Effect of syncytium structure of receptor systems on stochastic resonance induced by chaotic potential fluctuation.

To study a role of syncytium structure of sensory receptor systems in the detection of weak signals through stochastic resonance, we present a model of a receptor system with syncytium structure in which receptor cells are interconnected by gap junctions. The apical membrane of each cell includes two kinds of ion channels whose gating processes are described by the deterministic model. The membrane potential of each cell fluctuates chaotically or periodically, depending on the dynamical state of collective channel gating. The chaotic fluctuation of membrane potential acts as internal noise for the stochastic resonance. The detection ability of the system increases as the electric conductance between adjacent cells generated by the gap junction increases. This effect of gap junctions arises mainly from the fact that the synchronization of chaotic fluctuation of membrane potential between the receptor cells is strengthened as the density of gap junctions is increased.     

Stochastic resonance in the LIF models with input or threshold noise

In this paper, two stochastic versions of the LIF neural model are considered: one with the noise signal applied to the firing threshold, the other having it added to the input current. Then, adopting a discontinuous stepwise noise whose innovations are uncorrelated and gaussian distributed, the behaviours of the two models pertaining to the stochastic resonance (SR) are analysed and compared. Furthermore, it is shown that introducing a suitable time correlation into the noise signal brings us from the first model to the second one.     

Water as a structural element in a channel: gating in the Kcsa channel, and implications for voltage-gated ion channels

Water is becoming understood as a structural element in proteins. Here we are concerned with one particular type of protein, ion channels. The S. Lividans KcsA K(+) channel, the X-ray structure of which is known, is gated by protons (i.e, by a drop in pH). Ab initio calculations suggest that an H(5)O(2) group, partially charged, connects the E118 residues in the gating region, when the four residues have a -2 net charge, but that the hydrogen bonding is not strong enough to do this when the charge becomes -1. The H(5)O(2) group would block the channel, in the -2 state, and prevent motion of the four transmembrane (TM) segments of the protein, by binding them. With the weaker bond in the -1 state, the TM segments would be able to separate (as they have been found to do experimentally, opening the channel. Voltage gated channels have four additional TM segments for each of the four domains of the channel protein. These appear to allow motion of protons; in fact there is evidence that the initial step in gating must be the transfer of a proton. We have earlier shown that the transfer of a single proton between two methylamines under the influence of a field is possible, as proton tunneling. Subsequent steps are hypothesized to result from four proton transfer cascades of about three protons each, triggered by the initial proton transfer. We suggest that the extra 4 TM segments of the voltage gated channel act as a voltage to proton-current transducer. Water, held by hydrogen bonds, is also suggested as the source of the accessibility data found with MTS reagents, based largely on simulations, our earlier Monte Carlo simulations as well as molecular dynamics studies reported by others. These waters may also play a structural role in the protein.     

Stochastic amplification of calcium-activated potassium currents in Ca<Superscript>2+</Superscript> microdomains

Small conductance (SK) calcium-activated potassium channels are found in many tissues throughout the body and open in response to elevations in intracellular calcium. In hippocampal neurons, SK channels are spatially co-localized with L-Type calcium channels. Due to the restriction of calcium transients into microdomains, only a limited number of L-Type Ca²⁺ channels can activate SK and, thus, stochastic gating becomes relevant. Using a stochastic model with calcium microdomains, we predict that intracellular Ca²⁺ fluctuations resulting from Ca²⁺ channel gating can increase SK2 subthreshold activity by 1–2 orders of magnitude. This effectively reduces the value of the Hill coefficient. To explain the underlying mechanism, we show how short, high-amplitude calcium pulses associated with stochastic gating of calcium channels are much more effective at activating SK2 channels than the steady calcium signal produced by a deterministic simulation. This stochastic amplification results from two factors: first, a supralinear rise in the SK2 channel’s steady-state activation curve at low calcium levels and, second, a momentary reduction in the channel’s time constant during the calcium pulse, causing the channel to approach its steady-state activation value much faster than it decays. Stochastic amplification can potentially explain subthreshold SK2 activation in unified models of both sub- and suprathreshold regimes. Furthermore, we expect it to be a general phenomenon relevant to many proteins that are activated nonlinearly by stochastic ligand release.     

Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes

Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites.     

Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.     

A small fraction of strongly cooperative sodium channels boosts neuronal encoding of high frequencies

Generation of action potentials (APs) is a crucial step in neuronal information processing. Existing biophysical models for AP generation almost universally assume that individual voltage-gated sodium channels operate statistically independently, and their avalanche-like opening that underlies AP generation is coordinated only through the transmembrane potential. However, biological ion channels of various types can exhibit strongly cooperative gating when clustered. Cooperative gating of sodium channels has been suggested to explain rapid onset dynamics and large threshold variability of APs in cortical neurons. It remains however unknown whether these characteristic properties of cortical APs can be reproduced if only a fraction of channels express cooperativity, and whether the presence of cooperative channels has an impact on encoding properties of neuronal populations. To address these questions we have constructed a conductance-based neuron model in which we continuously varied the size of a fraction [Formula: see text] of sodium channels expressing cooperativity and the strength of coupling between cooperative channels [Formula: see text]. We show that starting at a critical value of the coupling strength [Formula: see text], the activation curve of sodium channels develops a discontinuity at which opening of all coupled channels becomes an all-or-none event, leading to very rapid AP onsets. Models with a small fraction, [Formula: see text], of strongly cooperative channels generate APs with the most rapid onset dynamics. In this regime APs are triggered by simultaneous opening of the cooperative channel fraction and exhibit a pronounced biphasic waveform often observed in cortical neurons. We further show that presence of a small fraction of cooperative Na+ channels significantly improves the ability of neuronal populations to phase-lock their firing to high frequency input fluctuation. We conclude that presence of a small fraction of strongly coupled sodium channels can explain characteristic features of cortical APs and has a functional impact of enhancing the spike encoding of rapidly varying signals.     

Stochastically Gating Ion Channels Enable Patterned Spike Firing through Activity-Dependent Modulation of Spike Probability

The transformation of synaptic input into patterns of spike output is a fundamental operation that is determined by the particular complement of ion channels that a neuron expresses. Although it is well established that individual ion channel proteins make stochastic transitions between conducting and non-conducting states, most models of synaptic integration are deterministic, and relatively little is known about the functional consequences of interactions between stochastically gating ion channels. Here, we show that a model of stellate neurons from layer II of the medial entorhinal cortex implemented with either stochastic or deterministically gating ion channels can reproduce the resting membrane properties of stellate neurons, but only the stochastic version of the model can fully account for perithreshold membrane potential fluctuations and clustered patterns of spike output that are recorded from stellate neurons during depolarized states. We demonstrate that the stochastic model implements an example of a general mechanism for patterning of neuronal output through activity-dependent changes in the probability of spike firing. Unlike deterministic mechanisms that generate spike patterns through slow changes in the state of model parameters, this general stochastic mechanism does not require retention of information beyond the duration of a single spike and its associated afterhyperpolarization. Instead, clustered patterns of spikes emerge in the stochastic model of stellate neurons as a result of a transient increase in firing probability driven by activation of HCN channels during recovery from the spike afterhyperpolarization. Using this model, we infer conditions in which stochastic ion channel gating may influence firing patterns in vivo and predict consequences of modifications of HCN channel function for in vivo firing patterns.     

Hysteresis in the voltage dependence of HCN channels: conversion between two modes affects pacemaker properties

Hyperpolarization-activated, cyclic nucleotide-gated (HCN) ion channels are important for rhythmic activity in the brain and in the heart. In this study, using ionic and gating current measurements, we show that cloned spHCN channels undergo a hysteresis in their voltage dependence during normal gating. For example, both the gating charge versus voltage curve, Q(V), and the conductance versus voltage curve, G(V), are shifted by about +60 mV when measured from a hyperpolarized holding potential compared with a depolarized holding potential. In addition, the kinetics of the tail current and the activation current change in parallel to the voltage shifts of the Q(V) and G(V) curves. Mammalian HCN1 channels display similar effects in their ionic currents, suggesting that the mammalian HCN channels also undergo voltage hysteresis. We propose a model in which HCN channels transit between two modes. The voltage dependence in the two modes is shifted relative to each other, and the occupancy of the two modes depends on the previous activation of the channel. The shifts in the voltage dependence are fast (tau approximately 100 ms) and are not accompanied by any apparent inactivation. In HCN1 channels, the shift in voltage dependence is slower in a 100 mM K extracellular solution compared with a 1 mM K solution. Based on these findings, we suggest that molecular conformations similar to slow (C-type) inactivation of K channels underlie voltage hysteresis in HCN channels. The voltage hysteresis results in HCN channels displaying different voltage dependences during different phases in the pacemaker cycle. Computer simulations suggest that voltage hysteresis in HCN channels decreases the risk of arrhythmia in pacemaker cells.     

A kinetic model of single and clustered IP3 receptors in the absence of Ca2+ feedback

Ca2+ liberation through inositol 1,4,5-trisphosphate receptor (IP3R) channels generates complex patterns of spatiotemporal cellular Ca2+ signals owing to the biphasic modulation of channel gating by Ca2+ itself. These processes have been extensively studied in Xenopus oocytes, where imaging studies have revealed local Ca2+ signals ("puffs") arising from clusters of IP3R, and patch-clamp studies on isolated oocyte nuclei have yielded extensive data on IP3R gating kinetics. To bridge these two levels of experimental data, we developed an IP3R model and applied stochastic simulation and transition matrix theory to predict the behavior of individual and clustered IP3R channels. The channel model consists of four identical, independent subunits, each of which has an IP3-binding site together with one activating and one inactivating Ca2+-binding site. The channel opens when at least three subunits undergo a conformational change to an "active" state after binding IP3 and Ca2+. The model successfully reproduces patch-clamp data; including the dependence of open probability, mean open duration, and mean closed duration on [IP3] and [Ca2+]. Notably, the biexponential distribution of open-time duration and the dependence of mean open time on [Ca2+] are explained by populations of openings involving either three or four active subunits. As a first step toward applying the single IP3R model to describe cellular responses, we then simulated measurements of puff latency after step increases of [IP3]. Assuming that stochastic opening of a single IP3R at basal cytosolic [Ca2+] and any given [IP3] has a high probability of rapidly triggering neighboring channels by calcium-induced calcium release to evoke a puff, optimal correspondence with experimental data of puff latencies after photorelease of IP3 was obtained when the cluster contained a total of 40-70 IP3Rs.     

Modeling ion channel regulation

Remarkable recent successes in structure determinations of voltage-gated channels, ligand-gated channels, mechanosensitive channels and proton channels have advanced our understanding of the molecular basis of ion channel gating substantially. Models have helped to clarify aspects of this process and are now being designed as sophisticated biomimetics for various technological applications.     

Consequences of phosphate-arginine complexes in voltage gated ion channels

There are two reasons for suspecting that phosphate complexes of arginine make it very difficult to derive gating charge in voltage gated potassium (and presumably sodium) channels from the motion of charged arginines. For one thing, the arginines should be complexed with phosphate, thereby neutralizing the charge, at least partially. Second, Li et al1 have shown that there is a large energy penalty for putting a charged arginine into a membrane. Ion channel gating current is generally attributed to S4 motion, in that the S4 segment of the voltage sensing domain (VSD) of these channels contains arginines, some of which are not (or at least not obviously) salt bridged, or otherwise charge compensated. There is, however, good reason to expect that there should be a complex of these arginines with phosphate, very probably from lipid headgroups. This has consequences for gating current; the complexed arginines, if they moved, would carry too little charge to provide the observed gating current, if they carried any net charge at all. This leads to the suggestion that an alternative to S4 physical motion, H+ transport, should be considered as a possible resolution of the apparent paradox. The consequences for a gating model that was proposed in our earlier work are discussed; there is one major difference in the model in the present form (a conformational change), but the proton cascade as gating current and the role of water in the closed state are reinforced.     

The mechanism of rectification of iK1 in canine Purkinje myocytes.

We have characterized the inward rectifying background potassium current, iK1, of canine cardiac Purkinje myocytes in terms of its reversal potential, voltage activation curve, and "steady-state" current-voltage relation. The latter parameter was defined from the difference current between holding currents in the presence and absence of 20 mM cesium. Our data suggest that iK1 rectification does not arise exclusively from voltage-dependent gating or exclusively from voltage-dependent blockade by internal magnesium ions. The voltage activation curve constructed from tail currents fit to a Boltzmann two-state model predicts less outward current than is actually observed. The magnesium-dependent rectification due to channel blockade is too fast to account for the time-dependent gating of iK1 that gives rise to the tail currents. We propose a new model of rectification that assumes that magnesium blockade of the channel occurs simultaneously with voltage-dependent gating. The new model incorporates the kinetic schema elaborated by Matsuda, H. (1988. J. Physiol. 397:237-258) to explain the appearance of subconducting states of the iK1 channel in the presence of blocking ions. That schema suggested that iK1 channels were composed of three parallel pores, each of which could be blocked independently. In our model we considered the consequences of partial blockade of the channel. If the channels are partially blocked at potentials where normally they are mostly gated closed, and if the partially blocked channels cannot close, then blockade will have the paradoxical result of enhancing the current carried by iK1.     

Electron paramagnetic resonance characterization of tetrahydrobiopterin radical formation in bacterial nitric oxide synthase compared to mammalian nitric oxide synthase

H(4)B is an essential catalytic cofactor of the mNOSs. It acts as an electron donor and activates the ferrous heme-oxygen complex intermediate during Arg oxidation (first step) and NOHA oxidation (second step) leading to nitric oxide and citrulline as final products. However, its role as a proton donor is still debated. Furthermore, its exact involvement has never been explored for other NOSs such as NOS-like proteins from bacteria. This article proposes a comparative study of the role of H(4)B between iNOS and bsNOS. In this work, we have used freeze-quench to stop the arginine and NOHA oxidation reactions and trap reaction intermediates. We have characterized these intermediates using multifrequency electron paramagnetic resonance. For the first time, to our knowledge, we report a radical formation for a nonmammalian NOS. The results indicate that bsNOS, like iNOS, has the capacity to generate a pterin radical during Arg oxidation. Our current electron paramagnetic resonance data suggest that this radical is protonated indicating that H(4)B may not transfer any proton. In the 2nd step, the radical trapped for iNOS is also suggested to be protonated as in the 1st step, whereas it was not possible to trap a radical for the bsNOS 2nd step. Our data highlight potential differences for the catalytic mechanism of NOHA oxidation between mammalian and bacterial NOSs.     

Consequences of phosphate-arginine complexes in voltage-gated ion channels

There are two reasons for suspecting that phosphate complexes of arginine make it very difficult to derive gating charge in voltage gated potassium (and presumably sodium) channels from the motion of charged arginines. For one thing, the arginines should be complexed with phosphate, thereby neutralizing the charge, at least partially. Second, Li et al.(1) have shown that there is a large energy penalty for putting a charged arginine into a membrane. on channel gating current is generally attributed to S4 motion, in that the S4 segment of the voltage sensing domain (VSD) of these channels contains arginines, some of which are not (or at least not obviously) salt bridged, or otherwise charge compensated. There is, however, good reason to expect that there should be a complex of these arginines with phosphate, very probably from lipid headgroups. This has consequences for gating current; the complexed arginines, if they moved, would carry too much of the membrane along. This leads to the suggestion that an alternative to S4 physical motion, H+ transport, should be considered as a possible resolution of the apparent paradox. The consequences for a gating model that was proposed in our earlier work are discussed; there is one major difference in the model in the present form (a conformational change), but the proton cascade as gating current and the role of water in the closed state are reinforced.     

LRET Determination of Molecular Distances during pH Gating of the Mammalian Inward Rectifier Kir1.1b

Gating of the mammalian inward rectifier Kir1.1 at the helix bundle crossing (HBC) by intracellular pH is believed to be mediated by conformational changes in the C-terminal domain (CTD). However, the exact motion of the CTD during Kir gating remains controversial. Crystal structures and single-molecule fluorescence resonance energy transfer of KirBac channels have implied a rigid body rotation and/or a contraction of the CTD as possible triggers for opening of the HBC gate. In our study, we used lanthanide-based resonance energy transfer on single-Cys dimeric constructs of the mammalian renal inward rectifier, Kir1.1b, incorporated into anionic liposomes plus PIP₂, to determine unambiguous, state-dependent distances between paired Cys residues on diagonally opposite subunits. Functionality and pH dependence of our proteoliposome channels were verified in separate electrophysiological experiments. The lanthanide-based resonance energy transfer distances measured in closed (pH 6) and open (pH 8) conditions indicated neither expansion nor contraction of the CTD during gating, whereas the HBC gate widened by 8.8 ± 4 Å, from 6.3 ± 2 to 15.1 ± 6 Å, during opening. These results are consistent with a Kir gating model in which rigid body rotation of the large CTD around the permeation axis is correlated with opening of the HBC hydrophobic gate, allowing permeation of a 7 Å hydrated K ion.     

Aperiodic stochastic resonance in neural information processing with Gaussian colored noise

The aim of this paper is to explore the phenomenon of aperiodic stochastic resonance in neural systems with colored noise. For nonlinear dynamical systems driven by Gaussian colored noise, we prove that the stochastic sample trajectory can converge to the corresponding deterministic trajectory as noise intensity tends to zero in mean square, under global and local Lipschitz conditions, respectively. Then, following forbidden interval theorem we predict the phenomenon of aperiodic stochastic resonance in bistable and excitable neural systems. Two neuron models are further used to verify the theoretical prediction. Moreover, we disclose the phenomenon of aperiodic stochastic resonance induced by correlation time and this finding suggests that adjusting noise correlation might be a biologically more plausible mechanism in neural signal processing.     

The rate of multi-step evolution in Moran and Wright-Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.     

Analysis of an electrostatic mechanism for ΔpH dependent gating of the voltage-gated proton channel,HV1, supports a contribution of protons to gating charge

Voltage-gated proton channels (H_V1) resemble the voltage-sensing domain of other voltage-gated ion channels, but differ in containing the conduction pathway. Essential to the functions of H_V1 channels in many cells and species is a unique feature called ΔpH dependent gating. The pH on both sides of the membrane strictly regulates the voltage range of channel opening, generally resulting in exclusively outward proton current. Two types of mechanisms could produce ΔpH dependent gating. The “countercharge” mechanism proposes that protons destabilize salt bridges between amino acids in the protein that stabilize specific gating configurations (closed or open). An “electrostatic” mechanism proposes that protons bound to the channel alter the electrical field sensed by the protein. Obligatory proton binding within the membrane electrical field would contribute to measured gating charge. Estimations on the basis of the electrostatic model explain ΔpH dependent gating, but quantitative modeling requires calculations of the electric field inside the protein which, in turn, requires knowledge of its structure. We conclude that both mechanisms operate and contribute to ΔpH dependent gating of H_V1.