Characterization of single channel currents using digital signal processing techniques based on Hidden Markov Models.

Techniques for extracting small, single channel ion currents from background noise are described and tested. It is assumed that single channel currents are generated by a first-order, finite-state, discrete-time, Markov process to which is added 'white' background noise from the recording apparatus (electrode, amplifiers, etc). Given the observations and the statistics of the background noise, the techniques described here yield a posteriori estimates of the most likely signal statistics, including the Markov model state transition probabilities, duration (open- and closed-time) probabilities, histograms, signal levels, and the most likely state sequence. Using variations of several algorithms previously developed for solving digital estimation problems, we have demonstrated that: (1) artificial, small, first-order, finite-state, Markov model signals embedded in simulated noise can be extracted with a high degree of accuracy, (2) processing can detect signals that do not conform to a first-order Markov model but the method is less accurate when the background noise is not white, and (3) the techniques can be used to extract from the baseline noise single channel currents in neuronal membranes. Some studies have been included to test the validity of assuming a first-order Markov model for biological signals. This method can be used to obtain directly from digitized data, channel characteristics such as amplitude distributions, transition matrices and open- and closed-time durations.     

大鼠大脑皮层神经元单离子钾通道门控动力学

采用膜片钳制技术,对新生大鼠大脑皮层神经元作了细胞贴附和内膜向外两种模式的单通道电流记录。通道开放和关闭事件的转换过程为一随机过程,开关时间服从指数分布。细胞膜单离子通道的时间常数和对离子通透性接近的构象组成构象集合态。由残差法获得模型参数初值,由非线性最小二乘法获得修正值。新生大鼠大脑皮层神经元在-40mV钳制电位和电极内外对称高钾溶液下,细胞贴附和内膜向外的两种膜片钳制模式的单通道电流的动力学特征有明显差异。通道开放时间分布接近一状态分布。细胞贴附时的通道平均开放时间为2.53ms。内膜向外时的通道平均开放时间为2.04ms。关闭时间分布接近三状态分布,细胞贴附时通道平均关闭时间为3.36ms,内膜向外时通道的平均关闭时间为7.58ms。细胞贴附下,通道关闭时主要处于第一和第二关闭态;内膜向外下,通道关闭时主要处于第一关闭态。经初值估计和参数修正,得到各状态间的转移概率密度常数。     

Yet another approach to the dwell-time omission problem of single-channel analysis.

Distortion of the open-time or closed-time distributions of single channel currents, due to limited time resolution of the recording system, has been addressed by many authors. The calculation of the modified distributions generally involves the numerical inversion of a Laplace transform and is difficult to apply in fitting multistate kinetic schemes to data. Our approach is to introduce "virtual states" into the kinetic scheme, as suggested by Blatz and Magleby (1986. Biophys. J. 49:967-980) to account for missed events. To simplify the assignment of rate constants in multistate schemes we make use of Kienker's (1989. Proc. R. Soc. Lond. 236:296-309) theory to first transform schemes to uncoupled form. Our approach provides a good approximation to the exact solution, while allowing the observable dwell-time distributions, and also the second-order probability density functions, to be computed by standard matrix techniques.     

Adaptive processing techniques based on hidden Markov models for characterizing very small channel currents buried in noise and deterministic interferences.

Techniques for characterizing very small single-channel currents buried in background noise are described and tested on simulated data to give confidence when applied to real data. Single channel currents are represented as a discrete-time, finite-state, homogeneous, Markov process, and the noise that obscures the signal is assumed to be white and Gaussian. The various signal model parameters, such as the Markov state levels and transition probabilities, are unknown. In addition to white Gaussian noise, the signal can be corrupted by deterministic interferences of known form but unknown parameters, such as the sinusoidal disturbance stemming from AC interference and a drift of the base line owing to a slow development of liquid-junction potentials. To characterize the signal buried in such stochastic and deterministic interferences, the problem is first formulated in the framework of a Hidden Markov Model and then the Expectation Maximization algorithm is applied to obtain the maximum likelihood estimates of the model parameters (state levels, transition probabilities), signals, and the parameters of the deterministic disturbances. Using fictitious channel currents embedded in the idealized noise, we first show that the signal processing technique is capable of characterizing the signal characteristics quite accurately even when the amplitude of currents is as small as 5-10 fA. The statistics of the signal estimated from the processing technique include the amplitude, mean open and closed duration, open-time and closed-time histograms, probability of dwell-time and the transition probability matrix. With a periodic interference composed, for example, of 50 Hz and 100 Hz components, or a linear drift of the baseline added to the segment containing channel currents and white noise, the parameters of the deterministic interference, such as the amplitude and phase of the sinusoidal wave, or the rate of linear drift, as well as all the relevant statistics of the signal, are accurately estimated with the algorithm we propose. Also, if the frequencies of the periodic interference are unknown, they can be accurately estimated. Finally, we provide a technique by which channel currents originating from the sum of two or more independent single channels are decomposed so that each process can be separately characterized. This process is also formulated as a Hidden Markov Model problem and solved by applying the Expectation Maximization algorithm. The scheme relies on the fact that the transition matrix of the summed Markov process can be construed as a tensor product of the transition matrices of individual processes.     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.     

On the admittance of membranes associated with channel conduction. Application to channels at non-equilibrium steady state

The small-signal admittance of membranes associated with channel conduction is derived for a general channel model. A general channel model is represented by a set of chemical reactions with each species of the reactions representing a channel state. The membrane admittance is shown to be related to the phenomenological relaxation matrix of the reactions. If the kinetic reactions are at a non-equilibrium steady state, the relaxation matrix may have complex eigenvalues and the equivalent circuit of the membrane admittance may contain RLC or RLC-like branches. For equilibrium kinetic systems, on the other hand, the equivalent circuit contains only RL or RC branches. Thus, the membrane admittance of equilibrium channels is quite different from that of non-equilibrium channels. In particular, we show that the low frequency feature in the admittance of squid axons as observed by Fishman, Poussart, Moore &; Siebenga (1977) can be obtained easily from a non-equilibrium cycling steady-state model.     

Calcium-activated endonexin II forms calcium channels across acidic phospholipid bilayer membranes

Human endonexin II (annexin V) and recombinant human endonexin II can be activated by Ca2+ to interact with acidic phospholipid bilayers formed at the tip of a patch pipette. Once associated with the bilayer, endonexin II forms voltage-gated channels which are selective for divalent cations according to the following series Ca2+ greater than Ba2+ greater than Sr2+ much greater than Mg2+. However, endonexin II also expresses a selective affinity for Ca2+ which is manifest by an observed reduced current through the open channel when Ca2+ is the charge carrier. La3+ blocks endonexin II channels, as it does synexin (annexin VII) and other types of Ca2+ channels. However, as with synexin, the dihydropyridine Ca2+ channel antagonist nifedipine does not affect endonexin II channel activity. Endonexin II channels are also permeant to Li+, Cs+, Na+, and to a lesser extent, K+, resembling in this manner Ca2+ release channels from sarcoplasmic reticulum. Indeed, the low affinity of endonexin II channels for such ions as Cs+ or Li+ have allowed us to use these cations for measurement of the kinetic properties of the channel, with minimal concerns for the ion/channel interactions observed with the physiological substrate, Ca+. Finally, we observed that endonexin II channel activity always occurred in bursts, making necessary the use of two exponential functions to fit open- and closed-time histograms. We conclude from these data that the domain responsible for endonexin II channel activity, first observed by ourselves in the homologue synexin, is probably the C-terminal tetrad repeat common to both molecules.     

Percolation model of ionic channel dynamics.

The nonexponential closed-time distributions observed for ionic channels have been explained recently by quasi-one-dimensional models of structural diffusion (Millhauser, G. L., E. E. Salpeter, and R. E. Oswald. 1988. Proc. Natl. Acad. Sci. USA. 85: 1503-1507; Condat, C. A., and J. J?ckle. 1989. Biophys. J. 55: 915-925; Levitt, D. G. 1989. Biophys. J. 55: 489-498). We generalize this treatment by allowing for more complex trajectories using percolation theory. We assume that the gating transition depends on marginally connected conformational states leading to the observed spread in time scales.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

Sequence of events underlying the allosteric transition of rod cyclic nucleotide-gated channels

Activation of cyclic nucleotide-gated (CNG) ion channels involves a conformational change in the channel protein referred to as the allosteric transition. The amino terminal region and the carboxyl terminal cyclic nucleotide-binding domain of CNG channels have been shown to be involved in the allosteric transition, but the sequence of molecular events occurring during the allosteric transition is unknown. We recorded single-channel currents from bovine rod CNG channels in which mutations had been introduced in the binding domain at position 604 and/or the rat olfactory CNG channel amino terminal region had been substituted for the bovine rod amino terminal region. Using a hidden Markov modeling approach, we analyzed the kinetics of these channels activated by saturating concentrations of cGMP, cIMP, and cAMP. We used thermodynamic mutant cycles to reveal an interaction during the allosteric transition between the purine ring of the cyclic nucleotides and the amino acid at position 604 in the binding site. We found that mutations at position 604 in the binding domain alter both the opening and closing rate constants for the allosteric transition, indicating that the interactions between the cyclic nucleotide and this amino acid are partially formed at the time of the transition state. In contrast, the amino terminal region affects primarily the closing rate constant for the allosteric transition, suggesting that the state-dependent stabilizing interactions between amino and carboxyl terminal regions are not formed at the time of the transition state for the allosteric transition. We propose that the sequence of events that occurs during the allosteric transition involves the formation of stabilizing interactions between the purine ring of the cyclic nucleotide and the amino acid at position 604 in the binding domain followed by the formation of stabilizing interdomain interactions.     

Molecular dynamics study of gating in the mechanosensitive channel of small conductance MscS

Mechanosensitive channels are a class of ubiquitous membrane proteins gated by mechanical strain in the cellular membrane. MscS, the mechanosensitive channel of small conductance, is found in the inner membrane of Escherichia coli and its crystallographic structure in an open form has been recently solved. By means of molecular dynamics simulations we studied the stability of the channel conformation suggested by crystallography in a fully solvated lipid (POPC) bilayer, the combined system encompassing 224,340 atoms. When restraining the backbone of the protein, the channel remained in the open form and the simulation revealed intermittent permeation of water molecules through the channel. Abolishing the restraints under constant pressure conditions led to spontaneous closure of the transmembrane channel, whereas abolishing the restraints when surface tension (20 dyn/cm) was applied led to channel widening. The large balloon-shaped cytoplasmic domain of MscS exhibited spontaneous diffusion of ions through its side openings. Interaction between the transmembrane domain and the cytoplasmic domain of MscS was observed and involved formation of salt bridges between residues Asp62 and Arg128; this interaction may be essential for the gating of MscS. K+ and Cl- ions showed distinctively different distributions in and around the channel.     

Two-dimensional probability density analysis of single channel currents from reconstituted acetylcholine receptors and sodium channels

Two-dimensional probability density analysis of single channel current recordings was applied to two purified channel proteins reconstituted in planar lipid bilayers: Torpedo acetylcholine receptors and voltage-sensitive sodium channels from rat brain. The information contained in the dynamic history of the gating process, i.e., the time sequence of opening and closing events was extracted from two-dimensional distributions of transitions between identifiable states. This approach allows one to identify kinetic models consistent with the observables. Gating of acetylcholine receptors expresses "memory" of the transition history: the receptor has two channel open (O) states; the residence time in each of them strongly depends on both the preceding open time and the intervening closed interval. Correspondingly, the residence time in the closed (C) states depends on both the preceding open time and the preceding closed time. This result confirms the scheme that considers, at least, two transition pathways between the open and closed states and extends the details of the model in that it defines that the short-lived open state is primarily entered from long-lived closed states while the long-lived open state is accessed mainly through short-lived closed states. Since ligand binding to the acetylcholine-binding sites is a reaction with channel closed states, we infer that the longest closed state (approximately 19 ms) is unliganded, the intermediate closed state (approximately 2 ms) is singly liganded and makes transitions to the short open state (approximately 0.5 ms) and the shortest closed state (approximately 0.4 ms) is doubly liganded and isomerizes to long open states (approximately 5 ms). This is the simplest interpretation consistent with available data. In contrast, sodium channels modified with batrachotoxin to eliminate inactivation show no correlation in the sequence of channel opening and closing events, i.e., have no memory of the transition history. This result is, therefore, consistent with any kinetic scheme that considers a single transition pathway between open and closed states, and confirms the C-C-O model previously inferred from one-dimensional distribution analysis. The strategy described should be of general validity in the analysis of single channel events from channel proteins in both natural and reconstituted membranes.     

Effects of anionic lipid and ion concentrations on the topology and segmental mobility of colicin Ia channel domain from solid-state NMR

Channel-forming colicins are bacterial toxins that spontaneously insert into the inner cell membrane of sensitive bacteria to form voltage-gated ion channels. It has been shown that the channel current and the conformational flexibility of colicin E1 channel domain depend on the membrane surface potential, which is regulated by the anionic lipid content and the ion concentration. To better understand the dependence of colicin structure and dynamics on the membrane surface potential, we have used solid-state NMR to investigate the topology and segmental motion of the closed state of colicin Ia channel-forming domain in membranes of different anionic lipid contents and ion concentrations. Colicin Ia channel domain was reconstituted into membranes with different POPG and KCl concentrations. 1H spin diffusion experiments indicate that the protein contains a small domain that inserts into the hydrophobic center of the 70% anionic membrane, similar to when it binds to the 25% anionic membrane. Measurements of C-H and N-H dipolar couplings indicate that, on the sub-microsecond time scale, the protein has the least segmental mobility under the high-salt and low-anionic lipid condition, which has the most physiological membrane surface potential. Measurement of millisecond time scale motions yielded similar results. These suggest that optimal channel activity requires the protein to have sufficient segmental rigidity so that entire helices can undergo cooperative conformational motions that are required for translocating the channel-forming helices across the lipid bilayer upon voltage activation.     

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.     

Protein Translocation across Planar Bilayers by the Colicin Ia Channel-Forming Domain: Where Will It End?

Colicin Ia, a 626-residue bactericidal protein, consists of three domains, with the carboxy-terminal domain (C domain) responsible for channel formation. Whole colicin Ia or C domain added to a planar lipid bilayer membrane forms voltage-gated channels. We have shown previously that the channel formed by whole colicin Ia has four membrane-spanning segments and an approximately 68-residue segment translocated across the membrane. Various experimental interventions could cause a longer or shorter segment within the C domain to be translocated, making us wonder why translocation normally stops where it does, near the amino-terminal end of the C domain (approximately residue 450). We hypothesized that regions upstream from the C domain prevent its amino-terminal end from moving into and across the membrane. To test this idea, we prepared C domain with a ligand attached near its amino terminus, added it to one side of a planar bilayer to form channels, and then probed from the opposite side with a water-soluble protein that can specifically bind the ligand. The binding of the probe had a dramatic effect on channel gating, demonstrating that the ligand (and hence the amino-terminal end of the C domain) had moved across the membrane. Experiments with larger colicin Ia fragments showed that a region of more than 165 residues, upstream from the C domain, can also move across the membrane. All of the colicin Ia carboxy-terminal fragments that we examined form channels that pass from a state of relatively normal conductance to a low-conductance state; we interpret this passage as a transition from a channel with four membrane-spanning segments to one with only three.