A kinetic model of the inositol trisphosphate receptor based on single-channel data

Structural titration reveals Ca2+-dependent conformational landscape of the IP3 receptor

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are endoplasmic reticulum Ca2+ channels whose biphasic dependence on cytosolic Ca2+ gives rise to Ca2+ oscillations that regulate fertilization, cell division and cell death. Despite the critical roles of IP3R-mediated Ca2+ responses, the structural underpinnings of the biphasic Ca2+ dependence that underlies Ca2+ oscillations are incompletely understood. Here, we collect cryo-EM images of an IP3R with Ca2+ concentrations spanning five orders of magnitude. Unbiased image analysis reveals that Ca2+ binding does not explicitly induce conformational changes but rather biases a complex conformational landscape consisting of resting, preactivated, activated, and inhibited states. Using particle counts as a proxy for relative conformational free energy, we demonstrate that Ca2+ binding at a high-affinity site allows IP3Rs to activate by escaping a low-energy resting state through an ensemble of preactivated states. At high Ca2+ concentrations, IP3Rs preferentially enter an inhibited state stabilized by a second, low-affinity Ca2+ binding site. Together, these studies provide a mechanistic basis for the biphasic Ca2+-dependence of IP3R channel activity.

Modeling IP3-induced Ca2+ signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

Simulation of calcium signaling in fine astrocytic processes: Effect of spatial properties on spontaneous activity

Astrocytes, a glial cell type of the central nervous system, have emerged as detectors and regulators of neuronal information processing. Astrocyte excitability resides in transient variations of free cytosolic calcium concentration over a range of temporal and spatial scales, from sub-microdomains to waves propagating throughout the cell. Despite extensive experimental approaches, it is not clear how these signals are transmitted to and integrated within an astrocyte. The localization of the main molecular actors and the geometry of the system, including the spatial organization of calcium channels IP3R, are deemed essential. However, as most calcium signals occur in astrocytic ramifications that are too fine to be resolved by conventional light microscopy, most of those spatial data are unknown and computational modeling remains the only methodology to study this issue. Here, we propose an IP3R-mediated calcium signaling model for dynamics in such small sub-cellular volumes. To account for the expected stochasticity and low copy numbers, our model is both spatially explicit and particle-based. Extensive simulations show that spontaneous calcium signals arise in the model via the interplay between excitability and stochasticity. The model reproduces the main forms of calcium signals and indicates that their frequency crucially depends on the spatial organization of the IP3R channels. Importantly, we show that two processes expressing exactly the same calcium channels can display different types of calcium signals depending on the spatial organization of the channels. Our model with realistic process volume and calcium concentrations successfully reproduces spontaneous calcium signals that we measured in calcium micro-domains with confocal microscopy and predicts that local variations of calcium indicators might contribute to the diversity of calcium signals observed in astrocytes. To our knowledge, this model is the first model suited to investigate calcium dynamics in fine astrocytic processes and to propose plausible mechanisms responsible for their variability.

IP3 receptors – lessons from analyses ex cellula

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are widely expressed intracellular channels that release Ca2+ from the endoplasmic reticulum (ER). We review how studies of IP3Rs removed from their intracellular environment (‘ex cellula’), alongside similar analyses of ryanodine receptors, have contributed to understanding IP3R behaviour. Analyses of permeabilized cells have demonstrated that the ER is the major intracellular Ca2+ store, and that IP3 stimulates Ca2+ release from this store. Radioligand binding confirmed that the 4,5-phosphates of IP3 are essential for activating IP3Rs, and facilitated IP3R purification and cloning, which paved the way for structural analyses. Reconstitution of IP3Rs into lipid bilayers and patch-clamp recording from the nuclear envelope have established that IP3Rs have a large conductance and select weakly between Ca2+ and other cations. Structural analyses are now revealing how IP3 binding to the N-terminus of the tetrameric IP3R opens the pore ∼7 nm away from the IP3-binding core (IBC). Communication between the IBC and pore passes through a nexus of interleaved domains contributed by structures associated with the pore and cytosolic domains, which together contribute to a Ca2+-binding site. These structural analyses provide evidence to support the suggestion that IP3 gates IP3Rs by first stimulating Ca2+ binding, which leads to pore opening and Ca2+ release.

The stretch to stray on time: Resonant length of random walks in a transient

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling

The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP₃ induced Ca²⁺signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation.

Inositol 1,4,5-trisphosphate receptors in the endoplasmic reticulum: A single-channel point of view

As an intracellular Ca(2+) release channel at the endoplasmic reticulum membrane, the ubiquitous inositol 1,4,5-trisphosphate (InsP3) receptor (InsP3R) plays a crucial role in the generation, propagation and regulation of intracellular Ca(2+) signals that regulate numerous physiological and pathophysiological processes. This review provides a concise account of the fundamental single-channel properties of the InsP3R channel: its conductance properties and its regulation by InsP3 and Ca(2+), its physiological ligands, studied using nuclear patch clamp electrophysiology.

A Computational Model of the Dendron of the GnRH Neuron

Gonadotropin-releasing hormone (GnRH) neurons have two major processes that have properties of both dendrites (they receive synaptic input from other neurons) and axons (they actively propagate action potentials to the synaptic terminal). These processes have thus been termed dendrons. We construct a stochastic spatiotemporal model of the dendron of the GnRH neuron, with the goal of studying how stochastic synaptic input along the length of the dendron affects the initiation and propagation of action potentials. We show (1) that synaptic inputs closer to the soma are effective controllers of action potential initiation and electrical bursting and (2) that although the effects on the amplitude and width of propagating action potentials are critically dependent on the timing and location of synaptic input addition, the effects remain small. We conclude that although stochastic synaptic input along the length of the dendron is likely to be a major determinant of action potential initiation, it is an unlikely mechanism for controlling whether or not action potentials reach the synaptic terminal. Thus, the role of synaptic inputs situated along the dendron a long way from the site of action potential initiation remains unclear. We also show that the actions of kisspeptin can result in significant modulation of the amount of calcium released by an action potential at the synaptic terminal. Furthermore, we show that the actions of kisspeptin are greatest when multiple effects operate together and that a kisspeptin-induced increase in firing rate is, by itself, less effective at increasing Ca2+ release than is a combination of an increased firing rate, an increase in Ca2+ influx, and an increase in inositol trisphosphate (IP3) production. We conclude that the inherent synergies in the various actions of kisspeptin make it a likely candidate for the precise control of Ca2+ transients at the synaptic terminal.

An allosteric model of the inositol trisphosphate receptor with nonequilibrium binding

The inositol trisphosphate receptor (IPR) is a crucial ion channel that regulates the Ca(2+) influx from the endoplasmic reticulum (ER) to the cytoplasm. A thorough study of the IPR channel contributes to a better understanding of calcium oscillations and waves. It has long been observed that the IPR channel is a typical biological system which performs adaptation. However, recent advances on the physical essence of adaptation show that adaptation systems with a negative feedback mechanism, such as the IPR channel, must break detailed balance and always operate out of equilibrium with energy dissipation. Almost all previous IPR models are equilibrium models assuming detailed balance and thus violate the dissipative nature of adaptation. In this article, we constructed a nonequilibrium allosteric model of single IPR channels based on the patch-clamp experimental data obtained from the IPR in the outer membranes of isolated nuclei of the Xenopus oocyte. It turns out that our model reproduces the patch-clamp experimental data reasonably well and produces both the correct steady-state and dynamic properties of the channel. Particularly, our model successfully describes the complicated bimodal [Ca(2+)] dependence of the mean open duration at high [IP3], a steady-state behavior which fails to be correctly described in previous IPR models. Finally, we used the patch-clamp experimental data to validate that the IPR channel indeed breaks detailed balance and thus is a nonequilibrium system which consumes energy.

Multiscale modelling of saliva secretion

We review a multiscale model of saliva secretion, describing in brief how the model is constructed and what we have so far learned from it. The model begins at the level of inositol trisphosphate receptors (IPR), and proceeds through the cellular level (with a model of acinar cell calcium dynamics) to the multicellular level (with a model of the acinus), finally to a model of a saliva production unit that includes an acinus and associated duct. The model at the level of the entire salivary gland is not yet completed. Particular results from the model so far include (i) the importance of modal behaviour of IPR, (ii) the relative unimportance of Ca(2+) oscillation frequency as a controller of saliva secretion, (iii) the need for the periodic Ca(2+) waves to be as fast as possible in order to maximise water transport, (iv) the presence of functional K(+) channels in the apical membrane increases saliva secretion, (v) the relative unimportance of acinar spatial structure for isotonic water transport, (vi) the prediction that duct cells are highly depolarised, (vii) the prediction that the secondary saliva takes at least 1mm (from the acinus) to reach ionic equilibrium. We end with a brief discussion of future directions for the model, both in construction and in the study of scientific questions.

Calcium and IP3 dynamics in cardiac myocytes: experimental and computational perspectives and approaches

Calcium plays a crucial role in excitation-contraction coupling (ECC), but it is also a pivotal second messenger activating Ca²⁺-dependent transcription factors in a process termed excitation-transcription coupling (ETC). Evidence accumulated over the past decade indicates a pivotal role of inositol 1,4,5-trisphosphate receptor (IP₃R)-mediated Ca²⁺ release in the regulation of cytosolic and nuclear Ca²⁺ signals. IP₃ is generated by stimulation of plasma membrane receptors that couple to phospholipase C (PLC), liberating IP₃ from phosphatidylinositol 4,5-bisphosphate (PIP₂). An intriguing aspect of IP₃ signaling is the presence of the entire PIP₂-PLC-IP₃ signaling cascade as well as the presence of IP₃Rs at the inner and outer membranes of the nuclear envelope (NE) which functions as a Ca²⁺ store. The observation that the nucleus is surrounded by its own putative Ca²⁺ store raises the possibility that nuclear IP₃-dependent Ca²⁺ release plays a critical role in ETC. This provides a potential mechanism of regulation that acts locally and autonomously from the global cytosolic Ca²⁺ signal underlying ECC. Moreover, there is evidence that: (i) the sarcoplasmic reticulum (SR) and NE are a single contiguous Ca²⁺ store; (ii) the nuclear pore complex is the major gateway for Ca²⁺ and macromolecules to pass between the cytosol and the nucleoplasm; (iii) the inner membrane of the NE hosts key Ca²⁺ handling proteins including the Na⁺/Ca²⁺ exchanger (NCX)/GM1 complex, ryanodine receptors (RyRs), nicotinic acid adenine dinucleotide phosphate receptors (NAADPRs), Na⁺/K⁺ ATPase, and Na⁺/H⁺ exchanger. Thus, it appears that the nucleus represents a Ca²⁺ signaling domain equipped with its own ion channels and transporters that allow for complex local Ca²⁺ signals. Many experimental and modeling approaches have been used for the study of intracellular Ca²⁺ signaling but the key to the understanding of the dual role of Ca²⁺ mediating ECC and ECT lays in quantitative differences of local [Ca²⁺] in the nuclear and cytosolic compartment. In this review, we discuss the state of knowledge regarding the origin and the physiological implications of nuclear Ca²⁺ transients in different cardiac cell types (adult atrial and ventricular myocytes) as well as experimental and mathematical approaches to study Ca²⁺ and IP₃ signaling in the cytosol and nucleus. In particular, we focus on the concept that highly localized Ca²⁺ signals are required to translocate and activate Ca²⁺-dependent transcription factors (e.g., nuclear factor of activated T-cells, NFAT; histone deacetylase, HDAC) through phosphorylation/dephosphorylation processes.

Mesoscopic behavior from microscopic Markov dynamics and its application to calcium release channels

A major challenge in biology is to understand how molecular processes determine phenotypic features. We address this fundamental problem in a class of model systems by developing a general mathematical framework that allows the calculation of mesoscopic properties from the knowledge of microscopic Markovian transition probabilities. We show how exact analytic formulae for the first and second moments of resident time distributions in mesostates can be derived from microscopic resident times and transition probabilities even for systems with a large number of microstates. We apply our formalism to models of the inositol trisphosphate receptor, which plays a key role in generating calcium signals triggering a wide variety of cellular responses. We demonstrate how experimentally accessible quantities, such as opening and closing times and the coefficient of variation of inter-spike intervals, and other, more elaborated, quantities can be analytically calculated from the underlying microscopic Markovian dynamics. A virtue of our approach is that we do not need to follow the detailed time evolution of the whole system, as we derive the relevant properties of its steady state without having to take into account the often extremely complicated transient features. We emphasize that our formulae fully agree with results obtained by stochastic simulations and approaches based on a full determination of the microscopic system's time evolution. We also illustrate how experiments can be devised to discriminate between alternative molecular models of the inositol trisphosphate receptor. The developed approach is applicable to any system described by a Markov process and, owing to the analytic nature of the resulting formulae, provides an easy way to characterize also rare events that are of particular importance to understand the intermittency properties of complex dynamic systems.

A stochastic model of calcium puffs based on single-channel data

Calcium puffs are local transient Ca(2+) releases from internal Ca(2+) stores such as the endoplasmic reticulum or the sarcoplasmic reticulum. Such release occurs through a cluster of inositol 1,4,5-trisphosphate receptors (IP3Rs). Based on the IP3R model (which is determined by fitting to stationary single-channel data) and nonstationary single-channel data, we construct a new IP3R model that includes time-dependent rates of mode switches. A point-source model of Ca(2+) puffs is then constructed based on the new IP3R model and is solved by a hybrid Gillespie method with adaptive timing. Model results show that a relatively slow recovery of an IP3R from Ca(2+) inhibition is necessary to reproduce most of the experimental outcomes, especially the nonexponential interpuff interval distributions. The number of receptors in a cluster could be severely underestimated when the recovery is sufficiently slow. Furthermore, we find that, as the number of IP3Rs increases, the average duration of puffs initially increases but then becomes saturated, whereas the average decay time keeps increasing linearly. This gives rise to the observed asymmetric puff shape.

Patch-clamp electrophysiology of intracellular Ca2+ channels

The modulation of cytoplasmic free Ca(2+) concentration ([Ca(2+)]i) is a universal intracellular signaling pathway that regulates numerous cellular physiological processes. Ubiquitous intracellular Ca(2+)-release channels localized to the endoplasmic/sarcoplasmic reticulum-inositol 1,4,5-trisphosphate receptor (InsP3R) and ryanodine receptor (RyR) channels-play a central role in [Ca(2+)]i signaling in all animal cells. Despite their intracellular localization, electrophysiological studies of the single-channel permeation and gating properties of these Ca(2+)-release channels using the powerful patch-clamp approach have been possible by application of this technique to isolated nuclei because the channels are present in membranes of the nuclear envelope. Here we provide a concise description of how nuclear patch-clamp experiments have been used to study single-channel properties of different InsP3R channels in the outer nuclear membrane. We compare this with other methods for studying intracellular Ca(2+) release. We also briefly describe application of the technique to InsP3R channels in the inner nuclear membrane and to channels in the outer nuclear membrane of HEK293 cells expressing recombinant RyR.

Regulation of electrical bursting in a spatiotemporal model of a GnRH neuron

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such as the properties of burst action potential firing and their associated Ca(2+) transients. These previous studies were based on experimental recordings taken from the soma of GnRH neurons. However, some research groups have shown that the dendrites of GnRH neurons play very important roles. In particular, it is now known that the site of action potential initiation in these neurons is often in the dendrite, over 100 μm from the soma. This raises an important question. Since some of the mechanisms for controlling the burst length and interburst interval are located in the soma, how can electrical bursting be controlled when initiated at a site located some distance from these controlling mechanisms? In order to answer this question, we construct a spatio-temporal mathematical model that includes both the soma and the dendrite. Our model shows that the diffusion coefficient for the spread of electrical potentials in the dendrite is large enough to coordinate burst firing of action potentials when the initiation site is located at some distance from the soma.

Comparison of models for IP3 receptor kinetics using stochastic simulations

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ubiquitous intracellular calcium (Ca²⁺) channel which has a major role in controlling Ca²⁺ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP₃R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP₃R when modeling local Ca²⁺ transients in dendrites or overall Ca²⁺ dynamics in large neuronal models. The goal of this study was to evaluate existing IP₃R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP₃R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP₃R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP₃R with different concentrations of Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.

A data-driven model of a modal gated ion channel: the inositol 1,4,5-trisphosphate receptor in insect Sf9 cells

The inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) channel is crucial for the generation and modulation of intracellular Ca(2+) signals in animal cells. To gain insight into the complicated ligand regulation of this ubiquitous channel, we constructed a simple quantitative continuous-time Markov-chain model from the data. Our model accounts for most experimentally observed gating behaviors of single native IP(3)R channels from insect Sf9 cells. Ligand (Ca(2+) and IP(3)) dependencies of channel activity established six main ligand-bound channel complexes, where a complex consists of one or more states with the same ligand stoichiometry and open or closed conformation. Channel gating in three distinct modes added one complex and indicated that three complexes gate in multiple modes. This also restricted the connectivity between channel complexes. Finally, latencies of channel responses to abrupt ligand concentration changes defined a model with specific network topology between 9 closed and 3 open states. The model with 28 parameters can closely reproduce the equilibrium gating statistics for all three gating modes over a broad range of ligand concentrations. It also captures the major features of channel response latency distributions. The model can generate falsifiable predictions of IP(3)R channel gating behaviors and provide insights to both guide future experiment development and improve IP(3)R channel gating analysis. Maximum likelihood estimates of the model parameters and of the parameters in the De Young-Keizer model yield strong statistical evidence in favor of our model. Our method is simple and easily applicable to the dynamics of other ion channels and molecules.

Extending the IP3 receptor model to include competition with partial agonists

The inositol 1,4,5-trisphosphate (IP(3)) receptor is a Ca(2+) channel located in the endoplasmic reticulum and is regulated by IP(3) and Ca(2+). This channel is critical to calcium signaling in cell types as varied as neurons and pancreatic beta cells to mast cells. De Young and Keizer (1992) created an eight-state, nine-variable model of the IP(3) receptor. In their model, they accounted for three binding sites, a site for IP(3), activating Ca(2+), and deactivating Ca(2+). The receptor is only open if IP(3) and activating Ca(2+) is bound. Li and Rinzel followed up this paper in 1994 by introducing a reduction that made it into a two variable system. A recent publication by Rossi et al. (2009) studied the effect of introducing IP(3)-like molecules, referred to as partial agonists (PA), into the cell to determine the structure-function relationship between IP(3) and its receptor. Initial results suggest a competitive model, where IP(3) and PA fight for the same binding site. We extend the original eight-state model to a 12-state model in order to illustrate this competition, and perform a similar reduction to that of Li and Rinzel in the first modeling study we are aware of considering PA effect on an IP(3) receptor. Using this reduction we solve for the equilibrium open probability for calcium release in the model. We replicate graphs provided by the Rossi paper, and find that optimizing the subunit affinities for IP(3) and PA yields a good fit to the data. We plug our extended reduced model into a full cell model, in order to analyze the effects PA have on whole cell properties specifically the propagation of calcium waves in two dimensions. We conclude that PA creates qualitatively different calcium dynamics than would simply reducing IP(3), but that effectively PA can act as an IP(3) knockdown.

A kinetic model for type I and II IP3R accounting for mode changes

Based upon an extensive single-channel data set, a Markov model for types I and II inositol trisphosphate receptors (IP(3)R) is developed. The model aims to represent accurately the kinetics of both receptor types of IP(3)R depending on the concentrations of inositol trisphosphate (IP(3)), adenosine trisphosphate (ATP), and intracellular calcium (Ca(2+)). In particular, the model takes into account that for some combinations of ligands the IP(3)R switches between extended periods of inactivity alternating with intervals of bursting activity (mode changes). In a first step, the inactive and active modes are modeled separately. It is found that, within modes, both receptor types are ligand-independent. In a second step, the submodels are connected by transition rates. Ligand-dependent regulation of the channel activity is achieved by modulating these transitions between active and inactive modes. As a result, a compact representation of the IP(3)R is obtained that accurately captures stochastic single-channel dynamics including mode changes in a model with six states and 10 rate constants, only two of which are ligand-dependent.

A multi-scale approach to airway hyperresponsiveness: from molecule to organ

Airway hyperresponsiveness (AHR), a characteristic of asthma that involves an excessive reduction in airway caliber, is a complex mechanism reflecting multiple processes that manifest over a large range of length and time scales. At one extreme, molecular interactions determine the force generated by airway smooth muscle (ASM). At the other, the spatially distributed constriction of the branching airways leads to breathing difficulties. Similarly, asthma therapies act at the molecular scale while clinical outcomes are determined by lung function. These extremes are linked by events operating over intermediate scales of length and time. Thus, AHR is an emergent phenomenon that limits our understanding of asthma and confounds the interpretation of studies that address physiological mechanisms over a limited range of scales. A solution is a modular computational model that integrates experimental and mathematical data from multiple scales. This includes, at the molecular scale, kinetics, and force production of actin-myosin contractile proteins during cross-bridge and latch-state cycling; at the cellular scale, Ca²⁺ signaling mechanisms that regulate ASM force production; at the tissue scale, forces acting between contracting ASM and opposing viscoelastic tissue that determine airway narrowing; at the organ scale, the topographic distribution of ASM contraction dynamics that determine mechanical impedance of the lung. At each scale, models are constructed with iterations between theory and experimentation to identify the parameters that link adjacent scales. This modular model establishes algorithms for modeling over a wide range of scales and provides a framework for the inclusion of other responses such as inflammation or therapeutic regimes. The goal is to develop this lung model so that it can make predictions about bronchoconstriction and identify the pathophysiologic mechanisms having the greatest impact on AHR and its therapy.

Fundamental properties of Ca2+ signals

BACKGROUND

Ca2+ is a ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration. Recent investigations changed basic ideas on the dynamic character of Ca2+ signals and challenge traditional ideas on information transmission.

SCOPE OF REVIEW

We present recent findings on key characteristics of the cytosolic Ca2+ dynamics and theoretical concepts that explain the wide range of experimentally observed Ca2+ signals. Further, we relate properties of the dynamical regulation of the cytosolic Ca2+ concentration to ideas about information transmission by stochastic signals.

MAJOR CONCLUSIONS

We demonstrate the importance of the hierarchal arrangement of Ca2+ release sites on the emergence of cellular Ca2+ spikes. Stochastic Ca2+ signals are functionally robust and adaptive to changing environmental conditions. Fluctuations of interspike intervals (ISIs) and the moment relation derived from ISI distributions contain information on the channel cluster open probability and on pathway properties.

GENERAL SIGNIFICANCE

Robust and reliable signal transduction pathways that entail Ca2+ dynamics are essential for eukaryotic organisms. Moreover, we expect that the design of a stochastic mechanism which provides robustness and adaptivity will be found also in other biological systems. Ca2+ dynamics demonstrate that the fluctuations of cellular signals contain information on molecular behavior. This article is part of a Special Issue entitled Biochemical, biophysical and genetic approaches to intracellular calcium signaling.

Analysis of IP3 receptors in and out of cells

BACKGROUND

Inositol 1,4,5-trisphosphate receptors (IP3R) are expressed in almost all animal cells. Three mammalian genes encode closely related IP3R subunits, which assemble into homo- or hetero-tetramers to form intracellular Ca2+ channels.

SCOPE OF THE REVIEW

In this brief review, we first consider a variety of complementary methods that allow the links between IP3 binding and channel gating to be defined. How does IP3 binding to the IP3-binding core in each IP3R subunit cause opening of a cation-selective pore formed by residues towards the C-terminal? We then describe methods that allow IP3, Ca2+ signals and IP3R mobility to be examined in intact cells. A final section briefly considers genetic analyses of IP3R signalling.

MAJOR CONCLUSIONS

All IP3R are regulated by both IP3 and Ca2+. This allows them to initiate and regeneratively propagate intracellular Ca2+ signals. The elementary Ca2+ release events evoked by IP3 in intact cells are mediated by very small numbers of active IP3R and the Ca2+-mediated interactions between them. The spatial organization of these Ca2+ signals and their stochastic dependence on so few IP3Rs highlight the need for methods that allow the spatial organization of IP3R signalling to be addressed with single-molecule resolution.

GENERAL SIGNIFICANCE

A variety of complementary methods provide insight into the structural basis of IP3R activation and the contributions of IP3-evoked Ca2+ signals to cellular physiology. This article is part of a Special Issue entitled Biochemical, biophysical and genetic approaches to intracellular calcium signaling.

A mathematical model of adult GnRH neurons in mouse brain and its bifurcation analysis

GnRH neurons are hypothalamic neurons that secrete gonadotropin-releasing hormone (GnRH) which stimulates the release of gonadotropins, one of the crucial hormones for sexual development, fertility and maturation. A mathematical model was built to help elucidate the mechanisms underlying electrical bursting and synchronous [Ca²(+)] transients in GnRH neurons (Lee et al., 2010). The model predicted that bursting in GnRH neurons (at least of the short-bursting type) requires the existence of a [Ca²(+)]-dependent slow after-hyperpolarisation current (sI(AHP-UCL)), and this predicted current was found experimentally. GnRH behaviour under a wide range of conditions (inhibition of Na(+) channels, IP₃ receptors, [Ca²(+)]-dependent K(+) channels, or Ca²(+) pumps, or in the presence of zero extracellular [Ca²(+)]) is successfully reproduced by the model. In this paper, a simplified version of the previous model, with the same qualitative behaviour, is constructed and studied using timescale separation techniques and bifurcation analysis.

Stochastic fire-diffuse-fire model with realistic cluster dynamics

Living organisms use waves that propagate through excitable media to transport information. Ca2+ waves are a paradigmatic example of this type of processes. A large hierarchy of Ca2+ signals that range from localized release events to global waves has been observed in Xenopus laevis oocytes. In these cells, Ca2+ release occurs trough inositol 1,4,5-trisphosphate receptors (IP3Rs) which are organized in clusters of channels located on the membrane of the endoplasmic reticulum. In this article we construct a stochastic model for a cluster of IP3R 's that replicates the experimental observations reported in [D. Fraiman, Biophys. J. 90, 3897 (2006)]. We then couple this phenomenological cluster model with a reaction-diffusion equation, so as to have a discrete stochastic model for calcium dynamics. The model we propose describes the transition regimes between isolated release and steadily propagating waves as the IP3 concentration is increased.

Spatio-temporal calcium dynamics in pacemaking units of the interstitial cells of Cajal

The interstitial cells of Cajal (ICC) are responsible for producing pacemaking signals that stimulate rhythmic contractions in the gastro-intestinal system. The pacemaking signals are generated by membrane depolarizations, which are in turn linked to the integrated transport of calcium between the endoplasmic reticulum (ER), through inositol-trisphosphate receptor (IP(3)R) release, and mitochondria, through the uniporter. A non-specific cation channel (NSCC) is associated with the membrane depolarizations, and is inhibited by intracellular calcium. One theory proposes that the integrated calcium transport occurs within specific regions of the ICC called "pacemaker units," and results in localized calcium concentration reductions within these units, which in turn activate the NSCC and depolarize the membrane. We have constructed a model of the spatio-temporal calcium dynamics within an ICC pacemaker unit to determine under what conditions the local calcium concentrations may reduce below baseline. We obtain reductions of calcium concentrations below baseline but only under certain conditions. Without strong and persistent stimulation of the IP(3)R, reductions of calcium below baseline occur only with a non-physiological, time-dependent uniporter. Alternatively, sufficient IP(3)R release leads to reductions of calcium below baseline, due to depletion of the ER calcium store over the time scale of seconds, although these reductions require strong mitochondrial and ER calcium uptake.

A dynamic model of saliva secretion

We construct a mathematical model of the parotid acinar cell with the aim of investigating how the distribution of K(+) and Cl(-) channels affects saliva production. Secretion of fluid is initiated by Ca(2+) signals acting on Ca(2+) dependent K(+) and Cl(-) channels. The opening of these channels facilitates the movement of Cl(-) ions into the lumen which water follows by osmosis. We use recent results into both the release of Ca(2+) from internal stores via the inositol (1,4,5)-trisphosphate receptor (IP(3)R) and IP(3) dynamics to create a physiologically realistic Ca(2+) model which is able to recreate important experimentally observed behaviours seen in parotid acinar cells. We formulate an equivalent electrical circuit diagram for the movement of ions responsible for water flow which enables us to calculate and include distinct apical and basal membrane potentials to the model. We show that maximum saliva production occurs when a small amount of K(+) conductance is located at the apical membrane, with the majority in the basal membrane. The maximum fluid output is found to coincide with a minimum in the apical membrane potential. The traditional model whereby all Cl(-) channels are located in the apical membrane is shown to be the most efficient Cl(-) channel distribution.

Inositol trisphosphate receptor and ion channel models based on single-channel data

The inositol trisphosphate receptor (IPR) plays an important role in controlling the dynamics of intracellular Ca(2+). Single-channel patch-clamp recordings are a typical way to study these receptors as well as other ion channels. Methods for analyzing and using this type of data have been developed to fit Markov models of the receptor. The usual method of parameter fitting is based on maximum-likelihood techniques. However, Bayesian inference and Markov chain Monte Carlo techniques are becoming more popular. We describe the application of the Bayesian methods to real experimental single-channel data in three ion channels: the ryanodine receptor, the K(+) channel, and the IPR. One of the main aims of all three studies was that of model selection with different approaches taken. We also discuss the modeling implications for single-channel data that display different levels of channel activity within one recording.