Calculation of local Hurst exponents in the Ca(2+)-activated K(+)-channel dwell time

A novel method based on the maximum overlap wavelet transform of dwell time series is proposed. Information on local multifractal properties of the series, namely local Hurst exponents or Holder exponents, was obtained. The results confirm the presence of multifractality and intrinsic correlations in the Ca(2+)-activated K+ channel dwell time series. The data on the local multifractal structure of the series can be interpreted in terms of processes having self-organized criticality. The proposed approach allows one to widen the store of methods for the analysis of single ion channel activity.     

Improving the accuracy of depth of anaesthesia using modified detrended fluctuation analysis method

This paper presents a modified detrended fluctuation analysis (MDFA) to improve the monitoring accuracy of the depth of anaesthesia (DoA). We first use MDFA to classify anaesthesia state levels into awake, light, moderate, deep and very deep states. Then we build up five zones using linear regression method from very deep anaesthesia state to awake state, corresponding with different box sizes. Finally, the Lagrange method is applied to compute the DoA. Comparing with the most popular Bispectral Index (BIS) method, our modified DFA method extends the ranges of the moderate anaesthesia, deep anaesthesia and very deep anaesthesia to provide more information about the DoA. This extension is very significant in the clinical perspective as these states are within the ranges for operations and need more attention. Simulation results demonstrate that the new technique monitors the DoA in all anaesthesia states accurately.     

Discriminating coding, non-coding and regulatory regions using rescaled range and detrended fluctuation analysis

In this paper we analyse the efficiency of two methods, rescaled range analysis and detrended fluctuation analysis, in distinguishing between coding DNA, regulatory DNA and non-coding non-regulatory DNA of Drosophila melanogaster. Both methods were used to estimate the degree of sequential dependence (or persistence) among nucleotides. We found that these three types of DNA can be discriminated by both methods, although rescaled range analysis performs slightly better than detrended fluctuation analysis. On average, non-coding, non-regulatory DNA has the highest degree of sequential persistence. Coding DNA could be characterised as being anti-persistent, which is in line with earlier findings of latent periodicity. Regulatory regions are shown to possess intermediate sequential dependency. Together with other available methods, rescaled range and detrended fluctuation analysis on the basis of a combined purine/pyrimidine and weak/strong classification of the nucleotides are useful tools for refined structural and functional segmentation of DNA.     

Existence of memory in membrane channels: analysis of ion current through a voltage‐dependent potassium single channel

Ion current fluctuation of voltage‐dependent potassium channel in LβT2 cells has been investigated by autocorrelation function and DFA (detrended fluctuation analysis) methods. The calculation of the autocorrelation function exponent and DFA exponent of the sample was based on the digital signals or the 0–1 series corresponding to closing and opening of channels after routine evolution, rather than the sequence of sojourn times. The persistent character of the correlation of the time series was evident from the slow decay of the autocorrelation function. DFA exponent α was significantly greater than 0.5. The main outcome has been the demonstration of the existence of memory in this ion channel. Thus, the ion channel current fluctuation provided information about the kinetics of the channel protein. The result suggests the correlation character of the ion channel protein non‐linear kinetics indicates whether the channel is open or not.     

Multichannel detrended fluctuation analysis reveals synchronized patterns of spontaneous spinal activity in anesthetized cats

The analysis of the interaction and synchronization of relatively large ensembles of neurons is fundamental for the understanding of complex functions of the nervous system. It is known that the temporal synchronization of neural ensembles is involved in the generation of specific motor, sensory or cognitive processes. Also, the intersegmental coherence of spinal spontaneous activity may indicate the existence of synaptic neural pathways between different pairs of lumbar segments. In this study we present a multichannel version of the detrended fluctuation analysis method (mDFA) to analyze the correlation dynamics of spontaneous spinal activity (SSA) from time series analysis. This method together with the classical detrended fluctuation analysis (DFA) were used to find out whether the SSA recorded in one or several segments in the spinal cord of the anesthetized cat occurs either in a random or in an organized manner. Our results are consistent with a non-random organization of the sets of neurons involved in the generation of spontaneous cord dorsum potentials (CDPs) recorded either from one lumbar segment (DFA-α mean = 1.04[Formula: see text]0.09) or simultaneously from several lumbar segments (mDFA-α mean = 1.01[Formula: see text]0.06), where α = 0.5 indicates randomness while α = 0.5 indicates long-term correlations. To test the sensitivity of the mDFA method we also examined the effects of small spinal lesions aimed to partially interrupt connectivity between neighboring lumbosacral segments. We found that the synchronization and correlation between the CDPs recorded from the L5 and L6 segments in both sides of the spinal cord were reduced when a lesion comprising the left dorsal quadrant was performed between the segments L5 and L6 (mDFA-[Formula: see text] = 0.992 as compared to initial conditions mDFA-α = 1.186). The synchronization and correlation were reduced even further after a similar additional right spinal lesion (mDFA-α = 0.924). In contrast to the classical methods, such as correlation and coherence quantification that define a relation between two sets of data, the mDFA method properly reveals the synchronization of multiple groups of neurons in several segments of the spinal cord. This method is envisaged as a useful tool to characterize the structure of higher order ensembles of cord dorsum spontaneous potentials after spinal cord or peripheral nerve lesions.     

Interactions of protons with single open L-type calcium channels. pH dependence of proton-induced current fluctuations with Cs+, K+, and Na+ as permeant ions

We studied the pH dependence of the proton-induced current fluctuations that appear in single open L-type Ca channels when monovalent ions are the charge carriers. We used different methods of analysis to obtain kinetic measurements even under conditions where the individual transitions were too fast to be resolved directly as discrete current steps between two conductance levels. The reciprocal of the dwell times at the high conductance level increased linearly with the pipette proton activity, with a slope that was similar for Cs, K, and Na as permeant ions. Contrary to the expectation for a simple model in which the high and low conductances represent the unprotonated and protonated states of the channel, respectively, the dwell times at the low conductance level were also pH dependent and lengthened with increasing proton activity. At all pH values the dwell times at the low conductance level were longest with Cs as permeant ion and shortened in the order Cs greater than K greater than Na. We introduce a more general model of the protonation cycle in which the channel is represented by four states and can be protonated and deprotonated both at the high and low conductance levels. The conductance change is represented by a conformational change of the channel protein. We discuss the validity of this model and its implications for the mechanism by which protons interact with ion permeation through L-type Ca channels.     

Gating of the squid sodium channel at positive potentials: II. Single channels reveal two open states.

Single-channel recordings from squid axon Na+ channels were made under conditions of reverse sodium gradient. In the range of potentials studied, +40-(+)120 mV, channels opened promptly after depolarization, closed and reopened several times during the pulse. In patches containing only one channel, the distributions of open dwell times showed two components showing the existence of a second open state. The ensemble average of single-channel records showed incomplete inactivation that became more pronounced at more positive potentials, showing that the maintained phase of the current is the result of only one type of sodium channel with two open states. Analysis of bursts indicated that the dwell times of the events at the onset of the depolarization are longer than those later in the pulse. The dwell open times of the first events could be fitted with a single exponential. This indicated that the channels open preferentially through the first open state, the access to the second open state happening subsequently. Maximum likelihood analysis was used to evaluate several possible kinetic schemes incorporating a second open state. The best model to fit the data from single channels, and consistent with the data from macroscopic and gating currents, has a second open state evolving from the inactivated state. A kinetic model is proposed that incorporates information obtained from dialyzed axons.     

Relationship between statistical properties of single ionic channel recordings and the thermodynamic state of the channels

Recordings of the electric conductivity of a single ionic channel usually exhibit two levels of conductance: a zero and a finite level. The channel may, however, be in a few states which have the same conductivity level, and the distribution of dwell time durations at this conductivity level is thus not monoexponential. It is shown that the joint probability p(tc,to) of the occurrence of a time interval tc during which the channel is not conducting, immediately followed by a time interval to during which the channel is conducting may or may not be equal to the joint probability pr(tc,to) of the occurrence of a non-conducting interval tc preceded by a conducting interval to. If the interconversions between the various states in which the channel can exist obey detailed balance, i.e., if the channel behaves like a system at thermodynamic equilibrium, then p(tc,to) = pr(tc,to). This should help to reveal whether irreversible processes, like metabolic reactions or flows of substances across the membrane, are coupled to the gating process of the ionic channels.     

Multifractal Properties of a Closed Contour: A Peek beyond the Shape Analysis

In recent decades multifractal analysis has been successfully applied to characterize the complex temporal and spatial organization of such diverse natural phenomena as heartbeat dynamics, the dendritic shape of neurons, retinal vessels, rock fractures, and intricately shaped volcanic ash particles. The characterization of multifractal properties of closed contours has remained elusive because applying traditional methods to their quasi-one-dimensional nature yields ambiguous answers. Here we show that multifractal analysis can reveal meaningful and sometimes unexpected information about natural structures with a perimeter well-defined by a closed contour. To this end, we demonstrate how to apply multifractal detrended fluctuation analysis, originally developed for the analysis of time series, to an arbitrary shape of a given study object. In particular, we show the application of the method to fish otoliths, calcareous concretions located in fish''s inner ear. Frequently referred to as the fish''s “black box", they contain a wealth of information about the fish''s life history and thus have recently attracted increasing attention. As an illustrative example, we show that a multifractal approach can uncover unexpected relationships between otolith contours and size and age of fish at maturity.     

Fatigue estimation using a novel multi-fractal detrended fluctuation analysis-based approach

This paper presents a novel multi-fractal detrended fluctuation analysis-based approach for fatigue estimation. This approach exploits the statistical self-similarity and long-range correlation of surface electromyography signals at different time scales in which the myoelectric manifestation of fatigue is more significant compared to the influence of varying force, muscle length (joint angle), and innervation zone. This approach provides a fatigue index which outperforms the conventional median frequency during cyclic and random contractions. This type of analysis is promising an efficient framework for analysis of surface electromyography signals with several potential applications.     

Brownian motion, fluctuation and life

The measurements of dynamic behaviors of biomolecules in relation to their functions have been allowed using single molecule measurements. Thermal Brownian motion causes random step motion of motor proteins and structural fluctuation of protein molecules between multiple states. In hierarchic structure of life, the fluctuation is modulated. Random fluctuation is biased to directional motion and reactions as a result of interaction of proteins. The fluctuation of kinetic state of signaling proteins results in polarization and localization of cells. A recognition process in brain is also explained by the equation analogous to biochemical reaction at the molecular level. Thus dynamic processes originated from thermal motion may play an important role in activation processes in life.     

Multifractal analysis of K+ channel activity

The Multifractal version of the Detrended Fluctuation Analysis was used for the study of non-stationary dwell time series of Ca²⁺-activated K⁺ channels (K_Ca channels) in cultured kidney Vero cells and of voltage-dependent K⁺ channels (K_v channels) in mollusc (Lymnaea stagnalis) neurons. The data obtained can briefly be summarized as follows: (i) The generalized fluctuation function F _q (l) strongly depends on the index (order) q; for monofractal time series, such dependence is nonexistent; (ii) The relationship between the scaling exponent τ commonly employed in standard multifractal analysis and q is characterized by two slopes and a transitory region, whereas monofractal processes are characterized by the linear dependence; (iii) The relationship between the singularity spectrum f(h) and the Hurst exponent h is bell-shaped, while in the case of monofractal processes it is represented by a single point f(h) = 1. Random mixing of the time series resulted in the narrowing of the spectrum f(h) and a shift of f(h) towards the value more characteristic of stochastic (monofractal) processes (h ～ 0.5). It is concluded that the activities of both K_Ca channels in kidney Vero cells and of K_V channels in mollusc (Lymnaea stagnalis) neurons can be characterized as multifractal processes.     

Theory of the kinetic analysis of patch-clamp data.

This paper describes a theory of the kinetic analysis of patch-clamp data. We assume that channel gating is a Markov process that can be described by a model consisting of n kinetic states and n(n - 1) rate constants at each voltage, and that patch-clamp data describe the occupancy of x different conductance levels over time. In general, all the kinetic information in a set of patch-clamp data is found in either two-dimensional dwell time histograms describing the frequency of observation of sequential dwell times of durations tau 1 and tau 2 (Fredkin, D. R., M. Montal, and J. A. Rice, 1985, Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. 1, 269-289) or in three-point joint probability functions describing the probability that a channel is in a given conductance at time t, and at time t + tau 1, and at time t + tau 1 + tau 2. For the special case of channels with a single open state plus multiple closed states, one-dimensional analyses provide all of the kinetic information. Stationary patch-clamp data have information that can be used to determine H rate constants, where H = n(n - 1) - G and G is the number of intraconductance rate constants. Thus, to calculate H rate constants, G rate constants must be fixed. In general there are multiple sets of G rate constants that can be fixed to allow the calculation of H rate constants although not every set of G rate constants will work. Arbitrary assignment of the G intraconductance rate constants equal to zero always provides a solution and the calculation of H rate constants. Nonstationary patch-clamp data have information for the determination of H rate constants at a reference voltage plus n(n - 1) rate constants at all test voltages. Thus, nonstationary data have extra information about the voltage dependencies of rate constants that can be used to rule out kinetic models that cannot be disqualified on the basis of stationary data.     

Cumulant analysis in fluorescence fluctuation spectroscopy

A novel technique for the analysis of fluorescence fluctuation experiments is introduced. Fluorescence cumulant analysis (FCA) exploits the factorial cumulants of the photon counts and resolves heterogeneous samples based on differences in brightness. A simple analytical model connects the cumulants of the photon counts with the brightness epsilon and the number of molecules N in the optical observation volume for each fluorescent species. To provide the tools for a rigorous error analysis of FCA, expressions for the variance of factorial cumulants are developed and tested. We compare theory with experiment by analyzing dye mixtures and simple fluorophore solutions with FCA. A comparison of FCA with photon-counting histogram (PCH) analysis, a related technique, shows that both methods give identical results within experimental uncertainty. Both FCA and PCH are restricted to data sampling times that are short compared to the diffusion time of molecules through the observation volume of the instrument. But FCA theory, in contrast to PCH, can be extended to treat arbitrary sampling times. Here, we derive analytical expressions for the second factorial cumulant as a function of the sampling time and demonstrate that the theory successfully models fluorescence fluctuation data.     

A model of ion channel kinetics using deterministic chaotic rather than stochastic processes.

Models of ion channel kinetics have previously assumed that the switching between the open and closed states is an intrinsically random process. Here, we present an alternative model based on a deterministic process. This model is a piecewise linear iterated map. We calculate the dwell time distributions, autocorrelation function, and power spectrum of this map. We also explore non-linear generalizations of this map. The chaotic nature of our model implies that its long-term behavior mimics the stochastic properties of a random process. In particular, the linear map produces an exponential probability distribution of dwell times in the open and closed states, the same as that produced by the two-state, closed in equilibrium open, Markov model. We show how deterministic and random models can be distinguished by their different phase space portraits. A test of some experimental data seems to favor the deterministic model, but further experimental evidence is needed for an unequivocal decision.     

Voltage-dependent block of anthrax toxin channels in planar phospholipid bilayer membranes by symmetric tetraalkylammonium ions. Single-channel analysis

Previous studies have shown that symmetric tetraalkylammonium ions affect, in a voltage-dependent manner, the conductance of membranes containing many channels formed by the PA65 fragment of anthrax toxin. In this paper we analyze this phenomenon at the single-channel level for tetrabutylammonium ion (Bu4N+). We find that Bu4N+ induces a flickery block of the PA65 channel when present on either side of the membrane, and this block is relieved by large positive voltages on the blocking-ion side. At high frequencies (greater than 2 kHz) we have resolved individual blocking events and measured the dwell times in the blocked and unblocked states. These dwell times have single-exponential distributions, with time constants tau b and tau u that are voltage dependent, consistent with the two-barrier, single-well potential energy diagram that we postulated in our previous paper. The fraction of time the channel spends unblocked [tau u/(tau u + tau b)] as a function of voltage is identical to the normalized conductance-voltage relation determined from macroscopic measurements of blocking, thus demonstrating that these single channels mirror the behavior seen with many (greater than 10,000) channels in the membrane. In going from large negative to large positive voltages (-100 to +160 mV) on the cis (PA65-containing) side of the membrane, one sees the mean blocked time (tau b) increase to a maximum at +60 mV and then steadily decline for voltages greater than +60 mV, thereby clearly demonstrating that Bu4N+ is driven through the channel by positive voltages on the blocking-ion side. In other words, the channel is permeable to Bu4N+. An interesting finding that emerges from analysis of the voltage dependence of mean blocked and unblocked times is that the blocking rate, with Bu4N+ present on the cis side of the membrane, plateaus at large positive cis voltages to a voltage-independent value consistent with the rate of Bu4N+ entry into the blocking site being diffusion limited.     

Glutamate Receptor Channel Kinetics: The Effect of Glutamate Concentration

Single channel recordings from the locust muscle D-glutamate receptor channel were obtained using glutamate concentrations ranging from 10^-6 to 10^-2 M. Channel kinetics were analyzed to aid in the development of a model for the gating mechanism. Analysis of channel dwell time histograms demonstrated that the channel possessed multiple open and closed states at concentrations of glutamate between 10^-5 and 10^-2 M. Correlations between successive dwell times showed that the gating mechanism was nonlinear (i.e., branched or cyclic) over the same glutamate concentration range. The glutamate concentration dependence of the channel open probability, and of the event frequency, was used to explore two possible allosteric gating mechanisms in more detail.     

Estimating transition rates in aggregated Markov models of ion channel gating with loops and with nearly equal dwell times

A typical task in the application of aggregated Markov models to ion channel data is the estimation of the transition rates between the states. Realistic models for ion channel data often have one or more loops. We show that the transition rates of a model with loops are not identifiable if the model has either equal open or closed dwell times. This non-identifiability of the transition rates also has an effect on the estimation of the transition rates for models which are not subject to the constraint of either equal open or closed dwell times. If a model with loops has nearly equal dwell times, the Hessian matrix of its likelihood function will be ill-conditioned and the standard deviations of the estimated transition rates become extraordinarily large for a number of data points which are typically recorded in experiments.     

Time-integrated fluorescence cumulant analysis in fluorescence fluctuation spectroscopy

We introduce a new analysis technique for fluorescence fluctuation data. Time-integrated fluorescence cumulant analysis (TIFCA) extracts information from the cumulants of the integrated fluorescence intensity. TIFCA builds on our earlier FCA theory, but in contrast to FCA or photon counting histogram (PCH) analysis is valid for arbitrary sampling times. The motivation for long sampling times lies in the improvement of the signal/noise ratio of the data. Because FCA and PCH theory are not valid in this regime, we first derive a theoretical model of cumulant functions for arbitrary sampling times. TIFCA is the first exact theory that describes the effects of sampling time on fluorescence fluctuation experiments. We calculate factorial cumulants of the photon counts for various sampling times by rebinning of the original data. Fits of the data to models determine the brightness, the occupation number, and the diffusion time of each species. To provide the tools for a rigorous error analysis of TIFCA, expressions for the variance of cumulants are developed and tested. We demonstrate that over a limited range rebinning reduces the relative error of higher order cumulants, and therefore improves the signal/noise ratio. The first four cumulant functions are explicitly calculated and are applied to simple dye systems to test the validity of TIFCA and demonstrate its ability to resolve species.