Superposition properties of interacting ion channels

Nonextensive realizations in interacting ion channels: Implications for mechano-electrical transducer mechanisms

We propose a theoretical model to investigate the thermodynamics of single and coupled two-state ion channels, associated with mechanoelectrical transduction (MET) and hair cell biophysics. The modeling was based on the Tsallis nonextensive statistical mechanics. The choice for a nonextensive framework in modeling ion channels is encouraged on the fact that we take into account the presence of interactions or long-range correlations in the dynamics of single and coupled ion channels. However, the basic assumptions that support Boltzmann-Gibbs statistics, traditionally used to model ion channel dynamics, state that the system is formed by independent or weakly interacting elements. Despite being well studied in many biological systems, the literature has not yet addressed the study of both entropy and mutual information related to isolated or physically interacting pairs of MET channels. Inspired by hair cell biophysics, we show how the presence of nonextensivity, or subadditivity and superadditivity modulates the nonextensive entropy and mutual information as functions of stereocilia displacements. We also observe that the magnitude of the interaction between the two channels, given by a nonextensive parameter, influences the amplitude of the nonextensive joint entropy and mutual information as functions of the hair cell displacements. Finally, we show how nonextensivity regulates the current versus displacement curve for a single and a pair of interacting two-state channels. The present findings shed light on the thermodynamic process involved in the molecular mechanisms of the auditory system.

Mechanisms and Physiological Implications of Cooperative Gating of Ion Channels Clusters

Ion channels play a central role in the regulation of nearly every cellular process. Dating back to the classic 1952 Hodgkin-Huxley model of the generation of the action potential, ion channels have always been thought of as independent agents. A myriad of recent experimental findings exploiting advances in electrophysiology, structural biology, and imaging techniques, however, have posed a serious challenge to this long-held axiom as several classes of ion channels appear to open and close in a coordinated, cooperative manner. Ion channel cooperativity ranges from variable-sized oligomeric cooperative gating in voltage-gated, dihydropyridine-sensitive Ca_v1.2 and Ca_v1.3 channels to obligatory dimeric assembly and gating of voltage-gated Na_v1.5 channels. Potassium channels, transient receptor potential channels, hyperpolarization cyclic nucleotide-activated channels, ryanodine receptors (RyRs), and inositol trisphosphate receptors (IP₃Rs) have also been shown to gate cooperatively. The implications of cooperative gating of these ion channels range from fine tuning excitation-contraction coupling in muscle cells to regulating cardiac function and vascular tone, to modulation of action potential and conduction velocity in neurons and cardiac cells, and to control of pace-making activity in the heart. In this review, we discuss the mechanisms leading to cooperative gating of ion channels, their physiological consequences and how alterations in cooperative gating of ion channels may induce a range of clinically significant pathologies.

Superposition of Interacting Aggregated Continuous-Time Markov Chains

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

Superposition of Interacting Aggregated Continuous-Time Markov Chains

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

Stochastic model of endothelial TRPV4 calcium sparklets: effect of bursting and cooperativity on EDH

We examined the endothelial transient receptor vanilloid 4 (TRPV4) channel's vasodilatory signaling using mathematical modeling. The model analyzes experimental data by Sonkusare and coworkers on TRPV4-induced endothelial Ca(2+) events (sparklets). A previously developed continuum model of an endothelial and a smooth muscle cell coupled through microprojections was extended to account for the activity of a TRPV4 channel cluster. Different stochastic descriptions for the TRPV4 channel flux were examined using finite-state Markov chains. The model also took into consideration recent evidence for the colocalization of intermediate-conductance calcium-activated potassium channels (IKCa) and TRPV4 channels near the microprojections. A single TRPV4 channel opening resulted in a stochastic localized Ca(2+) increase in a small region (i.e., few μm(2) area) close to the channel. We predict micromolar Ca(2+) increases lasting for the open duration of the channel sufficient for the activation of low-affinity endothelial KCa channels. Simulations of a cluster of four TRPV4 channels incorporating burst and cooperative gating kinetics provided quantal Ca(2+) increases (i.e., steps of fixed amplitude), similar to the experimentally observed Ca(2+) sparklets. These localized Ca(2+) events result in endothelium-derived hyperpolarization (and SMC relaxation), with magnitude that depends on event frequency. The gating characteristics (bursting, cooperativity) of the TRPV4 cluster enhance Ca(2+) spread and the distance of KCa channel activation. This may amplify the EDH response by the additional recruitment of distant KCa channels.

ATP-sensitive potassium channels exhibit variance in the number of open channels below the limit predicted for identical and independent gating

In small cells containing small numbers of ion channels, noise due to stochastic channel opening and closing can introduce a substantial level of variability into the cell's membrane potential. Negatively cooperative interactions that couple a channel's gating conformational change to the conformation of its neighbor(s) provide a potential mechanism for mitigating this variability, but such interactions have not previously been directly observed. Here we show that heterologously expressed ATP-sensitive potassium channels generate noise (i.e., variance in the number of open channels) below the level possible for identical and independent channels. Kinetic analysis with single-molecule resolution supports the interpretation that interchannel negative cooperativity (specifically, the presence of an open channel making a closed channel less likely to open) contributes to the decrease in noise. Functional coupling between channels may be important in modulating stochastic fluctuations in cellular signaling pathways.

Cooperative gating between single HCN pacemaker channels

HCN pacemaker channels (I(f), I(q), or I(h)) play a fundamental role in the physiology of many excitable cell types, including cardiac myocytes and central neurons. While cloned HCN channels have been studied extensively in macroscopic patch clamp experiments, their extremely small conductance has precluded single channel analysis to date. Nevertheless, there remain fundamental questions about HCN gating that can be resolved only at the single channel level. Here we present the first detailed single channel study of cloned mammalian HCN2. Excised patch clamp recordings revealed discrete hyperpolarization-activated, cAMP-sensitive channel openings with amplitudes of 150-230 fA in the activation voltage range. The average conductance of these openings was approximately 1.5 pS at -120 mV in symmetrical 160 mM K(+). Some traces with multiple channels showed unusual gating behavior, characterized by a variable long delay after a voltage step followed by runs of openings. Noise analysis on macroscopic currents revealed fluctuations whose magnitudes were systematically larger than predicted from the actual single channel current size, consistent with cooperativity between single HCN channels.

Evidence for non-independent gating of P2X2 receptors expressed in Xenopus oocytes

BACKGROUND

P2X2 receptor is an ATP-activated ion channel which is widely expressed in the nervous system, and mediates synaptic transmission.

RESULTS

We recorded currents of P2X2 receptors expressed in Xenopus oocytes from outside-out patches and have found that currents recorded from patches containing a single or multiple P2X2 channels differ in a manner suggesting positive cooperativity. First, the currents from multichannel patches exhibit simultaneous transitions more frequently than predicted from the activity of independent channels. Second, the mean open lifetime at the current level of a single channel in a multichannel burst is about six times longer than the open time of currents from single channel patches, a trend opposite to what is expected of independent channels. These results indicate that the channels have positive cooperativity and that the longer opening is due to a slower closing rate. Third, from kinetic analysis the likelihood of the cooperative model is significantly larger than that of the independent model. Fourth, the open channel noise of currents from patches containing multiple channels is less than half that from a single channel, which is consistent with the channel properties being different when they are active in groups.

CONCLUSION

Taken together, our results suggest that P2X2 receptors are non-independent, but interact with positive cooperativity.

Amplitude histograms can identify positively but not negatively coupled channels

We investigated the ability of amplitude distributions to determine if the gating of a pair of channels is coupled. These distributions are expressed as probability density amplitude histograms with peaks corresponding to zero, one, or two open channels. If the channels gate independently, the areas under these peaks (A, B, and C, respectively) determine the open probabilities of the two channels (p(1) and p(2)). Manivannan et al. (Biophys J 1994;61:216) showed that if Delta=B(2)/AC was less than 4 then the channel gating is coupled. We defined a similar parameter, D=(B(2)/4)-AC. If D<0 then channel gating is coupled. However, amplitude histograms with D0 are consistent with both independent and coupled gating. We further present a simple model in which channels are assumed to be identical and can be positively or negatively coupled. Here, amplitude histograms determine q=(B+2C)/2 (open probability of the coupled channels) and r=-D (the coupling parameter). Thus, positively coupled channels (r0) produce amplitude histograms with D<0 whereas negatively coupled channels (r<0) produce amplitude histograms with D0.

Evidence for cooperativity between nicotinic acetylcholine receptors in patch clamp records

It is often assumed that ion channels in cell membrane patches gate independently. However, in the present study nicotinic receptor patch clamp data obtained in cell-attached mode from embryonic chick myotubes suggest that the distribution of steady-state probabilities for conductance multiples arising from concurrent channel openings may not be binomial. In patches where up to four active channels were observed, the probabilities of two or more concurrent openings were greater than expected, suggesting positive cooperativity. For the case of two active channels, we extended the analysis by assuming that 1) individual receptors (not necessarily identical) could be modeled by a five-state (three closed and two open) continuous-time Markov process with equal agonist binding affinity at two recognition sites, and 2) cooperativity between channels could occur through instantaneous changes in specific transition rates in one channel following a change in conductance state of the neighboring channel. This allowed calculation of open and closed sojourn time density functions for either channel conditional on the neighboring channel being open or closed. Simulation studies of two channel systems, with channels being either independent or cooperative, nonidentical or identical, supported the discriminatory power of the optimization algorithm. The experimental results suggested that individual acetylcholine receptors were kinetically identical and that the open state of one channel increased the probability of opening of its neighbor.