A subsequent fit of time series and amplitude histogram of patch-clamp records reveals rate constants up to 1 per microsecond

2D-dwell-time analysis with simulations of ion-channel gating using high-performance computing

Single-channel patch-clamp recordings allow observing the action of a single protein complex in real time and hence the deduction of the underlying conformational changes in the ion-channel protein. Commonly, recordings are modeled using hidden Markov chains, connecting open and closed states in the experimental data with protein conformations. The rates between states denote transition probabilities that could be modified by membrane voltage or ligand binding. Modeling algorithms have to deal with limited recording bandwidth and a very noisy background. It was previously shown that the fit of two-dimensional (2D)-dwell-time histograms with simulations is very robust in that regard. Errors introduced by the low-pass filter or noise cancel out to a certain degree when comparing experimental and simulated data. In addition, the topology of models (that is, the chain of open and closed states) could be inferred from 2D-histograms. However, the 2D-fit was never applied to its full potential. A major reason may be the extremely time-consuming and often unreliable fitting process, due to the stochastic variability in the simulations. We have now solved these issues by introducing a message-passing interface (MPI) allowing massive parallel computing on a high-performance computing (HPC) cluster and obtaining ensemble solutions. With ensembles, we have demonstrated how important ranked solutions are for difficult tasks related to a noisy background, fast gating events beyond the corner frequency of the low-pass filter, and topology estimation of the underlying Markov model. Finally, we have shown that, by combining the objective function of the 2D-fit with the deviation of the current amplitude distributions, automatic determination of the current level of the conducting state is possible, even with an apparent current reduction due to low-pass filtering. Making use of an HPC cluster, the power of 2D-dwell-time analysis can be used to its fullest with minor input of the experimenter.

High bandwidth approaches in nanopore and ion channel recordings - A tutorial review

Transport processes through ion-channel proteins, protein pores, or solid-state nanopores are traditionally recorded with commercial patch-clamp amplifiers. The bandwidth of these systems is typically limited to 10 kHz by signal-to-noise-ratio (SNR) considerations associated with these measurement platforms. At high bandwidth, the input-referred current noise in these systems dominates, determined by the input-referred voltage noise of the transimpedance amplifier applied across the capacitance at the input of the amplifier. This capacitance arises from several sources: the parasitic capacitance of the amplifier itself; the capacitance of the lipid bilayer harboring the ion channel protein (or the membrane used to form the solid-state nanopore); and the capacitance from the interconnections between the electronics and the membrane. Here, we review state-of-the-art applications of high-bandwidth conductance recordings of both ion channels and solid-state nanopores. These approaches involve tightly integrating measurement electronics fabricated in complementary metal-oxide semiconductors (CMOS) technology with lipid bilayer or solid-state membranes. SNR improvements associated with this tight integration push the limits of measurement bandwidths, in some cases in excess of 10 MHz. Recent case studies demonstrate the utility of these approaches for DNA sequencing and ion-channel recordings. In the latter case, studies with extended bandwidth have shown the potential for providing new insights into structure-function relations of these ion-channel proteins as the temporal resolutions of functional recordings matches time scales achievable with state-of-the-art molecular dynamics simulations.

Extended beta distributions open the access to fast gating in bilayer experiments-assigning the voltage-dependent gating to the selectivity filter

Lipid bilayers provide many benefits for ion channel recordings, such as control of membrane composition and transport molecules. However, they suffer from high membrane capacitance limiting the bandwidth and impeding analysis of fast gating. This can be overcome by fitting the deviations of the open-channel noise from the baseline noise by extended beta distributions. We demonstrate this analysis step-by-step by applying it to the example of viral K+  channels (Kcv), from the choice of the gating model through the fitting process, validation of the results, and what kinds of results can be obtained. These voltage sensor-less channels show profoundly voltage-dependent gating with dwell times in the closed state of about 50 μs. Mutations assign it to the selectivity filter.

A stochastic mechanism for signal propagation in the brain: Force of rapid random fluctuations in membrane potentials of individual neurons

There are two functionally important factors in signal propagation in a brain structural network: the very first synaptic delay-a time delay about 1ms-from the moment when signals originate to the moment when observation on the signal propagation can begin; and rapid random fluctuations in membrane potentials of every individual neuron in the network at a timescale of microseconds. We provide a stochastic analysis of signal propagation in a general setting. The analysis shows that the two factors together result in a stochastic mechanism for the signal propagation as described below. A brain structural network is not a rigid circuit rather a very flexible framework that guides signals to propagate but does not guarantee success of the signal propagation. In such a framework, with the very first synaptic delay, rapid random fluctuations in every individual neuron in the network cause an "alter-and-concentrate effect" that almost surely forces signals to successfully propagate. By the stochastic mechanism we provide analytic evidence for the existence of a force behind signal propagation in a brain structural network caused by rapid random fluctuations in every individual neuron in the network at a timescale of microseconds with a time delay of 1ms.

Signal propagation in small-world biological networks with weak noise

An emerging notion in systems biology is that biological networks have evolved to function well while their components behave stochastically. Thus, the dynamics in a biological network consist of two parts, deterministic and stochastic. A fundamental question is to find a quantitative relation between the two parts. We term such a relation as a deterministic-stochastic principle (DSP) and propose a model for a DSP with regard to signal propagation in biological networks. In this model, (i) the dynamics in a biological network is supposed to be captured by a stochastic differential equation which has been a standard approach in modeling systems with internal noise; (ii) the internal noise of a biological network is weak as is apparent in experimental observations; and (iii) a biological network is organized as small-world as suggested by recent studies. We introduce the concept of a signaling sample path. Using this concept we relate the structure of a biological network to its dynamics. The network structure characterizes the deterministic part of the dynamics, which in turn ensures a probability for a signal to propagate. The weakness of the internal noise characterizes the stochastic part of the dynamics. Analysis of the proposed model yields a quantitative description as follows: In a small-world biological network with weak internal noise, the signaling pathways (induced by the network structure) for a signal may ensure a probability near 0 for the signal propagation. Despite such a small probability, a correct response to the signal will still occur with a probability close to 1 provided that this signal propagation can take a certain amount of time. Computer simulations are performed to illustrate this result. We also discuss how a recent study on the reconstruction of a transcription network in Saccharomyces cerevisiae has tested the proposed model against real data.

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.

Molecular Properties of Kcv, a Virus Encoded K+ Channel

The miniature viral K+ channel Kcv represents the pore module of all K+ channels. A synthetic gene of Kcv with an elevated GC content compared to that of the wild-type gene was expressed heterologously in Pichia pastoris, and the purified protein was functionally reconstituted into liposomes. Biochemical assays reveal a remarkable cation selective stability of the channel tetramer via SDS-PAGE. Only cations, which permeate Kcv, were able to protect the oligomer against disassembly into monomers at high temperatures. Electrophysiological characterization of the single Kcv channel reveals a saturating conductance (lambda(max)) of 360 pS; the single-channel current-voltage relation was strongly rectifying with a negative slope conductance at extreme voltages. The channel was highly selective for K+ and was blocked by Ba2+ and in a side specific manner by Na+ and Cs+ also. The channel conducted Rb+, but as a consequence, the channel was shifted into a hyperactive state. We conclude that specific binding interactions of cations in the conductive pathway are an important determinant of channel stability and function.

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.