Analysis of pressure-activated Piezo1 open and subconductance states at a single channel level.

Mechanically activated Piezo1 channels undergo transitions from closed to open-state in response to pressure and other mechanical stimuli. However, the molecular details of these mechanosensitive gating transitions are unknown. Here, we used cell-attached pressure-clamp recordings to acquire single channel data at steady-state conditions (where inactivation has settled down), at various pressures and voltages. Importantly, we identify and analyze subconductance states of the channel which were not reported before. Pressure-dependent activation of Piezo1 increases the occupancy of open and subconductance state at the expense of decreased occupancy of shut-states. No significant change in the mean open time of subconductance states was observed with increasing negative pipette pressure or with varying voltages (ranging from -40 to -100 mV). Using Markov-chain modeling, we identified a minimal four-states kinetic scheme, which recapitulates essential characteristics of the single channel data, including that of the subconductance level. This study advances our understanding of Piezo1-gating mechanism in response to discrete stimuli (such as pressure and voltage) and paves the path to develop cellular and tissue level models to predict Piezo1 function in various cell types.

Exploration of the Parameter Space of an Ion Channel Kinetic Model by a Markov-Chain-Based Methodology.

In this work, we propose a methodology based on Monte Carlo Markov chains to explore the parameter space of kinetic models for ion channels. The methodology allows the detection of potential parameter sets of a model that are compatible with experimentally obtained whole-cell currents, which could remain hidden when methods focus on obtaining the parameters that provide the best fit. To show its implementation and utility, we considered a four-state kinetic model proposed in the literature to describe the activation of the voltage-gated proton channel (Hv1), Biophysical Journal, 2014, 107, 1564. In that work, a set of values for the rate transitions that describe the channel kinetics at different intracellular H+ concentration (pHi) were obtained by the Simplex method. With our approach, we find that, in fact, there is more than one parameter set for each pHi, which renders the same open probability temporal course within the experimental error margin for all of the considered voltages. The large differences that we obtained for the values of some rate constants among the different solutions show that there is more than one possible interpretation of this channel behavior as a function of pHi. We also simulated a proposed new experimental condition where it is possible to observe that different sets of parameters yield different results. Our study highlights the importance of a comprehensive analysis of parameter space in kinetic models and the utility of the proposed methodology for finding potential solutions.

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons.

Two subpopulations of midbrain dopamine (DA) neurons are known to have different dynamic firing ranges in vitro that correspond to distinct projection targets: the originally identified conventional DA neurons project to the dorsal striatum and the lateral shell of the nucleus accumbens, whereas an atypical DA population with higher maximum firing frequencies projects to prefrontal regions and other limbic regions including the medial shell of nucleus accumbens. Using a computational model, we show that previously identified differences in biophysical properties do not fully account for the larger dynamic range of the atypical population and predict that the major difference is that originally identified conventional cells have larger occupancy of voltage-gated sodium channels in a long-term inactivated state that recovers slowly; stronger sodium and potassium conductances during action potential firing are also predicted for the conventional compared to the atypical DA population. These differences in sodium channel gating imply that longer intervals between spikes are required in the conventional population for full recovery from long-term inactivation induced by the preceding spike, hence the lower maximum frequency. These same differences can also change the bifurcation structure to account for distinct modes of entry into depolarization block: abrupt versus gradual. The model predicted that in cells that have entered depolarization block, it is much more likely that an additional depolarization can evoke an action potential in conventional DA population. New experiments comparing lateral to medial shell projecting neurons confirmed this model prediction, with implications for differential synaptic integration in the two populations.

Dynamical model of the CLC-2 ion channel reveals conformational changes associated with selectivity-filter gating.

This work reports a dynamical Markov state model of CLC-2 "fast" (pore) gating, based on 600 microseconds of molecular dynamics (MD) simulation. In the starting conformation of our CLC-2 model, both outer and inner channel gates are closed. The first conformational change in our dataset involves rotation of the inner-gate backbone along residues S168-G169-I170. This change is strikingly similar to that observed in the cryo-EM structure of the bovine CLC-K channel, though the volume of the intracellular (inner) region of the ion conduction pathway is further expanded in our model. From this state (inner gate open and outer gate closed), two additional states are observed, each involving a unique rotameric flip of the outer-gate residue GLUex. Both additional states involve conformational changes that orient GLUex away from the extracellular (outer) region of the ion conduction pathway. In the first additional state, the rotameric flip of GLUex results in an open, or near-open, channel pore. The equilibrium population of this state is low (∼1%), consistent with the low open probability of CLC-2 observed experimentally in the absence of a membrane potential stimulus (0 mV). In the second additional state, GLUex rotates to occlude the channel pore. This state, which has a low equilibrium population (∼1%), is only accessible when GLUex is protonated. Together, these pathways model the opening of both an inner and outer gate within the CLC-2 selectivity filter, as a function of GLUex protonation. Collectively, our findings are consistent with published experimental analyses of CLC-2 gating and provide a high-resolution structural model to guide future investigations.

Biophysically interpretable inference of single neuron dynamics.

Identification of key ionic channel contributors to the overall dynamics of a neuron is an important problem in experimental neuroscience. Such a problem is challenging since even in the best cases, identification relies on noisy recordings of membrane potential only, and strict inversion to the constituent channel dynamics is mathematically ill-posed. In this work, we develop a biophysically interpretable, learning-based strategy for data-driven inference of neuronal dynamics. In particular, we propose two optimization frameworks to learn and approximate neural dynamics from an observed voltage trajectory. In both the proposed strategies, the membrane potential dynamics are approximated as a weighted sum of ionic currents. In the first strategy, the ionic currents are represented using voltage dependent channel conductances and membrane potential in a parametric form, while in the second strategy, the currents are represented as a linear combination of generic basis functions. A library of channel activation/inactivation and time-constant curves describing prototypical channel kinetics are used to provide estimates of the channel variables to approximate the ionic currents. Finally, a linear optimization problem is solved to infer the weights/scaling variables in the membrane-potential dynamics. In the first strategy, the weights can be used to recover the channel conductances, and the reversal potentials while in the second strategy, using the estimated weights, active channels can be inferred and the trajectory of the gating variables are recovered, allowing for biophysically salient inference. Our results suggest that the complex nonlinear behavior of the neural dynamics over a range of temporal scales can be efficiently inferred in a data-driven manner from noisy membrane potential recordings.

Stochastic shielding and edge importance for Markov chains with timescale separation.

Nerve cells produce electrical impulses ("spikes") through the coordinated opening and closing of ion channels. Markov processes with voltage-dependent transition rates capture the stochasticity of spike generation at the cost of complex, time-consuming simulations. Schmandt and Galán introduced a novel method, based on the stochastic shielding approximation, as a fast, accurate method for generating approximate sample paths with excellent first and second moment agreement to exact stochastic simulations. We previously analyzed the mathematical basis for the method's remarkable accuracy, and showed that for models with a Gaussian noise approximation, the stationary variance of the occupancy at each vertex in the ion channel state graph could be written as a sum of distinct contributions from each edge in the graph. We extend this analysis to arbitrary discrete population models with first-order kinetics. The resulting decomposition allows us to rank the "importance" of each edge's contribution to the variance of the current under stationary conditions. In most cases, transitions between open (conducting) and closed (non-conducting) states make the greatest contributions to the variance, but there are exceptions. In a 5-state model of the nicotinic acetylcholine receptor, at low agonist concentration, a pair of "hidden" transitions (between two closed states) makes a greater contribution to the variance than any of the open-closed transitions. We exhaustively investigate this "edge importance reversal" phenomenon in simplified 3-state models, and obtain an exact formula for the contribution of each edge to the variance of the open state. Two conditions contribute to reversals: the opening rate should be faster than all other rates in the system, and the closed state leading to the opening rate should be sparsely occupied. When edge importance reversal occurs, current fluctuations are dominated by a slow noise component arising from the hidden transitions.

Activation gating in HCN2 channels.

Hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels control electrical rhythmicity in specialized brain and heart cells. We quantitatively analysed voltage-dependent activation of homotetrameric HCN2 channels and its modulation by the second messenger cAMP using global fits of hidden Markovian models to complex experimental data. We show that voltage-dependent activation is essentially governed by two separable voltage-dependent steps followed by voltage-independent opening of the pore. According to this model analysis, the binding of cAMP to the channels exerts multiple effects on the voltage-dependent gating: It stabilizes the open pore, reduces the total gating charge from ~8 to ~5, makes an additional closed state outside the activation pathway accessible and strongly accelerates the ON-gating but not the OFF-gating. Furthermore, the open channel has a much slower computed OFF-gating current than the closed channel, in both the absence and presence of cAMP. Together, these results provide detailed new insight into the voltage- and cAMP-induced activation gating of HCN channels.

Deciphering the regulation of P2X4 receptor channel gating by ivermectin using Markov models.

The P2X4 receptor (P2X4R) is a member of a family of purinergic channels activated by extracellular ATP through three orthosteric binding sites and allosterically regulated by ivermectin (IVM), a broad-spectrum antiparasitic agent. Treatment with IVM increases the efficacy of ATP to activate P2X4R, slows both receptor desensitization during sustained ATP application and receptor deactivation after ATP washout, and makes the receptor pore permeable to NMDG+, a large organic cation. Previously, we developed a Markov model based on the presence of one IVM binding site, which described some effects of IVM on rat P2X4R. Here we present two novel models, both with three IVM binding sites. The simpler one-layer model can reproduce many of the observed time series of evoked currents, but does not capture well the short time scales of activation, desensitization, and deactivation. A more complex two-layer model can reproduce the transient changes in desensitization observed upon IVM application, the significant increase in ATP-induced current amplitudes at low IVM concentrations, and the modest increase in the unitary conductance. In addition, the two-layer model suggests that this receptor can exist in a deeply inactivated state, not responsive to ATP, and that its desensitization rate can be altered by each of the three IVM binding sites. In summary, this study provides a detailed analysis of P2X4R kinetics and elucidates the orthosteric and allosteric mechanisms regulating its channel gating.

Connexin Hemichannels in Astrocytes: An Assessment of Controversies Regarding Their Functional Characteristics.

Astrocytes in the mammalian central nervous system are interconnected by gap junctions made from connexins of the subtypes Cx30 and Cx43. These proteins may exist as hemichannels in the plasma membrane in the absence of a 'docked' counterpart on the neighboring cell. A variety of stimuli are reported to open the hemichannels and thereby create a permeation pathway through the plasma membrane. Cx30 and Cx43 have, in their hemichannel configuration, been proposed to act as ion channels and membrane pathways for different molecules, such as fluorescent dyes, ATP, prostaglandins, and glutamate. Published studies about astrocyte hemichannel behavior, however, have been highly variable and/or contradictory. The field of connexin hemichannel research has been complicated by great variability in the experimental preparations employed, a lack of highly specific pharmacological inhibitors and by confounding changes associated with genetically modified animal models. This review attempts to critically assess the gating, inhibition and permeability of astrocytic connexin hemichannels and proposes that connexins in their hemichannel configuration act as gated pores with isoform-specific permeant selectivity. We expect that some, or all, of the controversies discussed here will be resolved by future research and sincerely hope that this review serves to motivate such clarifying investigations.

Optimal decoding and information transmission in Hodgkin-Huxley neurons under metabolic cost constraints.

Information theory quantifies the ultimate limits on reliable information transfer by means of the channel capacity. However, the channel capacity is known to be an asymptotic quantity, assuming unlimited metabolic cost and computational power. We investigate a single-compartment Hodgkin-Huxley type neuronal model under the spike-rate coding scheme and address how the metabolic cost and the decoding complexity affects the optimal information transmission. We find that the sub-threshold stimulation regime, although attaining the smallest capacity, allows for the most efficient balance between the information transmission and the metabolic cost. Furthermore, we determine post-synaptic firing rate histograms that are optimal from the information-theoretic point of view, which enables the comparison of our results with experimental data.

On the capabilities and computational costs of neuron models.

We review the Hodgkin-Huxley, Izhikevich, and leaky integrate-and-fire neuron models in regular spiking modes solved with the forward Euler, fourth-order Runge-Kutta, and exponential Euler methods and determine the necessary time steps and corresponding computational costs required to make the solutions accurate. We conclude that the leaky integrate-and-fire needs the least number of computations, and that the Hodgkin-Huxley and Izhikevich models are comparable in computational cost.

Evolutionary imprint of activation: the design principles of VSDs.

Voltage-sensor domains (VSDs) are modular biomolecular machines that transduce electrical signals in cells through a highly conserved activation mechanism. Here, we investigate sequence-function relationships in VSDs with approaches from information theory and probabilistic modeling. Specifically, we collect over 6,600 unique VSD sequences from diverse, long-diverged phylogenetic lineages and relate the statistical properties of this ensemble to functional constraints imposed by evolution. The VSD is a helical bundle with helices labeled S1-S4. Surrounding conserved VSD residues such as the countercharges and the S2 phenylalanine, we discover sparse networks of coevolving residues. Additional networks are found lining the VSD lumen, tuning the local hydrophilicity. Notably, state-dependent contacts and the absence of coevolution between S4 and the rest of the bundle are imprints of the activation mechanism on the VSD sequence ensemble. These design principles rationalize existing experimental results and generate testable hypotheses.

Computer simulation of voltage sensitive calcium ion channels in a dendritic spine.

Membrane current through voltage-sensitive calcium ion channels at the postsynaptic density of a dendritic spine is investigated. To simulate the ion channels that carry such current and the resulting temporal and spatial distribution of concentration, current, and voltage within the dendritic spine, the immersed boundary method with electrodiffusion is applied. In this simulation method a spatially continuous chemical potential barrier is used to simulate the influence of the membrane on each species of ion. The amplitudes of these barriers can be regulated to simulate channel gating. Here we introduce this methodology in a one-dimensional setting. First, we study the current-voltage relationship obtained with fixed chemical potential barriers. Next, we simulate stochastic ion-channel gating in a calcium channel with multiple subunits, and observe the diffusive wave of calcium entry within the dendritic spine that follows channel opening. This work lays the foundation for future three-dimensional studies of electrodiffusion and advection electrodiffusion in dendritic spines.

Stochastic Hodgkin-Huxley equations with colored noise terms in the conductances.

The excitability of cells is facilitated by voltage-gated ion channels. These channels accommodate a multiple number of gates individually. The possible impact of that gate multiplicity on the cell's function, specifically when the membrane area is of limited size, was investigated in the author's prior work (Güler, 2011 ). There, it was found that a nontrivially persistent correlation takes place between the transmembrane voltage fluctuations (also between the fluctuations in the gating variables) and the component of open channel fluctuations attributed to the gate multiplicity. This nontrivial phenomenon was found to be playing a major augmentative role for the elevation of excitability and spontaneous firing in small cells. In addition, the same phenomenon was found to be enhancing spike coherence significantly. Here we extend Fox and Lu's ( 1994 ) stochastic Hodgkin-Huxley equations by incorporating colored noise terms into the conductances there to obtain a formalism capable of capturing the addressed cross-correlations. Statistics of spike generation, spike coherence, firing efficiency, latency, and jitter from the articulated set of equations are found to be highly accurate in comparison with the corresponding statistics from the exact microscopic Markov simulations. This way, it is demonstrated vividly that our formulation overcomes the inherent inadequacy of the Fox and Lu equations. Finally, a recently proposed diffusion approximation method (Linaro, Storace, & Giugliano, 2011 ) is taken into consideration, and a discussion on its character is pursued.

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states.

BACKGROUND: The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. MAIN CONTRIBUTIONS: We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.

Gating properties of the P2X2a and P2X2b receptor channels: experiments and mathematical modeling.

Adenosine triphosphate (ATP)-gated P2X2 receptors exhibit two opposite activation-dependent changes, pore dilation and pore closing (desensitization), through a process that is incompletely understood. To address this issue and to clarify the roles of calcium and the C-terminal domain in gating, we combined biophysical and mathematical approaches using two splice forms of receptors: the full-size form (P2X2aR) and the shorter form missing 69 residues in the C-terminal domain (P2X2bR). Both receptors developed conductivity for N-methyl-D-glucamine within 2-6 s of ATP application. However, pore dilation was accompanied with a decrease rather than an increase in the total conductance, which temporally coincided with rapid and partial desensitization. During sustained agonist application, receptors continued to desensitize in calcium-independent and calcium-dependent modes. Calcium-independent desensitization was more pronounced in P2X2bR, and calcium-dependent desensitization was more pronounced in P2X2aR. In whole cell recording, we also observed use-dependent facilitation of desensitization of both receptors. Such behavior was accounted for by a 16-state Markov kinetic model describing ATP binding/unbinding and activation/desensitization. The model assumes that naive receptors open when two to three ATP molecules bind and undergo calcium-independent desensitization, causing a decrease in the total conductance, or pore dilation, causing a shift in the reversal potential. In calcium-containing media, receptor desensitization is facilitated and the use-dependent desensitization can be modeled by a calcium-dependent toggle switch. The experiments and the model together provide a rationale for the lack of sustained current growth in dilating P2X2Rs and show that receptors in the dilated state can also desensitize in the presence of calcium.

Fluctuations in the open time of synaptic channels: an application to noise analysis based on charge.

Synaptic channels are stochastic devices. Even recording from large ensembles of channels, the fluctuations, described by Markov transition matrices, can be used to extract single channel properties. Here we study fluctuations in the open time of channels, which is proportional to the charge flowing through the channel. We use the results to implement a novel type of noise analysis that uses the charge rather than the current to extract fundamental channel parameters. We show in simulations that this charge based noise analysis is more robust if the synapse is located on the dendrites and thus subject to cable filtering. However, we also demonstrate that when multiple synapses are distributed on the dendrites, noise analysis breaks down. We finally discuss applications of our results to other biological processes.

MCMC can detect nonidentifiable models.

Continuous-time Markov models have been considered the best representation for the stochastic dynamics of ion channels for more than thirty years. For most single-channel data sets, several open and closed states are required for accurately representing the dynamics. However, each data point only shows if the channel is open or closed but not in which state it is. Consequently, some model structures are inherently overparameterized and therefore, in principle, unsuitable for representing any data--those models are called "nonidentifiable". As of this writing, it seems to be poorly understood which continuous-time Markov models are identifiable and which are not, therefore the unconscious use of a nonidentifiable model is a considerable concern. To address this problem, an improved variant of a recently published Markov-chain Monte Carlo method is presented. The algorithm is tested using test data as well as experimental data. We demonstrate that, opposed to a widely used maximum-likelihood estimator, it gives clear warning signs when a nonidentifiable model is used for fitting. Furthermore, for test data that was generated from a nonidentifiable model, the Markov-chain Monte Carlo results recover much more information from the data than maximum-likelihood estimation.

Calcium-dependent block of P2X7 receptor channel function is allosteric.

Among purinergic P2X receptor (P2XR) channels, the P2X7R exhibits the most complex gating kinetics; the binding of orthosteric agonists at the ectodomain induces a conformational change in the receptor complex that favors a gating transition from closed to open and dilated states. Bath Ca(2+) affects P2X7R gating through a still uncharacterized mechanism: it could act by reducing the adenosine triphosphate(4-) (ATP(4-)) concentration (a form proposed to be the P2X7R orthosteric agonist), as an allosteric modulator, and/or by directly altering the selectivity of pore to cations. In this study, we combined biophysical and mathematical approaches to clarify the role of calcium in P2X7R gating. In naive receptors, bath calcium affected the activation permeability dynamics indirectly by decreasing the potency of orthosteric agonists in a concentration-dependent manner and independently of the concentrations of the free acid form of agonists and status of pannexin-1 (Panx1) channels. Bath calcium also facilitated the rates of receptor deactivation in a concentration-dependent manner but did not affect a progressive delay in receptor deactivation caused by repetitive agonist application. The effects of calcium on the kinetics of receptor deactivation were rapid and reversible. A438079, a potent orthosteric competitive antagonist, protected the rebinding effect of 2'(3')-O-4-benzoylbenzoyl)ATP on the kinetics of current decay during the washout period, but in the presence of A438079, calcium also increased the rate of receptor deactivation. The corresponding kinetic (Markov state) model indicated that the decrease in binding affinity leads to a decrease in current amplitudes and facilitation of receptor deactivation, both in an extracellular calcium concentration-dependent manner expressed as a Hill function. The results indicate that calcium in physiological concentrations acts as a negative allosteric modulator of P2X7R by decreasing the affinity of receptors for orthosteric ligand agonists, but not antagonists, and not by affecting the permeability dynamics directly or indirectly through Panx1 channels. We expect these results to generalize to other P2XRs.