A model of IP3 receptor with a luminal calcium binding site: stochastic simulations and analysis

Analyzing optical imaging of Ca2+ signals via TIRF microscopy: The limits on resolution due to chemical rates and depth of the channels

High resolution total internal reflection (TIRF) microscopy (TIRFM) together with detailed computational modeling provides a powerful approach towards the understanding of a wide range of Ca²⁺ signals mediated by the ubiquitous inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R) channel. Exploiting this fruitful collaboration further requires close agreement between the models and observations. However, elementary Ca²⁺ release events, puffs, imaged through TIRFM do not show the rapid single-channel openings and closings during and between puffs as are present in simulated puffs using data-driven single channel models. TIRFM also shows a rapid equilibration of 10ms after a channel opens or closes which is not achievable in simulation using standard Ca²⁺ diffusion coefficients and reaction rates between indicator dye and Ca²⁺. Furthermore, TIRFM imaging cannot decipher the depth of the channel with respect to the microscope, which will affect the change in fluorescence that the microscope detects, thereby affecting its sensitivity to fast single-channel activity. Using the widely used Ca²⁺ diffusion coefficients and reaction rates, our simulations show equilibration rates that are eight times slower than TIRFM imaging. We show that to get equilibrium rates consistent with observed values, the diffusion coefficients and reaction rates have to be significantly higher than the values reported in the literature, and predict the channel depth to be 200-250nm. Finally, we show that with the addition of noise, short events due to 1-2ms opening and closing of channels that are observed in computational models can be missed in TIRFM.

Luminal Ca(2+) dynamics during IP3R mediated signals

The role of cytosolic Ca(2+) on the kinetics of Inositol 1,4,5-triphosphate receptors (IP3Rs) and on the dynamics of IP3R-mediated Ca(2+) signals has been studied at large both experimentally and by modeling. The role of luminal Ca(2+) has not been investigated with that much detail although it has been found that it is relevant for signal termination in the case of Ca(2+) release through ryanodine receptors. In this work we present the results of observing the dynamics of luminal and cytosolic Ca(2+) simultaneously in Xenopus laevis oocytes. Combining observations and modeling we conclude that there is a rapid mechanism that guarantees the availability of free Ca(2+) in the lumen even when a relatively large Ca(2+) release is evoked. Comparing the dynamics of cytosolic and luminal Ca(2+) during a release, we estimate that they are consistent with a 80% of luminal Ca(2+) being buffered. The rapid availability of free luminal Ca(2+) correlates with the observation that the lumen occupies a considerable volume in several regions across the images.

Modeling analysis of inositol 1,4,5-trisphosphate receptor-mediated Ca2+ mobilization under the control of glucagon-like peptide-1 in mouse pancreatic β-cells

Glucagon-like peptide-1 (GLP-1) is an intestinally derived blood glucose-lowering hormone that potentiates glucose-stimulated insulin secretion from pancreatic β-cells. The secretagogue action of GLP-1 is explained, at least in part, by its ability to stimulate cAMP production so that cAMP may facilitate the release of Ca(2+) from inositol trisphosphate receptor (IP3R)-regulated Ca(2+) stores. However, a quantitative model has yet to be provided that explains the molecular mechanisms and dynamic processes linking GLP-1-stimulated cAMP production to Ca(2+) mobilization. Here, we performed simulation studies to investigate how GLP-1 alters the abilities of Ca(2+) and IP3 to act as coagonists at IP3R Ca(2+) release channels. A new dynamic model was constructed based on the Kaftan model, which demonstrates dual steady-state allosteric regulation of the IP3R by Ca(2+) and IP3. Data obtained from β-cells were then analyzed to understand how GLP-1 facilitates IP3R-mediated Ca(2+) mobilization when UV flash photolysis is used to uncage Ca(2+) and IP3 intracellularly. When the dynamic model for IP3R activation was incorporated into a minimal cell model, the Ca(2+) transients and oscillations induced by GLP-1 were successfully reconstructed. Simulation studies indicated that transient and oscillatory responses to GLP-1 were produced by sequential positive and negative feedback regulation due to fast activation and slow inhibition of the IP3R by Ca(2+). The slow rate of Ca(2+)-dependent inhibition was revealed to provide a remarkable contribution to the time course of the decay of cytosolic Ca(2+) transients. It also served to drive and pace Ca(2+) oscillations that are significant when evaluating how GLP-1 stimulates insulin secretion.

Differential Regulation of Multiple Steps in Inositol 1,4,5-Trisphosphate Signaling by Protein Kinase C Shapes Hormone-stimulated Ca2+ Oscillations

How Ca(2+) oscillations are generated and fine-tuned to yield versatile downstream responses remains to be elucidated. In hepatocytes, G protein-coupled receptor-linked Ca(2+) oscillations report signal strength via frequency, whereas Ca(2+) spike amplitude and wave velocity remain constant. IP3 uncaging also triggers oscillatory Ca(2+) release, but, in contrast to hormones, Ca(2+) spike amplitude, width, and wave velocity were dependent on [IP3] and were not perturbed by phospholipase C (PLC) inhibition. These data indicate that oscillations elicited by IP3 uncaging are driven by the biphasic regulation of the IP3 receptor by Ca(2+), and, unlike hormone-dependent responses, do not require PLC. Removal of extracellular Ca(2+) did not perturb Ca(2+) oscillations elicited by IP3 uncaging, indicating that reloading of endoplasmic reticulum stores via plasma membrane Ca(2+) influx does not entrain the signal. Activation and inhibition of PKC attenuated hormone-induced Ca(2+) oscillations but had no effect on Ca(2+) increases induced by uncaging IP3. Importantly, PKC activation and inhibition differentially affected Ca(2+) spike frequencies and kinetics. PKC activation amplifies negative feedback loops at the level of G protein-coupled receptor PLC activity and/or IP3 metabolism to attenuate IP3 levels and suppress the generation of Ca(2+) oscillations. Inhibition of PKC relieves negative feedback regulation of IP3 accumulation and, thereby, shifts Ca(2+) oscillations toward sustained responses or dramatically prolonged spikes. PKC down-regulation attenuates phenylephrine-induced Ca(2+) wave velocity, whereas responses to IP3 uncaging are enhanced. The ability to assess Ca(2+) responses in the absence of PLC activity indicates that IP3 receptor modulation by PKC regulates Ca(2+) release and wave velocity.

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.

Termination of calcium puffs and coupled closings of inositol trisphosphate receptor channels

Calcium puffs are localized Ca(2+) signals mediated by Ca(2+) release from the endoplasmic reticulum (ER) through clusters of inositol trisphosphate receptor (IP3R) channels. The recruitment of IP3R channels during puffs depends on Ca(2+)-induced Ca(2+) release, a regenerative process that must be terminated to maintain control of cell signaling and prevent Ca(2+) cytotoxicity. Here, we studied puff termination using total internal reflection microscopy to resolve the gating of individual IP3R channels during puffs in intact SH-SY5Y neuroblastoma cells. We find that the kinetics of IP3R channel closing differ from that expected for independent, stochastic gating, in that multiple channels tend to remain open together longer than predicted from their individual open lifetimes and then close in near-synchrony. This behavior cannot readily be explained by previously proposed termination mechanisms, including Ca(2+)-inhibition of IP3Rs and local depletion of Ca(2+) in the ER lumen. Instead, we postulate that the gating of closely adjacent IP3Rs is coupled, possibly via allosteric interactions, suggesting an important mechanism to ensure robust puff termination in addition to Ca(2+)-inactivation.

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.

Buffer regulation of calcium puff sequences

Puffs are localized Ca(2 +) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP3). They are the result of the liberation of Ca(2 +) from the endoplasmic reticulum through the coordinated opening of IP3 receptor/channels clustered at a functional release site. The presence of buffers that trap Ca(2 +) provides a mechanism that enriches the spatio-temporal dynamics of cytosolic calcium. The expression of different types of buffers along the cell's life provides a tool with which Ca(2 +) signals and their responses can be modulated. In this paper we extend the stochastic model of a cluster of IP3R-Ca(2 +) channels introduced previously to elucidate the effect of buffers on sequences of puffs at the same release site. We obtain analytically the probability laws of the interpuff time and of the number of channels that participate of the puffs. Furthermore, we show that under typical experimental conditions the effect of buffers can be accounted for in terms of a simple inhibiting function. Hence, by exploring different inhibiting functions we are able to study the effect of a variety of buffers on the puff size and interpuff time distributions. We find the somewhat counter-intuitive result that the addition of a fast Ca(2 +) buffer can increase the average number of channels that participate of a puff.

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.

Permeant calcium ion feed-through regulation of single inositol 1,4,5-trisphosphate receptor channel gating

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) Ca(2+) release channel plays a central role in the generation and modulation of intracellular Ca(2+) signals, and is intricately regulated by multiple mechanisms including cytoplasmic ligand (InsP(3), free Ca(2+), free ATP(4-)) binding, posttranslational modifications, and interactions with cytoplasmic and endoplasmic reticulum (ER) luminal proteins. However, regulation of InsP(3)R channel activity by free Ca(2+) in the ER lumen ([Ca(2+)](ER)) remains poorly understood because of limitations of Ca(2+) flux measurements and imaging techniques. Here, we used nuclear patch-clamp experiments in excised luminal-side-out configuration with perfusion solution exchange to study the effects of [Ca(2+)](ER) on homotetrameric rat type 3 InsP(3)R channel activity. In optimal [Ca(2+)](i) and subsaturating [InsP(3)], jumps of [Ca(2+)](ER) from 70 nM to 300 µM reduced channel activity significantly. This inhibition was abrogated by saturating InsP(3) but restored when [Ca(2+)](ER) was raised to 1.1 mM. In suboptimal [Ca(2+)](i), jumps of [Ca(2+)](ER) (70 nM to 300 µM) enhanced channel activity. Thus, [Ca(2+)](ER) effects on channel activity exhibited a biphasic dependence on [Ca(2+)](i). In addition, the effect of high [Ca(2+)](ER) was attenuated when a voltage was applied to oppose Ca(2+) flux through the channel. These observations can be accounted for by Ca(2+) flux driven through the open InsP(3)R channel by [Ca(2+)](ER), raising local [Ca(2+)](i) around the channel to regulate its activity through its cytoplasmic regulatory Ca(2+)-binding sites. Importantly, [Ca(2+)](ER) regulation of InsP(3)R channel activity depended on cytoplasmic Ca(2+)-buffering conditions: it was more pronounced when [Ca(2+)](i) was weakly buffered but completely abolished in strong Ca(2+)-buffering conditions. With strong cytoplasmic buffering and Ca(2+) flux sufficiently reduced by applied voltage, both activation and inhibition of InsP(3)R channel gating by physiological levels of [Ca(2+)](ER) were completely abolished. Collectively, these results rule out Ca(2+) regulation of channel activity by direct binding to the luminal aspect of the channel.

Mean field strategies induce unrealistic non-linearities in calcium puffs

Mean field models are often useful approximations to biological systems, but sometimes, they can yield misleading results. In this work, we compare mean field approaches with stochastic models of intracellular calcium release. In particular, we concentrate on calcium signals generated by the concerted opening of several clustered channels (calcium puffs). To this end we simulate calcium puffs numerically and then try to reproduce features of the resulting calcium distribution using mean field models were all the channels open and close simultaneously. We show that an unrealistic non-linear relationship between the current and the number of open channels is needed to reproduce the simulated puffs. Furthermore, a single channel current which is five times smaller than the one of the stochastic simulations is also needed. Our study sheds light on the importance of the stochastic kinetics of the calcium release channel activity to estimate the release fluxes.

The role of agonist-independent conformational transformation (AICT) in IP₃ cluster behavior

Inositol 1,4,5-trisphosphate (IP(3)) receptor is a central unit in intracellular Ca(2+) signaling. Regulation of the IP₃ receptor by calcium is well characterized. High open probability values are reported for a single IP₃ receptor in nuclear patch clamp experiments. These experimental observations are in contrast with the lower open probability values of the lipid bilayer experiments. Most theoretical models do not account for high open probabilities of the receptor. But more recently, new models of the IP₃ receptor have been put forward which are constrained by single-channel nuclear patch clamp recordings, which generate the larger open probability with the aid of an additional agonist-independent conformational transformation (AICT)-'active' state. The main aim of this work is to constrain the AICT models with a wealth of experimental data characterizing calcium release from IP₃ receptor clusters. Our results suggest that consistency of cluster release between theory and experiments constrains the kinetics of the agonist-independent conformational transition rates (AICT) to values which lead to small open probabilities for the IP₃ receptor inconsistent with nuclear patch clamp experimental data.

Observable effects of Ca2+ buffers on local Ca2+ signals

Calcium signals participate in a large variety of physiological processes. In many instances, they involve calcium entry through inositol 1,4,5-trisphosphate (IP(3)) receptors (IP(3)Rs), which are usually organized in clusters. Recent high-resolution optical experiments by Smith & Parker have provided new information on Ca(2+) release from clustered IP(3)Rs. In the present paper, we use the model recently introduced by Solovey & Ponce Dawson to determine how the distribution of the number of IP(3)Rs that become open during a localized release event may change by the presence of Ca(2+) buffers, substances that react with Ca(2+), altering its concentration and transport properties. We then discuss how buffer properties could be extracted from the observation of local signals.

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.

Intra-cluster percolation of calcium signals

Calcium signals are involved in a large variety of physiological processes. Their versatility relies on the diversity of spatio-temporal behaviors that the calcium concentration can display. Calcium entry through inositol 1,4,5-trisphosphate (IP) receptors (IPR's) is a key component that participates in both local signals such as “puffs” and in global waves. IPR's are usually organized in clusters on the membrane of the endoplasmic reticulum and their spatial distribution has important effects on the resulting signal. Recent high resolution observations [1] of Ca puffs offer a window to study intra-cluster organization. The experiments give the distribution of the number of IPR's that open during each puff without much processing. Here we present a simple model with which we interpret the experimental distribution in terms of two stochastic processes: IP binding and unbinding and Ca-mediated inter-channel coupling. Depending on the parameters of the system, the distribution may be dominated by one or the other process. The transition between both extreme cases is similar to a percolation process. We show how, from an analysis of the experimental distribution, information can be obtained on the relative weight of the two processes. The largest distance over which Ca-mediated coupling acts and the density of IP-bound IPR's of the cluster can also be estimated. The approach allows us to infer properties of the interactions among the channels of the cluster from statistical information on their emergent collective behavior.

An investigation of models of the IP3R channel in Xenopus oocyte

We consider different models of inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) channels in order to fit nuclear membrane patch clamp data of the stationary open probability, mean open time, and mean close time of channels in the Xenopus oocyte. Our results indicate that rather than to treat the tetrameric IP(3)R as four independent and identical subunits, one should assume sequential binding-unbinding processes of Ca(2+) ions and IP(3) messengers. Our simulations also favor the assumption that a channel opens through a conformational transition from a close state to an active state.

Stochastic modeling of calcium in 3D geometry

Release of inflammatory mediators by mast cells in type 1 immediate-hypersensitivity allergic reactions relies on antigen-dependent increases in cytosolic calcium. Here, we used a series of electron microscopy images to build a 3D reconstruction representing a slice through a rat tumor mast cell, which then served as a basis for stochastic modeling of inositol-trisphosphate-mediated calcium responses. The stochastic approach was verified by reaction-diffusion modeling within the same geometry. Local proximity of the endoplasmic reticulum to either the plasma membrane or mitochondria is predicted to differentially impact local inositol trisphosphate receptor transport. The explicit consideration of organelle spatial relationships represents an important step toward building a comprehensive, realistic model of cellular calcium dynamics.

Glial-neuronal interactions--implications for plasticity and drug addiction

Among neuroscientists, astrocytes have for long played Cinderella to their neuron stepsisters. While the importance of glia in regulating brain activity was predicted by Ramon y Cajal more than a century ago (Garcia-Marin et al., Trends. Neurosci. 30:479-787, 2007), these cells, until recently, have been thought to play mainly a passive part in synaptic signaling. Results obtained over the last decade have begun to suggest otherwise. Experiments carried out in a number of labs have shown that glial cells, especially astrocytes, directly participate in synaptic signaling and potentially regulate synaptic plasticity and network excitability. The presence of signaling pathways on astrocytes that are analogous to those at presynaptic terminals suggests a role for these cells in network plasticity. Findings that the same signaling pathways can be activated by receptors for drugs of abuse present on astrocytes suggest a role for these cells in the addictive process. In this review, we summarize current understanding of astrocytic role in synaptic signaling and suggest that a complete understanding of the process of addiction requires a better understanding of the functional role of these cells.

Simplified model of cytosolic Ca2+ dynamics in the presence of one or several clusters of Ca2+ -release channels

Calcium release from intracellular stores plays a key role in the regulation of a variety of cellular activities. In various cell types this release occurs through inositol-triphosphate (IP3) receptors which are Ca2+ channels whose open probability is modulated by the cytosolic Ca2+ concentration itself. Thus, the combination of Ca2+ release and Ca2+ diffusion evokes a variety of Ca2+ signals depending on the number and relative location of the channels that participate of them. In fact, a hierarchy of Ca2+ signals has been observed in Xenopus laevis oocytes, ranging from very localized events (puffs and blips) to waves that propagate throughout the cell. In this cell type channels are organized in clusters. The behavior of individual channels within a cluster cannot be resolved with current optical techniques. Therefore, a combination of experiments and mathematical modeling is unavoidable to understand these signals. However, the numerical simulation of a detailed mathematical model of the problem is very hard given the large range of spatial and temporal scales that must be covered. In this paper we present an alternative model in which the cluster region is modeled using a relatively fine grid but where several approximations are made to compute the cytosolic Ca2+ concentration ([Ca;{2+}]) distribution. The inner-cluster [Ca;{2+}] distribution is used to determine the openings and closings of the channels of the cluster. The spatiotemporal [Ca;{2+}] distribution outside the cluster is determined using a coarser grid in which each (active) cluster is represented by a point source whose current is proportional to the number of open channels determined before. A full reaction-diffusion system is solved on this coarser grid.

Modeling Ca2+ feedback on a single inositol 1,4,5-trisphosphate receptor and its modulation by Ca2+ buffers

The inositol 1,4,5-trisphosphate receptor/channel (IP(3)R) is a major regulator of intracellular Ca(2+) signaling, and liberates Ca(2+) ions from the endoplasmic reticulum in response to binding at cytosolic sites for both IP(3) and Ca(2+). Although the steady-state gating properties of the IP(3)R have been extensively studied and modeled under conditions of fixed [IP(3)] and [Ca(2+)], little is known about how Ca(2+) flux through a channel may modulate the gating of that same channel by feedback onto activating and inhibitory Ca(2+) binding sites. We thus simulated the dynamics of Ca(2+) self-feedback on monomeric and tetrameric IP(3)R models. A major conclusion is that self-activation depends crucially on stationary cytosolic Ca(2+) buffers that slow the collapse of the local [Ca(2+)] microdomain after closure. This promotes burst-like reopenings by the rebinding of Ca(2+) to the activating site; whereas inhibitory actions are substantially independent of stationary buffers but are strongly dependent on the location of the inhibitory Ca(2+) binding site on the IP(3)R in relation to the channel pore.

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

Mode switching is the major mechanism of ligand regulation of InsP3 receptor calcium release channels

The inositol 1,4,5-trisphosphate (InsP₃) receptor (InsP₃R) plays a critical role in generation of complex Ca²⁺ signals in many cell types. In patch clamp recordings of isolated nuclei from insect Sf9 cells, InsP₃R channels were consistently detected with regulation by cytoplasmic InsP₃ and free Ca²⁺ concentrations ([Ca²⁺]_i) very similar to that observed for vertebrate InsP₃R. Long channel activity durations of the Sf9-InsP₃R have now enabled identification of a novel aspect of InsP₃R gating: modal gating. Using a novel algorithm to analyze channel modal gating kinetics, InsP₃R gating can be separated into three distinct modes: a low activity mode, a fast kinetic mode, and a burst mode with channel open probability (P_o) within each mode of 0.007 ± 0.002, 0.24 ± 0.03, and 0.85 ± 0.02, respectively. Channels reside in each mode for long periods (tens of opening and closing events), and transitions between modes can be discerned with high resolution (within two channel opening and closing events). Remarkably, regulation of channel gating by [Ca²⁺]_i and [InsP₃] does not substantially alter channel P_o within a mode. Instead, [Ca²⁺]_i and [InsP₃] affect overall channel P_o primarily by changing the relative probability of the channel being in each mode, especially the high and low P_o modes. This novel observation therefore reveals modal switching as the major mechanism of physiological regulation of InsP₃R channel activity, with implications for the kinetics of Ca²⁺ release events in cells.

Rapid ligand-regulated gating kinetics of single inositol 1,4,5-trisphosphate receptor Ca2+ release channels

The ubiquitous inositol 1,4,5-trisphosphate receptor (InsP(3)R) intracellular Ca(2+) release channel is engaged by thousands of plasma membrane receptors to generate Ca(2+) signals in all cells. Understanding how complex Ca(2+) signals are generated has been hindered by a lack of information on the kinetic responses of the channel to its primary ligands, InsP(3) and Ca(2+), which activate and inhibit channel gating. Here, we describe the kinetic responses of single InsP(3)R channels in native endoplasmic reticulum membrane to rapid ligand concentration changes with millisecond resolution, using a new patch-clamp configuration. The kinetics of channel activation and deactivation showed novel Ca(2+) regulation and unexpected ligand cooperativity. The kinetics of Ca(2+)-mediated channel inhibition showed the single-channel bases for fundamental Ca(2+) release events and Ca(2+) release refractory periods. These results provide new insights into the channel regulatory mechanisms that contribute to complex spatial and temporal features of intracellular Ca(2+) signals.

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.

Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.

Simple data-driven models of intracellular calcium dynamics with predictive power

Biology is complex. However, it is not clear how much of this complexity must necessarily translate into complicated mathematical models of biological processes. Simple models can be appealing to physicists but are usually deceiving for biologists. Complicated models, on the other hand, depend on too many parameters whose values are frequently unknown. Therefore, complicated models, although in principle more realistic, can lead to erroneous results if they are sensitive to these unknown parameter values. Intracellular calcium signals provide an example of utmost biological importance in which the issue of "simple vs complex" can be explored. In this paper we show that simple models describing the dynamics of intracellular calcium can be directly inferred from experimental data, without no a priori information on unknown parameters. A similar approach can be followed to study other reaction-diffusion systems. In spite of their simplicity, these models can provide quantitative information on some of the processes that shape calcium signals, such as the calcium current that underlies an experimental observation. This shows that simple models of biological systems are not limited to qualitative descriptions.

Analysis of puff dynamics in oocytes: interdependence of puff amplitude and interpuff interval

Puffs are localized Ca(2+) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP(3)). They are analogous to the sparks of myocytes and are believed to be the result of the liberation of Ca(2+) from the endoplasmic reticulum through the coordinated opening of IP(3) receptor/channels clustered at a functional release site. In this article, we analyze sequences of puffs that occur at the same site to help elucidate the mechanisms underlying puff dynamics. In particular, we show a dependence of the interpuff time on the amplitude of the preceding puff, and of the amplitude of the following puff on the preceding interval. These relationships can be accounted for by an inhibitory role of the Ca(2+) that is liberated during puffs. We construct a stochastic model for a cluster of IP(3) receptor/channels that quantitatively replicates the observed behavior, and we determine that the characteristic time for a channel to escape from the inhibitory state is of the order of seconds.

The effect of residual on the stochastic gating of -regulated channel models

Single-channel models of intracellular Ca(2+) channels such as the inositol 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by a constant background [Ca(2+)] as opposed to a dynamic [Ca(2+)] representing the formation and collapse of a localized Ca(2+) domain. This assumption neglects the fact that Ca(2+) released by open intracellular Ca(2+) channels may influence subsequent gating through the processes of Ca(2+)-activation or -inactivation. We study the effect of such "residual Ca(2+)" from previous channel opening on the stochastic gating of minimal and realistic single-channel models coupled to a restricted cytoplasmic compartment. Using Monte Carlo simulation as well as analytical and numerical solution of a system of advection-reaction equations for the probability density of the domain [Ca(2+)] conditioned on the state of the channel, we determine how the steady-state open probability (p(open)) of single-channel models of Ca(2+)-regulated Ca(2+) channels depends on the time constant for Ca(2+) domain formation and collapse. As expected, p(open) for a minimal model including Ca(2+) activation increases as the domain time constant becomes large compared to the open and closed dwell times of the channel, that is, on average the channel is activated by residual Ca(2+) from previous openings. Interestingly, p(open) for a channel model that is inactivated by Ca(2+) also increases as a function of the domain time constant when the maximum domain [Ca(2+)] is fixed, because slow formation of the Ca(2+) domain attenuates Ca(2+)-mediated inactivation. Conversely, when the source amplitude of the channel is fixed, increasing the domain time constant leads to elevated domain [Ca(2+)] and decreased open probability. Consistent with these observations, a realistic De Young-Keizer-like IP(3)R model responds to residual Ca(2+) with a steady-state open probability that is a monotonic function of the domain time constant, though minimal models that include both Ca(2+)-activation and -inactivation show more complex behavior. We show how the probability density approach described here can be generalized for arbitrarily complex channel models and for any value of the domain time constant. In addition, we present a comparatively simple numerical procedure for estimating p(open) for models of Ca(2+)-regulated Ca(2+) channels in the limit of a very fast or very slow Ca(2+) domain. When the ordinary differential equation for the [Ca(2+)] in a restricted cytoplasmic compartment is replaced by a partial differential equation for the buffered diffusion of intracellular Ca(2+) in a homogeneous isotropic cytosol, we find the dependence of p(open) on the buffer time constant is qualitatively similar to the above-mentioned results.

Gating mechanisms of the type-1 inositol trisphosphate receptor

A large amount of data and observations on inositol 1,4,5-trisphosphate (IP(3)) binding to the IP(3) receptor/Ca(2+) channel, the steady-state activity of the channel, and its inactivation by IP(3) can be explained by assuming one activation and one inhibition module, both allosterically operated by Ca(2+), IP(3), and ATP, and one adaptation element, driven by IP(3), Ca(2+), and the interconversion between two possible conformations of the receptor. The adaptation module becomes completely insensitive to a second IP(3) pulse within 80 s. Observed kinetic responses are well reproduced if, in addition, two module open states are rendered inactive by the current charge carrier Mn(2+). The inactivation time constants are 59 s in the activation, and 0.75 s in the adaptation module. The in vivo open probability of the channel is predicted to be almost in coincidence with the behavior in lipid bilayers for IP(3) levels of 0.2 and 2 microM and one-order-higher at 0.02 microM IP(3), whereas at 180 microM IP(3) the maximal in vivo activity may be 2.5-orders higher than in bilayers and restricted to a narrower Ca(2+) domain (approximately 10 microM-wide versus approximately 100 microM-wide). IP(3) is likely to inhibit channel activity at < or =120 nM Ca(2+) in vivo.

Using independent open-to-closed transitions to simplify aggregated Markov models of ion channel gating kinetics

Deducing plausible reaction schemes from single-channel current traces is time-consuming and difficult. The goal is to find the simplest scheme that fits the data, but there are many ways to connect even a small number of states (>2 million schemes with four open and four closed states). Many schemes make identical predictions. An exhaustive search over model space does not address the many equivalent schemes that will result. We have found a canonical form that can express all reaction schemes for binary channels. This form has the minimal number of rate constants for any rank (number of independent open-closed transitions), unlike other canonical forms such as the well established "uncoupled" scheme. Because all of the interconductance transitions in the new form are independent, we refer to it as the manifest interconductance rank (MIR) form. In the case of four open and four closed states, there are four MIR form schemes, corresponding to ranks 1-4. For many models proposed in the literature for specific ion channels, the equivalent MIR form has dramatically fewer links than the uncoupled form. By using the MIR form we prove that all rank 1 topologies with a given number of open and closed states make identical predictions in steady state, thus narrowing the search space for simple models. Moreover, we prove that fitting to canonical form preserves detailed balance. We also propose an efficient hierarchical algorithm for searching for the simplest possible model consistent with a given data set.

Modulation of mammalian inositol 1,4,5-trisphosphate receptor isoforms by calcium: a role of calcium sensor region

In the accompanying article, we compared main functional properties of the three mammalian inositol 1,4,5-trisphosphate receptors (InsP3R) isoforms. In this article we focused on modulation of mammalian InsP3R isoforms by cytosolic Ca2+. We found that: 1), when recorded in the presence of 2 microM InsP3 and 0.5 mM ATP all three mammalian InsP3R isoforms display bell-shaped Ca2+ dependence in physiological range of Ca2+ concentrations (pCa 8-5); 2), in the same experimental conditions InsP3R3 is most sensitive to modulation by Ca2+ (peak at 107 nM Ca2+), followed by InsP3R2 (peak at 154 nM Ca2+), and then by InsP3R1 (peak at 257 nM Ca2+); 3), increase in ATP concentration to 5 mM had no significant effect of Ca2+ dependence of InsP3R1 and InsP3R2; 4), increase in ATP concentration to 5 mM converted Ca2+ dependence of InsP3R3 from "narrow" shape to "square" shape; 5), ATP-induced change in the shape of InsP3R3 Ca2+ dependence was mainly due to an >200-fold reduction in the apparent affinity of the Ca2+-inhibitory site; 6), the apparent Ca2+ affinity of the Ca2+ sensor region (Cas) determined in biochemical experiments is equal to 0.23 microM Ca2+ for RT1-Cas, 0.16 microM Ca2+ for RT2-Cas, and 0.10 microM Ca2+ for RT3-Cas; and 7), Ca2+ sensitivity of InsP3R1 and InsP3R3 isoforms recorded in the presence of 2 microM InsP3 and 0.5 mM ATP or 2 microM InsP3 and 5 mM ATP can be exchanged by swapping their Cas regions. Obtained results provide novel information about functional properties of mammalian InsP3R isoforms and support the importance of the Ca2+ sensor region (Cas) in determining the sensitivity of InsP3R isoforms to modulation by Ca2+.