The power of two-dimensional dwell-time analysis for model discrimination, temporal resolution, multichannel analysis and level detection

Two-dimensional (2D) dwell-time analysis of time series of single-channel patch-clamp current was improved by employing a Hinkley detector for jump detection, introducing a genetic fit algorithm, replacing maximum likelihood by a least square criterion, averaging over a field of 9 or 25 bins in the 2D plane and normalizing per measuring time, not per events. Using simulated time series for the generation of the "theoretical" 2D histograms from assumed Markov models enabled the incorporation of the measured filter response and noise. The effects of these improvements were tested with respect to the temporal resolution, accuracy of the determination of the rate constants of the Markov model, sensitivity to noise and requirement of open time and length of the time series. The 2D fit was better than the classical hidden Markov model (HMM) fit in all tested fields. The temporal resolution of the two most efficient algorithms, the 2D fit and the subsequent HMM/beta fit, enabled the determination of rate constants 10 times faster than the corner frequency of the low-pass filter. The 2D fit was much less sensitive to noise. The requirement of computing time is a problem of the 2D fit (100 times that of the HMM fit) but can now be handled by personal computers. The studies revealed a fringe benefit of 2D analysis: it can reveal the "true" single-channel current when the filter has reduced the apparent current level by averaging over undetected fast gating.     

Using a five-state model for fitting amplitude histograms from MaxiK channels: β-distributions reveal more than expected

Fast gating of ion channels with rate constants higher than the corner frequency of the recording set-up can be evaluated by fitting so-called beta distributions to measured amplitude histograms. Up to now, this was preferentially done for O–C Markov sub-models with one open and one closed state. Here, a fit of the amplitude histograms from MaxiK (BK) single-channel records was achieved with a five-state model with two open and three closed states including three open–close transitions with rate constants higher than the corner frequency (20 kHz) of the inevitable low-pass filter of the recording system. The numerical values of the rate constants of these transitions enabled a nearly one-to-one relationship between typical regions of the histograms and the reactions in the Markov model. These characteristic features are the width of the peak at the apparent single-channel current, the side slopes at the open and at the closed peak, and the depth of the valley between the two peaks. However, the simplex routine alone was incapable of finding the solution but could do so if guided by hand along a suggested strategy.     

Interference of Shot Noise of Open-Channel Current with Analysis of Fast Gating: Patchers do not (Yet) Have to Care

Microsecond gating of ion channels can be evaluated by fitting beta distributions to amplitude histograms of measured time series. The shape of these histograms is determined not only by the rate constants of the gating process (in relation to the filter frequency) but also by baseline noise and shot noise, resulting from the stochastic nature of ion flow. Under normal temporal resolution, the small shot noise can be ignored. This simplification may no longer be legitimate when rate constants reach the range above 1 μs⁻¹. Here, the influence of shot noise is studied by means of simulated time series for several values of single-channel current of the fully open state and baseline noise. Under realistic optimal conditions (16 pA current, 1 pA noise, 50 kHz bandwidth), ignoring the shot noise leads to an underestimation of the rate constants above 1 μs⁻¹ by a factor of about 2.5. However, in that range, the scatter of the evaluated rate constants is at least of the same magnitude, obscuring the systematic error. The incorporation of shot noise into the analysis will become more important when amplifiers with significantly reduced noise become available.     

A Subsequent Fit of Time Series and Amplitude Histogram of Patch-Clamp Records Reveals Rate Constants up to 1 per Microsecond

Fast gating in time series of patch-clamp current demands powerful tools to reveal the rate constants of the adequate Hidden Markov model. Here, two approaches are presented to improve the temporal resolution of the direct fit of the time series. First, the prediction algorithm is extended to include intermediate currents between the nominal levels as caused by the anti-aliasing filter. This approach can reveal rate constants that are about 4 times higher than the corner frequency of the anti-aliasing filter. However, this approach is restricted to time series with very low noise. Second, the direct fit of the time series is combined with a beta fit, i.e., a fit of the deviations of the amplitude histogram from the Gaussian distribution. Since the “theoretical” amplitude histograms for higher-order Bessel filters cannot be calculated by analytical tools, they are generated from simulated time series. In a first approach, a simultaneous fit of the time series and of the Beta fit is tested. This simultaneous fit, however, inherits the drawbacks of both approaches, not the benefits. More successful is a subsequent fit: The fit of the time series yields a set of rate constants. The subsequent Beta fit uses the slow rate constants of the fit of the time series as fixed parameters and the optimization algorithm is restricted to the fast ones. The efficiency of this approach is illustrated by means of time series obtained from simulation and from the dominant K⁺ channel in Chara. This shows that temporal resolution can reach the microsecond range.     

The anomalous mole fraction effect in Chara: gating at the edge of temporal resolution

The anomalous mole fraction effect (AMFE) of the K(+) channel in excised patches of the tonoplast of Chara showed a minimum of apparent open-channel current at 20 mM Tl(+) and 230 mM K(+). Time series obtained at a sampling rate of 100 kHz (filter 25 kHz) were analyzed by three methods to find out whether the AMFE results from an effect on gating or on the conductivity of the open state. Fitting the amplitude histograms by a superposition of gaussians showed a broadening in the presence of Tl(+). Dwell-time analysis based on an O-O-C-C-C model failed to evaluate rate constants above the filter frequency. Thus, the absence of any reduction of apparent open-channel current in time series simulated with the evaluated rate constants could not be taken as evidence against the hypothesis of gating. Finally, a direct fit of the measured time series using five different 5-state Hidden Markov models revealed that the presence of Tl(+) changed the rate constants in such a way that the number of transitions into the short-lived open state (30 micros) increased strongly compared to those in the absence of Tl(+). These models explain 25% reduction of apparent single-channel current amplitude through a rapid gating mechanism.     

Strengths and Limits of Beta Distributions as a Means of Reconstructing the True Single-Channel Current in Patch Clamp Time Series with Fast Gating

Single-channel current seems to be one of the most obvious characteristics of ion transport. But in some cases, its determination is more complex than anticipated at first glance. Problems arise from fast gating in time series of patch-clamp current, which can lead to a reduced apparent (measured) single-channel current. Reduction is caused by undetected averaging over closed and open intervals in the anti-aliasing filter. Here it is shown that fitting the measured amplitude histograms by Beta distributions is an efficient tool of reconstructing the true current level from measured data. This approach becomes even more powerful when it is applied to amplitude distributions-per-level. Simulated time series are employed to show that the error sum is a good guideline for finding the correct current level. Furthermore, they show that a Markov model smaller than the one used for gating analysis can be used for current determination (mostly O-C, i.e., open-closed). This increases the reliability of the Beta fit. The knowledge of the true current level is not only important for the understanding of the biophysical properties of the channel. It is also a prerequisite for the correct determination of the rate constants of gating. The approach is applied to measured data. The examples reveal the limits of the analysis imposed by the signal-to-noise ratio and the shape of the amplitude distribution. One application shows that the negative slope of the I-V curve of the human MaxiK channel expressed in HEK293 cells is caused by fast gating.     

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

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

Time-Dependent Molecular Memory in Single Voltage-Gated Sodium Channel

Excitability in neurons is associated with firing of action potentials and requires the opening of voltage-gated sodium channels with membrane depolarization. Sustained membrane depolarization, as seen in pathophysiological conditions like epilepsy, can have profound implications on the biophysical properties of voltage-gated ion channels. Therefore, we sought to characterize the effect of sustained membrane depolarization on single voltage-gated Na⁺ channels. Single-channel activity was recorded in the cell-attached patch-clamp mode from the rNa_v1.2α channels expressed in CHO cells. Classical statistical analysis revealed complex nonlinear changes in channel dwell times and unitary conductance of single Na⁺ channels as a function of conditioning membrane depolarization. Signal processing tools like weighted wavelet Z (WWZ) and discrete Fourier transform analyses attributed a “pseudo-oscillatory” nature to the observed nonlinear variation in the kinetic parameters. Modeling studies using the hidden Markov model (HMM) illustrated significant changes in kinetic states and underlying state transition rate constants upon conditioning depolarization. Our results suggest that sustained membrane depolarization induces novel nonlinear properties in voltage-gated Na⁺ channels. Prolonged membrane depolarization also induced a “molecular memory” phenomenon, characterized by clusters of dwell time events and strong autocorrelation in the dwell time series similar to that reported recently for single enzyme molecules. The persistence of such molecular memory was found to be dependent on the duration of depolarization. Voltage-gated Na⁺ channel with the observed time-dependent nonlinear properties and the molecular memory phenomenon may determine the functional state of the channel and, in turn, the excitability of a neuron.     

Modeling of the actomyosin ATPase activity

In the rapid “quench” kientics of myosin, the “initial phosphate burst” is the excess inorganic phosphate that is produced during the early time-course of ATP hydrolysis by myosin subfragment-1 (S-1) or HMM. In general, the existence of a Pi burst implies a rapid (i.e., generally an order of magnitude faster than the steady-state hydrolysis rate) lysis of the phospho-anhydride bond within the ATP molecule, followed by one or more slower steps that are rate limiting for the process. Thus, the presence of a Pi burst can provide an important clue to the mechanism of the reaction. However, in the case of actomyosin, this clue as long been the subject of controversy and misunderstanding. To measure the (initial) Pi burst, myosin S-1 (or HMM) is rapidly mixed with ATP and then the mixture is acid quenched after a specific time period. The medium produced contains free Pi generated from hydrolysis of the ATP. The quantitative measure of the phosphate generated in this way has always been significantly greater than that expected by steady-state “release” of Pi alone, and it is that very difference between this measured Pi after the quench and that amount of Pi expected to be released by steady-state considerations in that same time period that has been referred to as the “initial Pi burst”. Recent investigations of the kinetics of Pi release have used an entirely new method that directly measures the release of Pi from the enzyme-product complex. These studies have made reference to the properties of the “initial Pi burst” in the presence of actin, as well as to a new kinetic entity: the “burst of Pi release”, and have been often vague concerning the true nature of the initial Pi burst, as well as the properties of Pi release as predicted by the current models of the actin activation of the myosin ATPase activity. The purpose of the current article is to correct this oversight, to discuss the “burst” in some detail, and to display the kinetics predicted by the current models for the actin activation of myosin. Furthermore, predictions for the kinetics of the new “burst of Pi release” are discussed in terms of its ability to discriminate between the two current competing models for actin activation of the myosin ATPase activity.     

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.     

Mechanotransduction by Membrane Proteins: The speed of the hair cell mechanotransducer channel revealed by fluctuation analysis

Although mechanoelectrical transducer (MET) channels have been extensively studied, uncertainty persists about their molecular architecture and single-channel conductance. We made electrical measurements from mouse cochlear outer hair cells (OHCs) to reexamine the MET channel conductance comparing two different methods. Analysis of fluctuations in the macroscopic currents showed that the channel conductance in apical OHCs determined from nonstationary noise analysis was about half that of single-channel events recorded after tip link destruction. We hypothesized that this difference reflects a bandwidth limitation in the noise analysis, which we tested by simulations of stochastic fluctuations in modeled channels. Modeling indicated that the unitary conductance depended on the relative values of the channel activation time constant and the applied low-pass filter frequency. The modeling enabled the activation time constant of the channel to be estimated for the first time, yielding a value of only a few microseconds. We found that the channel conductance, assayed with both noise and recording of single-channel events, was reduced by a third in a new deafness mutant, Tmc1 p.D528N. Our results indicate that noise analysis is likely to underestimate MET channel amplitude, which is better characterized from recordings of single-channel events.     

Gating Models of the Anomalous Mole-Fraction Effect of Single-Channel Current in <Emphasis Type="Italic">Chara</Emphasis>

The dependence of single-channel current on the Tl+/K+ mole fraction exhibiting a minimum at [Tl+]/[K+] of about 1:15 is proportional to open probability in bursts. Five models are suggested to explain modulation of gating by the Tl+/K+ ratio. Three models start from a channel with 4 identical subunits, each with an allosteric binding site for K+ or Tl+. In the first model, ion binding is directly observable as a transition from one Markov state to another. This model can explain the dependence of the apparent single-channel current on Tl+ concentrations. However, the predicted linear dependence on ion concentrations of the apparent rate constants was not observed in measurements in 25 or 250 mM KNO3 and 250 mM Tl NO3. The second model can overcome this problem by introducing saturation kinetics for ion binding. In the third model, gating is caused by inherent vibrations of the protein, and the rate constants of the related transitions depend on the occupation of the allosteric sites. The fourth model is based on the foot-in-the-door approach with the essential feature that two K+ ions in the selectivity filter are necessary to keep the pore radius suitable for K+ ions. The fifth model is also a foot-in-the-door model, but non-Markovian because, similar to model 3, it is assumed that the conformation of the protein (and thus the rate constants of the Markov model of the time series) depends on the force exerted by the temporal average over the states of a Markov model of ion occupation. These ions may reside in the pore itself or outside.     

Tl+-induced micros gating of current indicates instability of the MaxiK selectivity filter as caused by ion/pore interaction

Patch clamp experiments on single MaxiK channels expressed in HEK293 cells were performed at high temporal resolution (50-kHz filter) in asymmetrical solutions containing 0, 25, 50, or 150 mM Tl+ on the luminal or cytosolic side with [K+] + [Tl+] = 150 mM and 150 mM K+ on the other side. Outward current in the presence of cytosolic Tl+ did not show fast gating behavior that was significantly different from that in the absence of Tl+. With luminal Tl+ and at membrane potentials more negative than -40 mV, the single-channel current showed a negative slope resistance concomitantly with a flickery block, resulting in an artificially reduced apparent single-channel current I(app). The analysis of the amplitude histograms by beta distributions enabled the estimation of the true single-channel current and the determination of the rate constants of a simple two-state O-C Markov model for the gating in the bursts. The voltage dependence of the gating ratio R = I(true)/I(app) = (k(CO) + k(OC))/k(CO) could be described by exponential functions with different characteristic voltages above or below 50 mM Tl(+). The true single-channel current I(true) decreased with Tl+ concentrations up to 50 mM and stayed constant thereafter. Different models were considered. The most likely ones related the exponential increase of the gating ratio to ion depletion at the luminal side of the selectivity filter, whereas the influence of [Tl+] on the characteristic voltage of these exponential functions and of the value of I(true) were determined by [Tl+] at the inner side of the selectivity filter or in the cavity.     

Fast activation of a time-dependent outward current in protoplasts derived from coats of developing Phaseolus vulgaris seeds

An outward current that appeared to activate instantaneously in response to depolarising voltage pulses at low sampling frequencies predominated in the plasma membrane of ground-parenchyma protoplasts derived from coats of developing Phaseolus vulgaris L. (cv. Redland Pioneer) seeds. However, the outward current showed time-dependent activation when higher sampling frequencies were used to measure the current. Activation of the current was best described as a double-exponential time course with the fast and slow time constants being 1 and 20 ms, respectively. The current also exhibited a rapid deactivation that followed a double-exponential time course with time constants of approximately 2 and 30 ms, respectively. “Tail-current” analysis allowed us to show that this current exhibited a low selectivity between K⁺ and Cl⁻ (P _K:Cl=1.8). Such a fast-activating current may account for some of the reports of time-independent, instantaneous currents that have been observed in plasma membranes of plant cells digitised at low sampling frequencies. Therefore, when “instantaneous” currents appear it is advisable to characterise these currents using higher sampling frequencies with correspondingly higher filtering frequency cut-offs. Received: 12 May 2000 / Accepted: 26 June 2000     

Effects of animal density,volume, and the use of 2D/3D recording on behavioral studies of copepods

Studies on the behavior of copepods require both an appropriate experimental design and the means to perform objectively verifiable numerical analysis. Despite the growing number of publications on copepod behavior, it has been difficult to compare these studies. In this study, we studied two species of copepods, Eurytemora affinis and Pseudodiaptomus annandalei, and employed recently developed scaling and non-scaling methodology to investigate the effects of density and volume on the swimming behavior of individual organisms in still water. We also compared the results of two- and three-dimensional projections of the swimming tracks. A combination of scale-dependent and scale-independent analysis was found to characterize a number of behavioral observations very effectively. We discovered that (i) density has no effect except to increase the time spent in the swimming state of “breaking”, (ii) smaller volumes resulted in more complex trajectories, and larger volumes, like density, increased the time spent in the swimming state “breaking”, and (iii) three-dimensional projections gave a more accurate estimation of speed and the time spent cruising. When only a vertical 2D projection was used, “cruising” could be confused with “sinking”. These results indicate that both experimental conditions and the selection of 2D or 3D projection have important implications regarding the study of copepod behavior. The development of standardized procedures with which to compare the observations made in different studies is an issue of particular urgency.     

Improved hidden Markov models for molecular motors, part 1: basic theory

Hidden Markov models (HMMs) provide an excellent analysis of recordings with very poor signal/noise ratio made from systems such as ion channels which switch among a few states. This method has also recently been used for modeling the kinetic rate constants of molecular motors, where the observable variable—the position—steadily accumulates as a result of the motor's reaction cycle. We present a new HMM implementation for obtaining the chemical-kinetic model of a molecular motor's reaction cycle called the variable-stepsize HMM in which the quantized position variable is represented by a large number of states of the Markov model. Unlike previous methods, the model allows for arbitrary distributions of step sizes, and allows these distributions to be estimated. The result is a robust algorithm that requires little or no user input for characterizing the stepping kinetics of molecular motors as recorded by optical techniques.     

Hidden Markov modeling for single channel kinetics with filtering and correlated noise

Hidden Markov modeling (HMM) can be applied to extract single channel kinetics at signal-to-noise ratios that are too low for conventional analysis. There are two general HMM approaches: traditional Baum's reestimation and direct optimization. The optimization approach has the advantage that it optimizes the rate constants directly. This allows setting constraints on the rate constants, fitting multiple data sets across different experimental conditions, and handling nonstationary channels where the starting probability of the channel depends on the unknown kinetics. We present here an extension of this approach that addresses the additional issues of low-pass filtering and correlated noise. The filtering is modeled using a finite impulse response (FIR) filter applied to the underlying signal, and the noise correlation is accounted for using an autoregressive (AR) process. In addition to correlated background noise, the algorithm allows for excess open channel noise that can be white or correlated. To maximize the efficiency of the algorithm, we derive the analytical derivatives of the likelihood function with respect to all unknown model parameters. The search of the likelihood space is performed using a variable metric method. Extension of the algorithm to data containing multiple channels is described. Examples are presented that demonstrate the applicability and effectiveness of the algorithm. Practical issues such as the selection of appropriate noise AR orders are also discussed through examples.     

Sensor fusion by neural networks using spatially represented information

 A neural network model based on a lateral-inhibition-type feedback layer is analyzed with regard to its capabilities to fuse signals from two different sensors reporting the same event (“multisensory convergence”). The model consists of two processing stages. The input stage holds spatial representations of the sensor signals and transmits them to the second stage where they are fused. If the input signals differ, the model exhibits two different processing modes: with small differences it produces a weighted average of the input signals, whereas with large differences it enters a decision mode where one of the two signals is suppressed. The dynamics of the network can be described by a series of two first-order low-pass filters, whose bandwidth depends nonlinearly on the level of concordance of the input signals. The network reduces sensor noise by means of both its averaging and filtering properties. Hence noise suppression, too, depends on the level of concordance of the inputs. When the network's neurons have internal noise, sensor noise suppression is reduced but still effective as long as the input signals do not differ strongly. The possibility of extending the scheme to three and more inputs is discussed. Received: 2 August 2000 / Accepted in revised form: 3 May 2001