight channels, ensemble fluctuation analysis was performed to obtain an estimate of 17.4 pS for the single-channel conductance. Averaged currents obtained from these multichannel patches had a time course of activation similar to that of macroscopic K currents recorded from perfused squid giant axons. In patches where single events could be recorded, it was possible to find channels with conductances of 10, 20, and 40 pS. The channel most frequently encountered was the 20-pS channel; for a pulse to 50 mV, this channel had a probability of being open of 0.9. In other single-channel patches, a channel with a conductance of 40 pS was present. The activity of this channel varied from patch to patch. In some patches, it showed a very low probability of being open (0.16 for a pulse to 50 mV) and had a pronounced lag in its activation time course. In other patches, the 40-pS channel had a much higher probability of being open (0.75 at a holding potential of 50 mV). The 40-pS channel was found to be quite selective for K over Na. In some experiments, the cut-open axon was exposed to a solution containing no K for several minutes. A channel with a conductance of 10 pS was more frequently observed after this treatment. Our study shows that the macroscopic K conductance is a composite of several K channel types, but the relative contribution of each type is not yet clear. The time course of activation of the 20-pS channel and the ability to render it refractory to activation only by holding the membrane potential at a positive potential for several seconds makes it likely that it is the predominant channel contributing to the delayed rectifier conductance.     

Finite Sample Properties of the Maximum Likelihood Estimator and of Likelihood Ratio Tests in Hidden Markov Models

Hidden Markov models were successfully applied in various fields of time series analysis, especially for analyzing ion channel recordings. The maximum likelihood estimator (MLE) has recently been proven to be asymptotically normally distributed. Here, we investigate finite sample properties of the MLE and of different types of likelihood ratio tests (LRTs) by means of simulation studies. The MLE is shown to reach the asymptotic behavior within sample sizes that are common for various applications. Thus, reliable estimates and confidence intervals can be obtained. We give an approximative scaling function for the estimation error for finite samples, and investigate the power of different LRTs suitable for applications to ion channels, including tests for superimposed hidden Markov processes. Our results are applied to physiological sodium channel data.     

Monazomycin-induced single channels. II. Origin of the voltage dependence of the macroscopic conductance

The voltage dependence of the conductance induced induced in thin lipid membranes by monazomycin is shown here to be caused by voltage- dependent variations in the frequency of channel openings. We also experimentally demonstrate certain interesting properties of the channel activity that are predicted by a chemical kinetic model (Muller and Peskin, 1981), which successfully describes the macroscopic conductance. We conclude that two parallel mechanisms--one autocatalytic, the other simple mass action--exist that allow monazomycin to enter (or leave) the membrane so that the monazomycin molecules can be in a position to form channels.     

Voltage-dependent modulation of single N-Type Ca2+ channel kinetics by receptor agonists in IMR32 cells.

The voltage-dependent inhibition of single N-type Ca(2+) channels by noradrenaline (NA) and the delta-opioid agonist D-Pen(2)-D-Pen (5)-enkephalin (DPDPE) was investigated in cell-attached patches of human neuroblastoma IMR32 cells with 100 mM Ba(2+) and 5 microM nifedipine to block L-type channels. In 70% of patches, addition of 20 microM NA + 1 microM DPDPE delayed markedly the first channel openings, causing a four- to fivefold increase of the first latency at +20 mV. The two agonists or NA alone decreased also by 35% the open probability (P(o)), prolonged partially the mean closed time, and increased the number of null sweeps. In contrast, NA + DPDPE had little action on the single-channel conductance (19 versus 19.2 pS) and minor effects on the mean open time. Similarly to macroscopic Ba(2+) currents, the ensemble currents were fast activating at control but slowly activating and depressed with the two agonists. Inhibition of single N-type channels was effectively removed (facilitated) by short and large depolarizations. Facilitatory pre-pulses increased P(o) significantly and decreased fourfold the first latency. Ensemble currents were small and slowly activating before pre-pulses and became threefold larger and fast decaying after facilitation. Our data suggest that slowdown of Ca(2+) channel activation by transmitters is mostly due to delayed transitions from a modified to a normal (facilitated) gating mode. This single-channel gating modulation could be well simulated by a Monte Carlo method using previously proposed kinetic models predicting marked prolongation of first channel openings.     

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.     

Use of the covariance matrix in directly fitting kinetic parameters: application to GABAA receptors

A new method of analysis is described that begins to explore the relationship between the phases of ion channel desensitization and the underlying states of the channel. The method, referred to as covariance fitting (CVF), couples Q-matrix calculations with a maximum likelihood algorithm to fit macroscopic desensitization data directly to kinetic models. Unlike conventional sum-of-squares minimization, CVF fits both the magnitude of the recorded current and the strength of the correlations between different time points. When applied to simulated data generated using various kinetic models with up to 11 free parameters, CVF leads to reasonable parameter estimates. Coupled with the likelihood ratio test, it accurately discriminates between models with different numbers of states, discriminates between most models with the same number but a different arrangement of states, and extracts meaningful information on the relationship between the desensitized states and the phases of macroscopic desensitization. When applied to GABA(A) receptor traces (outside out patches, alpha 1 beta 2 gamma 2S, 1 mM GABA, >2.5 s), a model with two open states and three desensitized states is favored. When applied to simulated data generated using a consensus model, CVF leads to reasonable parameter estimates and accurately discriminates between this and other models.     

Pharmacological and kinetic analysis of K channel gating currents

We have measured gating currents from the squid giant axon using solutions that preserve functional K channels and with experimental conditions that minimize Na channel contributions to these currents. Two pharmacological agents were used to identify a component of gating current that is associated with K channels. Low concentrations of internal Zn2+ that considerably slow K channel ionic currents with no effect on Na channel currents altered the component of gating current associated with K channels. At low concentrations (10-50 microM) the small, organic, dipolar molecule phloretin has several reported specific effects on K channels: it reduces K channel conductance, shifts the relationship between channel conductance and membrane voltage (Vm) to more positive potentials, and reduces the voltage dependence of the conductance-Vm relation. The K channel gating charge movements were altered in an analogous manner by 10 microM phloretin. We also measured the dominant time constants of the K channel ionic and gating currents. These time constants were similar over part of the accessible voltage range, but at potentials between -40 and 0 mV the gating current time constants were two to three times faster than the corresponding ionic current values. These features of K channel function can be reproduced by a simple kinetic model in which the channel is considered to consist of two, two-state, nonidentical subunits.     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Estimation of kinetic rate constants from multi-channel recordings by a direct fit of the time series.

The maximum-likelihood technique for the direct estimation of rate constants from the measured patch clamp current is extended to the analysis of multi-channel recordings, including channels with subconductance levels. The algorithm utilizes a simplified approach for the calculation of the matrix exponentials of the probability matrix from the rate constants of the Markov model of the involved channel(s) by making use of the Kronecker sum and product. The extension to multi-channel analysis is tested by the application to simulated data. For these tests, three different channel models were selected: a two-state model, a three-state model with two open states of different conductance, and a three-state model with two closed states. For the simulations, time series of these models were calculated from the related first-order, finite-state, continuous-time Markov processes. Blue background noise was added, and the signals were filtered by a digital filter similar to the anti-aliasing low-pass. The tests showed that the fit algorithm revealed good estimates of the original rate constants from time series of simulated records with up to four independent and identical channels even in the case of signal-to-noise ratios being as low as 2. The number of channels in a record can be determined from the dependence of the likelihood on channel number. For large enough data sets, it takes on a maximum when the assumed channel number is equal to the "true" channel number.     

Low single channel conductance of the major skeletal muscle chloride channel, ClC-1.

We expressed the skeletal muscle chloride channel, ClC-1, in HEK293 cells and investigated it with the patch-clamp technique. Macroscopic properties are similar to those obtained after expression in Xenopus oocytes, except that faster gating kinetics are observed in mammalian cells. Nonstationary noise analysis revealed that both rat and human ClC-1 have a low single channel conductance of about 1 pS. This finding may explain the lack of single-channel data for chloride channels from skeletal muscle despite its high macroscopic chloride conductance.     

Extra- and Intracellular Proton-Binding Sites of Volume-Regulated Anion Channels

We have investigated the effects of extracellular and intracellular pH on single channel and macroscopic (macropatches) currents through volume-regulated anion channels (VRAC) in endothelial cells. Protonation of extracellular binding sites with an apparent pK of 4.6 increased voltage independent of the single-channel amplitude. Cytosolic acidification had a dual effect on VRAC currents: on the one hand, it increased single channel conductance by ∼20% due to protonation of a group with an apparent pK of 6.5 and a Hill coefficient of 2. On the other hand, it reduced channel activity due to protonation of a group with an apparent pK of 6.3 and a Hill coefficient of 2.1. This dual effect enhances the macroscopic current at a slightly acidic pH but inhibits it at more acidic pH. Cytosolic alkalization also reduced channel activity with a pK of 8.4 and a Hill coefficient of 1.9, but apparently did not affect single-channel conductance. These data show that VRAC channels are maintained in an active state in a narrow pH range around the normal physiological pH and shut down outside this range. They also show that HEPES-buffered pipette solutions do not effectively buffer pH in the vicinity of the VRAC channels. Received: 31 January 2000/Revised: 21 April 2000