Distributions-per-level: a means of testing level detectors and models of patch-clamp data

Ca2+ block and flickering both contribute to the negative slope of the IV curve in BK channels

Single-channel current–voltage (IV) curves of human large-conductance, voltage- and Ca²⁺-activated K⁺ (BK) channels are quite linear in 150 mM KCl. In the presence of Ca²⁺ and/or Mg²⁺, they show a negative slope conductance at high positive potentials. This is generally explained by a Ca²⁺/Mg²⁺ block as by Geng et al. (2013. J. Gen. Physiol. http://dx.doi.org/10.1085/jgp.201210955) in this issue. Here, we basically support this finding but add a refinement: the analysis of the open-channel noise by means of β distributions reveals what would be found if measurements were done with an amplifier of sufficient temporal resolution (10 MHz), namely that the block by 2.5 mM Ca²⁺ and 2.5 mM Mg²⁺ per se would only cause a saturating curve up to +160 mV. Further bending down requires the involvement of a second process related to flickering in the microsecond range. This flickering is hardly affected by the presence or absence of Ca²⁺/Mg²⁺. In contrast to the experiments reported here, previous experiments in BK channels (Schroeder and Hansen. 2007. J. Gen. Physiol. http://dx.doi.org/10.1085/jgp.200709802) showed saturating IV curves already in the absence of Ca²⁺/Mg²⁺. The reason for this discrepancy could not be identified so far. However, the flickering component was very similar in the old and new experiments, regardless of the occurrence of noncanonical IV curves.

Statistical assessment of change point detectors for single molecule kinetic analysis

Change point detectors (CPDs) are used to segment recordings of single molecules for the purpose of kinetic analysis. The assessment of the accuracy of CPD algorithms has usually been based on testing them with simulated data. However, there have not been methods to assess the output of CPDs from real data independent of simulation. Here we present one method to do this based on the assumption that the elementary kinetic unit is a stationary period (SP) with a normal distribution of samples, separated from other SPs by change points (CPs). Statistical metrics of normality can then be used to assess SPs detected by a CPD algorithm (detected SPs, DSPs). Two statistics in particular were found to be useful, the z-transformed skew (S(Z)) and z-transformed kurtosis (K(Z)). K(Z)(S(Z)) plots of DSP from noise, simulated data and single ion channel recordings showed that DSPs with false negative CP could be distinguished. Also they showed that filtering had a significant effect on the normality of data and so filtering should be taken into account when calculating statistics. This method should be useful for analyzing single molecule recordings where there is no simple model for the data.

Test of normality for integrated change point detection and mixture modeling

Single-molecule data often show step-like changes in the quantity measured between constant levels. Analysis of this data consists of detecting the steps, i.e., change point detection (CPD), and determining the levels, i.e., clustering. We describe a novel algorithm which integrates these two analyses, based on a statistical test of a normal distribution. The test of normality (TON) algorithm integrates statistical CPD with gaussian mixture model clustering. We used TON with both simulated data and ion channel patch-clamp recordings. It performed well with simulated data except at a high signal-to-noise ratio and when the frequency of steps was high compared to the sampling frequency. TON has advantages over separate CPD and mixture modeling algorithms, especially for complex single-molecule data. This was illustrated by its application to the maxichannel, an ion channel with multiple subconductance states.

Modulation of enrofloxacin binding in OmpF by Mg2+ as revealed by the analysis of fast flickering single-porin current

One major determinant of the efficacy of antibiotics on Gram-negative bacteria is the passage through the outer membrane. During transport of the fluoroquinolone enrofloxacin through the trimeric outer membrane protein OmpF of Escherichia coli, the antibiotic interacts with two binding sites within the pore, thus partially blocking the ionic current. The modulation of one affinity site by Mg²⁺ reveals further details of binding sites and binding kinetics. At positive membrane potentials, the slow blocking events induced by enrofloxacin in Mg²⁺-free media are converted to flickery sojourns at the highest apparent current level (all three pores flickering). This indicates weaker binding in the presence of Mg²⁺. Analysis of the resulting amplitude histograms with β distributions revealed the rate constants of blocking (k_OB) and unblocking (k_BO) in the range of 1,000 to 120,000 s⁻¹. As expected for a bimolecular reaction, k_OB was proportional to blocker concentration and k_BO independent of it. k_OB was approximately three times lower for enrofloxacin coming from the cis side than from the trans side. The block was not complete, leading to a residual conductivity of the blocked state being ∼25% of that of the open state. Interpretation of the results has led to the following model: fast flickering as caused by interaction of Mg²⁺ and enrofloxacin is related to the binding site at the trans side, whereas the cis site mediates slow blocking events which are also found without Mg²⁺. The difference in the accessibility of the binding sites also explains the dependency of k_OB on the side of enrofloxacin addition and yields a means of determining the most plausible orientation of OmpF in the bilayer. The voltage dependence suggests that the dipole of the antibiotic has to be adequately oriented to facilitate binding.

Fast and slow gating are inherent properties of the pore module of the K+ channel Kcv

Kcv from the chlorella virus PBCV-1 is a viral protein that forms a tetrameric, functional K⁺ channel in heterologous systems. Kcv can serve as a model system to study and manipulate basic properties of the K⁺ channel pore because its minimalistic structure (94 amino acids) produces basic features of ion channels, such as selectivity, gating, and sensitivity to blockers. We present a characterization of Kcv properties at the single-channel level. In symmetric 100 mM K⁺, single-channel conductance is 114 ± 11 pS. Two different voltage-dependent mechanisms are responsible for the gating of Kcv. “Fast” gating, analyzed by β distributions, is responsible for the negative slope conductance in the single-channel current–voltage curve at extreme potentials, like in MaxiK potassium channels, and can be explained by depletion-aggravated instability of the filter region. The presence of a “slow” gating is revealed by the very low (in the order of 1–4%) mean open probability that is voltage dependent and underlies the time-dependent component of the macroscopic current.

Using a five-state model for fitting amplitude histograms from MaxiK channels: beta-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 mus(-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 mus(-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.

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 β 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.

Saturation and microsecond gating of current indicate depletion-induced instability of the MaxiK selectivity filter

Patch clamp experiments on single MaxiK channels expressed in HEK293 cells were performed with a high temporal resolution (50-kHz filter) in symmetrical solutions with 50, 150, or 400 mM KCl and 2.5 mM CaCl(2) and 2.5 mM MgCl(2). At membrane potentials >+100 mV, the single-channel current showed a negative slope resistance, concomitantly with a flickery block, which was not influenced by Ca(2+) or Mg(2+). The analysis of the amplitude histograms by beta distributions revealed that current in this voltage range was reduced by two effects: rate limitation at the cytosolic side of the pore and gating with rate constants 10-20-fold higher than the cutoff frequency of the filter (i.e., dwell times in the microsecond range). The data were analyzed in terms of a model that postulates a coupling between both effects; if the voltage over the selectivity filter withdraws ions from the cavity at a higher rate than that of refilling from the cytosol, the selectivity filter becomes instable because of ion depletion, and current is interrupted by the resulting flickering. The fit of the IV curves revealed a characteristic voltage of 35 mV. In contrast, the voltage dependence of the gating factor R, i.e., the ratio between true and apparent single-channel current, could be fitted by exponentials with a characteristic voltage of 60 mV, suggesting that only part of the transmembrane potential is felt by the flux through the selectivity filter.

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.

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.

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.