An investigation of models of the IP3R channel in Xenopus oocyte

Wave failure at strong coupling in intracellular Ca^{2+} signaling system with clustered channels

As an important intracellular signal, Ca^{2+} ions control diverse cellular functions. In this paper, we discuss the Ca^{2+} signaling with a two-dimensional model in which the inositol 1,4,5-trisphosphate (IP_{3}) receptor channels are distributed in clusters on the endoplasmic reticulum membrane. The wave failure at large Ca^{2+} diffusion coupling is discussed in detail in the model. We show that with varying model parameters the wave failure is a robust behavior with either deterministic or stochastic channel dynamics. We suggest that the wave failure should be a general behavior in inhomogeneous diffusing systems with clustered excitable regions and may occur in biological Ca^{2+} signaling systems.

Single Ca2+ transients vs oscillatory Ca2+ signaling for assisted oocyte activation: limitations and benefits

Oocyte activation is a calcium (Ca²⁺)-dependent process that has been investigated in depth, in particular, regarding its impact on assisted reproduction technology (ART). Following a standard model of signal transduction, Ca²⁺ drives the meiotic progression upon fertilization in all species studied to date. However, Ca²⁺ changes during oocyte activation are species specific, and they can be classified in two modalities based on the pattern defined by the Ca²⁺ signature: a single Ca²⁺ transient (e.g. amphibians) or repetitive Ca²⁺ transients called Ca²⁺ oscillations (e.g. mammals). Interestingly, assisted oocyte activation (AOA) methods have highlighted the ability of mammalian oocytes to respond to single Ca²⁺ transients with normal embryonic development. In this regard, there is evidence supporting that cellular events during the process of oocyte activation are initiated by different number of Ca²⁺ oscillations. Moreover, it was proposed that oocyte activation and subsequent embryonic development are dependent on the total summation of the Ca²⁺ peaks, rather than to a specific frequency pattern of Ca²⁺ oscillations. The present review aims to demonstrate the complexity of mammalian oocyte activation by describing the series of Ca²⁺-linked physiological events involved in mediating the egg-to-embryo transition. Furthermore, mechanisms of AOA and the limitations and benefits associated with the application of different activation agents are discussed.

Importance of Altered Levels of SERCA, IP3R, and RyR in Vascular Smooth Muscle Cell

Calcium cycling between the sarcoplasmic reticulum (SR) and the cytosol via the sarco-/endoplasmic reticulum Ca-ATPase (SERCA) pump, inositol-1,4,5-triphosphate receptor (IP₃R), and Ryanodine receptor (RyR), plays a major role in agonist-induced intracellular calcium ([Ca²⁺]_cyt) dynamics in vascular smooth muscle cells (VSMC). Levels of these calcium handling proteins in SR get altered under disease conditions. We have developed a mathematical model to understand the significance of altered levels of SERCA, IP₃R, and RyR on the intracellular calcium dynamics of VSMC and to understand how variation in protein levels that arise due to diabetes contribute to different VSMC behavior and thus vascular disease. SR is modeled as a single continuous entity with homogeneous intra-SR calcium. Model results show that agonist-induced intracellular calcium dynamics can be modified by changing the levels of SERCA, IP₃R, and/or RyR. Lowering SERCA level will enable intracellular calcium oscillations at low agonist concentrations whereas lowered levels of IP₃R and RyR need higher agonist concentration for intracellular calcium oscillations. This research suggests that reduced SERCA level is the main factor responsible for the reduced intracellular calcium transients and contractility in VSMCs.

Comparison of gating dynamics of different IP3R channels with immune algorithm searching for channel parameter distributions

The gating properties of the inositol 1, 4, 5-trisphosphate (IP₃) receptor (IP₃R) are determined by the binding and unbinding capability of Ca²⁺ ions and IP₃ messengers. With the patch clamp experiments, the stationary properties have been discussed for Xenopus oocyte type-1 IP₃R (Oo-IP₃R1), type-3 IP₃R (Oo-IP₃R3) and Spodoptera frugiperda IP₃R (Sf-IP₃R). In this paper, in order to provide insights about the relation between the observed gating characteristics and the gating parameters in different IP₃Rs, we apply the immune algorithm to fit the parameters of a modified DeYoung-Keizer model. By comparing the fitting parameter distributions of three IP₃Rs, we suggest that the three types of IP₃Rs have the similar open sensitivity in responding to IP₃. The Oo-IP₃R3 channel is easy to open in responding to low Ca²⁺ concentration, while Sf-IP₃R channel is easily inhibited in responding to high Ca²⁺ concentration. We also show that the IP₃ binding rate is not a sensitive parameter for stationary gating dynamics for three IP₃Rs, but the inhibitory Ca²⁺ binding/unbinding rates are sensitive parameters for gating dynamics for both Oo-IP₃R1 and Oo-IP₃R3 channels. Such differences may be important in generating the spatially and temporally complex Ca²⁺ oscillations in cells. Our study also demonstrates that the immune algorithm can be applied for model parameter searching in biological systems.

Emergence of ion channel modal gating from independent subunit kinetics

Many ion channels exhibit a slow stochastic switching between distinct modes of gating activity. This feature of channel behavior has pronounced implications for the dynamics of ionic currents and the signaling pathways that they regulate. A canonical example is the inositol 1,4,5-trisphosphate receptor (IP3R) channel, whose regulation of intracellular Ca(2+) concentration is essential for numerous cellular processes. However, the underlying biophysical mechanisms that give rise to modal gating in this and most other channels remain unknown. Although ion channels are composed of protein subunits, previous mathematical models of modal gating are coarse grained at the level of whole-channel states, limiting further dialogue between theory and experiment. Here we propose an origin for modal gating, by modeling the kinetics of ligand binding and conformational change in the IP3R at the subunit level. We find good agreement with experimental data over a wide range of ligand concentrations, accounting for equilibrium channel properties, transient responses to changing ligand conditions, and modal gating statistics. We show how this can be understood within a simple analytical framework and confirm our results with stochastic simulations. The model assumes that channel subunits are independent, demonstrating that cooperative binding or concerted conformational changes are not required for modal gating. Moreover, the model embodies a generally applicable principle: If a timescale separation exists in the kinetics of individual subunits, then modal gating can arise as an emergent property of channel behavior.

Frequency and relative prevalence of calcium blips and puffs in a model of small IP₃R clusters

In this work, we model the local calcium release from clusters with a few inositol 1,4,5-trisphosphate receptor (IP3R) channels, focusing on the stochastic process in which an open channel either triggers other channels to open (as a puff) or fails to cause any channel to open (as a blip). We show that there are linear relations for the interevent interval (including blips and puffs) and the first event latency against the inverse cluster size. However, nonlinearity is found for the interpuff interval and the first puff latency against the inverse cluster size. Furthermore, the simulations indicate that the blip fraction among all release events and the blip frequency are increasing with larger basal [Ca(2+)], with blips in turn giving a growing contribution to basal [Ca(2+)]. This result suggests that blips are not just lapses to trigger puffs, but they may also possess a biological function to contribute to the initiation of calcium waves by a preceding increase of basal [Ca(2+)] in cells that have small IP3R clusters.

Modulation of elementary calcium release mediates a transition from puffs to waves in an IP3R cluster model

The oscillating concentration of intracellular calcium is one of the most important examples for collective dynamics in cell biology. Localized releases of calcium through clusters of inositol 1,4,5-trisphosphate receptor channels constitute elementary signals called calcium puffs. Coupling by diffusing calcium leads to global releases and waves, but the exact mechanism of inter-cluster coupling and triggering of waves is unknown. To elucidate the relation of puffs and waves, we here model a cluster of IP3R channels using a gating scheme with variable non-equilibrium IP3 binding. Hybrid stochastic and deterministic simulations show that puffs are not stereotyped events of constant duration but are sensitive to stimulation strength and residual calcium. For increasing IP3 concentration, the release events become modulated at a timescale of minutes, with repetitive wave-like releases interspersed with several puffs. This modulation is consistent with experimental observations we present, including refractoriness and increase of puff frequency during the inter-wave interval. Our results suggest that waves are established by a random but time-modulated appearance of sustained release events, which have a high potential to trigger and synchronize activity throughout the cell.

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.

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.

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.

Termination of Ca²+ release for clustered IP₃R channels

In many cell types, release of calcium ions is controlled by inositol 1,4,5-trisphosphate () receptor channels. Elevations in concentration after intracellular release through receptors (R) can either propagate in the form of waves spreading through the entire cell or produce spatially localized puffs. The appearance of waves and puffs is thought to implicate random initial openings of one or a few channels and subsequent activation of neighboring channels because of an “autocatalytic” feedback. It is much less clear, however, what determines the further time course of release, particularly since the lifetime is very different for waves (several seconds) and puffs (around 100 ms). Here we study the lifetime of signals and their dependence on residual microdomains. Our general idea is that microdomains are dynamical and mediate the effect of other physiological processes. Specifically, we focus on the mechanism by which binding proteins (buffers) alter the lifetime of signals. We use stochastic simulations of channel gating coupled to a coarse-grained description for the concentration. To describe the concentration in a phenomenological way, we here introduce a differential equation, which reflects the buffer characteristics by a few effective parameters. This non-stationary model for microdomains gives deep insight into the dynamical differences between puffs and waves. It provides a novel explanation for the different lifetimes of puffs and waves and suggests that puffs are terminated by inhibition while unbinding is responsible for termination of waves. Thus our analysis hints at an additional role of and shows how cells can make use of the full complexity in R gating behavior to achieve different signals.

Calcium signals are important for a host of cellular processes such as neurotransmitter release, cell contraction and gene expression. While the principles of activation and spreading of calcium signals have been largely understood, it is much less clear how their spatio-temporal appearance is shaped. This issue is of high relevance since the spatio-temporal signature is thought to carry the information content. In our paper we study the dynamical mechanisms that determine the time course of calcium release from receptor channels. We use a stochastic channel description combined with a recently developed model for the distribution of released calcium in a microdomain. The simulations uncover a complex control mechanism, which allows for the tuning of release from short frequent puffs to extended and less frequent wave-like release. Unexpectedly, the model predicts that for wave-like release the dissociation of from the receptors leads to termination of the calcium signal. This effect relies on a well-known gating property of R channels, which earlier has been regarded as superfluous in studies for groups of channels. Our results also provide a missing link to understand cellular response to calcium-binding proteins and present a novel mechanism for information processing by R channels.

Calcium oscillations

Calcium signaling results from a complex interplay between activation and inactivation of intracellular and extracellular calcium permeable channels. This complexity is obvious from the pattern of calcium signals observed with modest, physiological concentrations of calcium-mobilizing agonists, which typically present as sequential regenerative discharges of stored calcium, a process referred to as calcium oscillations. In this review, we discuss recent advances in understanding the underlying mechanism of calcium oscillations through the power of mathematical modeling. We also summarize recent findings on the role of calcium entry through store-operated channels in sustaining calcium oscillations and in the mechanism by which calcium oscillations couple to downstream effectors.

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.

Calcium domains around single and clustered IP3 receptors and their modulation by buffers

We study Ca(2+) release through single and clustered IP(3) receptor channels on the ER membrane under presence of buffer proteins. Our computational scheme couples reaction-diffusion equations and a Markovian channel model and allows our investigating the effects of buffer proteins on local calcium concentrations and channel gating. We find transient and stationary elevations of calcium concentrations around active channels and show how they determine release amplitude. Transient calcium domains occur after closing of isolated channels and constitute an important part of the channel's feedback. They cause repeated openings (bursts) and mediate increased release due to Ca(2+) buffering by immobile proteins. Stationary domains occur during prolonged activity of clustered channels, where the spatial proximity of IP(3)Rs produces a distinct [Ca(2+)] scale (0.5-10 microM), which is smaller than channel pore concentrations (>100 microM) but larger than transient levels. While immobile buffer affects transient levels only, mobile buffers in general reduce both transient and stationary domains, giving rise to Ca(2+) evacuation and biphasic modulation of release amplitude. Our findings explain recent experiments in oocytes and provide a general framework for the understanding of calcium signals.

Law of mass action, detailed balance, and the modeling of calcium puffs

Using deterministic-stochastic simulations we show that for intracellular calcium puffs the mixing assumption for reactants does not hold within clusters of receptor channels. Consequently, the law of mass action does not apply and useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels, thus eliminating detailed balance in the reaction scheme. A minimal Markovian model can be inferred, describing well calcium puffs in neuronal cells and allowing insight into the functioning of calcium puffs.

Introduction to focus issue: intracellular Ca2+ dynamics--a change of modeling paradigm?

Intracellular Ca(2+) concentration dynamics have been perceived as a prototypical deterministic intracellular reaction-diffusion system in biophysics for many years. Recent experimental findings challenge that view and suggest them to be fluctuation driven. That renders this system interesting for nonlinear physics, in general, since we can study the emergence of macroscopic behavior from mesoscopic dynamics. In particular, we can observe the random elemental events, called puffs, and the macroscopic pattern with the same experimental means. Here, we give a short introduction to the current discussion on theoretical and modeling concepts, and this Focus Issue reflecting it.