Optimal ion channel clustering for intracellular calcium signaling

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.

Synchronization and Enhanced Catalysis of Mechanically Coupled Enzymes

We examine the stochastic dynamics of two enzymes that are mechanically coupled to each other, e.g., through an elastic substrate or a fluid medium. The enzymes undergo conformational changes during their catalytic cycle, which itself is driven by stochastic steps along a biased chemical free energy landscape. We find conditions under which the enzymes can synchronize their catalytic steps, and discover that the coupling can lead to a significant enhancement in their overall catalytic rate. Both effects can be understood as arising from a global bifurcation in the underlying dynamical system at sufficiently strong coupling. Our findings suggest that, despite their molecular scale, enzymes can be cooperative and improve their performance in metabolic clusters.

Effects of temperature and ion channel blocks on propagation of action potential in myelinated axons

Potassium ion and sodium ion channels play important roles in the propagation of action potentials along a myelinated axon. The random opening and closing of ion channels can cause the fluctuation of action potentials. In this paper, an improved Hodgkin-Huxley chain network model is proposed to study the effects of ion channel blocks, temperature, and ion channel noise on the propagation of action potentials along the myelinated axon. It is found that the chain network has minimum coupling intensity threshold and maximum tolerance temperature threshold that allow the action potentials to pass along the whole axon, and the blockage of ion channels can change these two thresholds. A striking result is that the simulated value of the optimum membrane size (inversely proportional to noise intensity) coincides with the area range of feline thalamocortical relay cells in biological experiments.

Calcium-dependent insulin resistance in hepatocytes: mathematical model

Hepatocyte insulin resistance is one of the early factors of developing type II diabetes. If insulin resistance is treated early, type II diabetes could be prevented. In recent years, scientists have been conducting extensive research on the underlying issues on a cellular and molecular level. It was found that the modulation of IP₃-receptors, the mitochondrial ability to form the mitochondria-associated membranes (MAMs) and the endoplasmic reticulum stress during Ca²⁺ signaling play a key role in hepatocyte being able to maintain euglycemia and provide metabolic flexibility. However, researchers cannot agree on what factor is the key one in resulting in insulin resistance. In this work, we propose a mathematical model of Ca²⁺ signaling. We included in the model all the major contributors of a proper Ca²⁺ signaling during both the fasting and the postprandial state. Our modeling results are in good agreement with available experimental data. The analysis of modeling results suggests that MAMs dysfunction alone cannot result in abnormal Ca²⁺ signaling and the wrong modulation of IP₃-receptors is a more definite reason. However, both the MAMs dysfunction and the IP₃ signaling dysregulation combined can lead to a robust Ca²⁺ signal and improper glucose release. In addition, our model results suggest a strong dependence of Ca²⁺ oscillations pattern on morphological characteristics of the ER and the mitochondria.

Collective Dynamics of Ion Channels on Bilayer Lipid Membranes

Ion channels self-organize on cellular and organelle membranes as clusters and mutually modulate their gating behavior. It has been reported that the efficient information transfer is achieved by cooperative clustering of ion channels. To address the origin and nature of collective dynamics in ion channel clusters, a statistical mechanical model, namely, the Zimm-Bragg-type model in two dimensions with unequal weight distribution in channel-channel interactions, has been proposed. Nearest neighbor interaction along with next-nearest neighbor interaction has been considered, assuming symmetric spatial organization. The multichannel bilayer electrophysiology recordings of the voltage-dependent anion channel (VDAC) from rat brain mitochondria have been analyzed in order to test and further extend the model. The model successfully describes the multichannel gating behavior and self-organization of the VDAC cluster.

The Oscillation Amplitude, Not the Frequency of Cytosolic Calcium, Regulates Apoptosis Induction

Although a rising concentration of cytosolic Ca²⁺ has long been recognized as an essential signal for apoptosis, the dynamical mechanisms by which Ca²⁺ regulates apoptosis are not clear yet. To address this, we constructed a computational model that integrates known biochemical reactions and can reproduce the dynamical behaviors of Ca²⁺-induced apoptosis as observed in experiments. Model analysis shows that oscillating Ca²⁺ signals first convert into gradual signals and eventually transform into a switch-like apoptotic response. Via the two processes, the apoptotic signaling pathway filters the frequency of Ca²⁺ oscillations effectively but instead responds acutely to their amplitude. Collectively, our results suggest that Ca²⁺ regulates apoptosis mainly via oscillation amplitude, rather than frequency, modulation. This study not only provides a comprehensive understanding of how oscillatory Ca²⁺ dynamically regulates the complex apoptotic signaling network but also presents a typical example of how Ca²⁺ controls cellular responses through amplitude modulation.

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca²⁺) concentration in a cell in the present excitation may affect the Ca²⁺ dynamics in the next excitation via bi-directional coupling between voltage and Ca²⁺, forming a delayed feedback loop. Since the Ca²⁺ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca²⁺alternans in cardiac myocytes or spatially concordant or discordant Ca²⁺ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Understanding the mechanisms of pattern formation in biological systems is of great importance. Here we investigate the dynamical mechanisms by which delayed global feedback affects excitation pattern formation and stability in periodically-paced biological excitable media, such as cardiac or neural cells and tissue. We focus on the formation and stability of the temporal period-2 and spatially in-phase and out-of-phase excitation patterns. Using models of different levels of complexity, we show that when the delayed global feedback is negative, only the spatially in-phase patterns are stable. When the feedback is positive, both spatially in-phase and out-of-phase patterns are stable, and the out-of-phase patterns are spatially random but satisfy that the global signals are temporal period-1 solutions.

A Statistical View on Calcium Oscillations

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

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.

Spatial-temporal patterning of Ca2+ signals by the subcellular distribution of IP3 and IP3 receptors

The patterning of cytosolic Ca²⁺ signals in space and time underlies their ubiquitous ability to specifically regulate numerous cellular processes. Signals mediated by liberation of Ca²⁺ sequestered in the endoplasmic reticulum (ER) through inositol trisphosphate receptor (IP₃R) channels constitute a hierarchy of events; ranging from openings of individual IP₃ channels, through the concerted openings of several clustered IP₃Rs to generate local Ca²⁺ puffs, to global Ca²⁺ waves and oscillations that engulf the entire cell. Here, we review recent progress in elucidating how this hierarchy is shaped by an interplay between the functional gating properties of IP₃Rs and their spatial distribution within the cell. We focus in particular on the subset of IP₃Rs that are organized in stationary clusters and are endowed with the ability to preferentially liberate Ca²⁺.

Role of inositol 1,4,5-trisphosphate receptor type 1 in ATP-induced nuclear Ca2+ signal and hypertrophy in atrial myocytes

Inositol 1,4,5-trisphosphate receptor type 1 (IP₃R1) is expressed in atrial muscle, but not in ventricle, and they are abundant in the perinucleus. We investigated the role of IP₃R1 in the regulations of local Ca²⁺ signal and cell size in HL-1 atrial myocytes under stimulation by IP₃-generating chemical messenger, ATP. Assessment of nuclear and cytosolic Ca²⁺ signal using confocal Ca²⁺ imaging revealed that IP₃ generation by ATP (1 mM) induced monophasic nuclear Ca²⁺ increase, followed by cytosolic Ca²⁺ oscillation. Genetic knock-down (KD) of IP₃R1 eliminated the monophasic nuclear Ca²⁺ signal and slowed the cytosolic Ca²⁺ oscillation upon ATP exposure. Prolonged application of ATP as well as other known hypertrophic agonists (endothelin-1 and phenylephrine) increased cell size in wild-type cells, but not in IP₃R1 KD cells. Our data indicate that IP₃R1 mediates sustained elevation in nuclear Ca²⁺ level and facilitates cytosolic Ca²⁺ oscillation upon external ATP increase, and further suggests possible role of nuclear IP₃R1 in atrial hypertrophy.

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.

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.

CK2β regulates thrombopoiesis and Ca2+-triggered platelet activation in arterial thrombosis

CK2β is critically required for thrombopoiesis by regulating tubulin polymerization, MK fragmentation, and proplatelet formation. CK2β facilitates inositol triphosphate–mediated increase of cytosolic Ca2+ and is essential for platelet activation in arterial thrombosis in vivo.

Calcium waves in a grid of clustered channels with synchronous IP3 binding and unbinding

Calcium signals in cells occur at multiple spatial scales and variable temporal duration. However, a physical explanation for transitions between long-lasting global oscillations and localized short-term elevations (puffs) of cytoplasmic Ca²⁺ is still lacking. Here we introduce a phenomenological, coarse-grained model for the calcium variable, which is represented by ordinary differential equations. Due to its small number of parameters, and its simplicity, this model allows us to numerically study the interplay of multi-scale calcium concentrations with stochastic ion channel gating dynamics even in larger systems. We apply this model to a single cluster of inositol trisphosphate (IP ₃) receptor channels and find further evidence for the results presented in earlier work: a single cluster may be capable of producing different calcium release types, where long-lasting events are accompanied by unbinding of IP ₃ from the receptor (Rückl et al., PLoS Comput. Biol. 11, e1003965 (2015)). Finally, we show the practicability of the model in a grid of 64 clusters which is computationally intractable with previous high-resolution models. Here long-lasting events can lead to synchronized oscillations and waves, while short events stay localized. The frequency of calcium releases as well as their coherence can thereby be regulated by the amplitude of IP ₃ stimulation. Finally the model allows for a new explanation of oscillating [IP ₃], which is not based on metabolic production and degradation of IP ₃.

A Biochemical Oscillator Using Excitatory Molecules for Nanonetworks

For nanonetworks to be able to achieve large-scale functionality, such as to respond collectively to a trigger, synchrony between nanomachines is essential. However, to facilitate synchronization, some sort of physical clocking mechanism is required, such as the oscillators driven by auto-inhibitory molecules or by auto-inducing molecules. In this study, taking inspiration from the widely studied biological oscillatory phenomena called Calcium (Ca²⁺) oscillations, we undertake a different approach to design an oscillator. Our model employs three different types of excitatory molecules that work in tandem to generate oscillatory phenomenon in the concentration levels of the molecule of interest. The main objective of the study is to model a high frequency biochemical oscillator, along with the investigations to identify and determine the parameters that affect the period of the oscillations. The investigations entail and highlight the design of the reserve unit, a reservoir of the molecule of interest, as a key factor in realizing a high frequency stable biochemical oscillator.

Fluid limit theorems for stochastic hybrid systems with application to neuron models

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

Fluid limit theorems for stochastic hybrid systems with application to neuron models

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

A ROS-Assisted Calcium Wave Dependent on the AtRBOHD NADPH Oxidase and TPC1 Cation Channel Propagates the Systemic Response to Salt Stress

Plants exhibit rapid, systemic signaling systems that allow them to coordinate physiological and developmental responses throughout the plant body, even to highly localized and quickly changing environmental stresses. The propagation of these signals is thought to include processes ranging from electrical and hydraulic networks to waves of reactive oxygen species (ROS) and cytoplasmic Ca(2+) traveling throughout the plant. For the Ca(2+) wave system, the involvement of the vacuolar ion channel TWO PORE CHANNEL1 (TPC1) has been reported. However, the precise role of this channel and the mechanism of cell-to-cell propagation of the wave have remained largely undefined. Here, we use the fire-diffuse-fire model to analyze the behavior of a Ca(2+) wave originating from Ca(2+) release involving the TPC1 channel in Arabidopsis (Arabidopsis thaliana). We conclude that a Ca(2+) diffusion-dominated calcium-induced calcium-release mechanism is insufficient to explain the observed wave transmission speeds. The addition of a ROS-triggered element, however, is able to quantitatively reproduce the observed transmission characteristics. The treatment of roots with the ROS scavenger ascorbate and the NADPH oxidase inhibitor diphenyliodonium and analysis of Ca(2+) wave propagation in the Arabidopsis respiratory burst oxidase homolog D (AtrbohD) knockout background all led to reductions in Ca(2+) wave transmission speeds consistent with this model. Furthermore, imaging of extracellular ROS production revealed a systemic spread of ROS release that is dependent on both AtRBOHD and TPC1 These results suggest that, in the root, plant systemic signaling is supported by a ROS-assisted calcium-induced calcium-release mechanism intimately involving ROS production by AtRBOHD and Ca(2+) release dependent on the vacuolar channel TPC1.

Long-Lasting Sparks: Multi-Metastability and Release Competition in the Calcium Release Unit Network

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength.

Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Phase transitions and entropies for synchronizing oscillators

We study a generic model of coupled oscillators. In the model there is competition between phase synchronization and diffusive effects. For a model with a finite number of states we derive how a phase transition occurs when the coupling parameter is varied. The phase transition is characterized by a symmetry breaking and a discontinuity in the first derivative of the order parameter. We quantitatively account for how the synchronized pulse is a low-entropy structure that facilitates the production of more entropy by the system as a whole. For a model with many states we apply a continuum approximation and derive a potential Burgers' equation for a propagating pulse. No phase transition occurs in that case. However, positive entropy production by diffusive effects still exceeds negative entropy production by the shock formation.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Microtubule-Associated Protein EB3 Regulates IP3 Receptor Clustering and Ca(2+) Signaling in Endothelial Cells

The mechanisms by which the microtubule cytoskeleton regulates the permeability of endothelial barrier are not well understood. Here, we demonstrate that microtubule-associated end-binding protein 3 (EB3), a core component of the microtubule plus-end protein complex, binds to inositol 1,4,5-trisphosphate receptors (IP3Rs) through an S/TxIP EB-binding motif. In endothelial cells, α-thrombin, a pro-inflammatory mediator that stimulates phospholipase Cβ, increases the cytosolic Ca(2+) concentration and elicits clustering of IP3R3s. These responses, and the resulting Ca(2+)-dependent phosphorylation of myosin light chain, are prevented by depletion of either EB3 or mutation of the TxIP motif of IP3R3 responsible for mediating its binding to EB3. We also show that selective EB3 gene deletion in endothelial cells of mice abrogates α-thrombin-induced increase in endothelial permeability. We conclude that the EB3-mediated interaction of IP3Rs with microtubules controls the assembly of IP3Rs into effective Ca(2+) signaling clusters, which thereby regulate microtubule-dependent endothelial permeability.

Resonance induced by a spatially periodic force in the reaction-diffusion system

The stimulus-dynamic response is an important topic in physics. In this work, we study the dynamics in the reaction-diffusion system subjected to a weak signal and a spatially periodic force. We find that the response of the system to the weak signal is enhanced largely by the spatially periodic force, which is termed spatially periodic-force-induced resonance. In particular, the response becomes stronger when the spatial frequency is chosen such that the system synchronizes with spatially periodic force. This combinative behavior, i.e., the spatially periodic-force-induced resonance and the spatial-synchronization-enhanced resonance, is of great interest and may shed light on our understanding of the dynamics of nonlinear systems subjected to spatially periodic force in responding to a weak signal.

Single-molecule tracking of inositol trisphosphate receptors reveals different motilities and distributions

Puffs are local Ca(2+) signals that arise by Ca(2+) liberation from the endoplasmic reticulum through the concerted opening of tightly clustered inositol trisphosphate receptors/channels (IP3Rs). The locations of puff sites observed by Ca(2+) imaging remain static over several minutes, whereas fluorescence recovery after photobleaching (FRAP) experiments employing overexpression of fluorescently tagged IP3Rs have shown that the majority of IP3Rs are freely motile. To address this discrepancy, we applied single-molecule imaging to locate and track type 1 IP3Rs tagged with a photoswitchable fluorescent protein and expressed in COS-7 cells. We found that ∼ 70% of the IP3R1 molecules were freely motile, undergoing random walk motility with an apparent diffusion coefficient of ∼ 0.095 μm s(-1), whereas the remaining molecules were essentially immotile. A fraction of the immotile IP3Rs were organized in clusters, with dimensions (a few hundred nanometers across) comparable to those previously estimated for the IP3R clusters underlying functional puff sites. No short-term (seconds) changes in overall motility or in clustering of immotile IP3Rs were apparent following activation of IP3/Ca(2+) signaling. We conclude that stable clusters of small numbers of immotile IP3Rs may underlie local Ca(2+) release sites, whereas the more numerous motile IP3Rs appear to be functionally silent.

The role of IP3 receptor channel clustering in Ca2+ wave propagation during oocyte maturation

During oocyte maturation, the calcium-signaling machinery undergoes a dramatic remodeling resulting in distinctly different calcium-release patterns on all organizational scales from puffs to waves. The dynamics of the Ca(2+) release wave in mature as compared to immature oocytes are defined by a slower propagation speed and longer duration of the high Ca(2+) plateau. In this chapter, we use computational modeling to identify the changes in the signaling machinery, which contribute most significantly to the alterations observed in Ca(2+) wave propagation during Xenopus oocyte maturation. In addition to loss of store-operated calcium entry and internalization of plasma membrane pumps, we propose that spatial reorganization of the IP3 receptors in the plane of the ER membrane is a key factor for the observed signaling changes in Ca(2+) wave propagation.

The influence of Ca²⁺ buffers on free [Ca²⁺] fluctuations and the effective volume of Ca²⁺ microdomains

Intracellular calcium (Ca(2+)) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca(2+) binding proteins (Ca(2+) buffers) play an important role in regulating Ca(2+) concentration ([Ca(2+)]). Buffers typically slow [Ca(2+)] temporal dynamics and increase the effective volume of Ca(2+) domains. Because fluctuations in [Ca(2+)] decrease in proportion to the square-root of a domain's physical volume, one might conjecture that buffers decrease [Ca(2+)] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca(2+) signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca(2+) influx with passive exchange of both Ca(2+) and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca(2+) and buffer and use these stochastic differential equations to determine the magnitude of [Ca(2+)] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca(2+) signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca(2+)] fluctuations during periods of Ca(2+) influx, whereas stationary (immobile) Ca(2+) buffers do not. Also contrary to expectations, we find that in the absence of Ca(2+) influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca(2+)] fluctuations. We derive an analytical formula describing the influence of rapid Ca(2+) buffers on [Ca(2+)] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca(2+) buffers alter the dynamics of [Ca(2+)] fluctuations in a nonintuitive manner. The finding that Ca(2+) buffers do not suppress intrinsic domain [Ca(2+)] fluctuations raises the intriguing question of whether or not [Ca(2+)] fluctuations are a physiologically significant aspect of local Ca(2+) signaling.

Optimal microdomain crosstalk between endoplasmic reticulum and mitochondria for Ca2+ oscillations

A Ca²⁺ signaling model is proposed to consider the crosstalk of Ca²⁺ ions between endoplasmic reticulum (ER) and mitochondria within microdomains around inositol 1, 4, 5-trisphosphate receptors (IP₃R) and the mitochondrial Ca²⁺ uniporter (MCU). Our model predicts that there is a critical IP₃R-MCU distance at which 50% of the ER-released Ca²⁺ is taken up by mitochondria and that mitochondria modulate Ca²⁺ signals differently when outside of this critical distance. This study highlights the importance of the IP₃R-MCU distance on Ca²⁺ signaling dynamics. The model predicts that when MCU are too closely associated with IP₃Rs, the enhanced mitochondrial Ca²⁺ uptake will produce an increase of cytosolic Ca²⁺ spike amplitude. Notably, the model demonstrates the existence of an optimal IP₃R-MCU distance (30–85 nm) for effective Ca²⁺ transfer and the successful generation of Ca²⁺ signals in healthy cells. We suggest that the space between the inner and outer mitochondria membranes provides a defense mechanism against occurrences of high [Ca²⁺]_Cyt. Our results also hint at a possible pathological mechanism in which abnormally high [Ca²⁺]_Cyt arises when the IP₃R-MCU distance is in excess of the optimal range.

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Effect of internal noise on the oscillation of N2O decomposition over Cu-ZSM-5 zeolites using a stochastic description

When considering stochastic oscillations of heterogeneous catalyst systems, most researches have focused on the surface of a metal or its oxide catalysts, but there have been few studies on porous catalysts. In this work, the effects of internal noise on oscillations of N2O decomposition over Cu-ZSM-5 zeolites are investigated, using the chemical Langevin equation and a mesoscopic stochastic model. Considering that Cu-ZSM-5 particles are finely divided particles, the number of Cu ions (Ns) is proportional to the particle size at a certain Cu/Al, and the internal noise is inversely proportional to Ns. Stochastic oscillations can be observed outside the deterministic oscillatory region. Furthermore, the performance of the oscillation characterized by the signal-to-noise ratio has a maximum within the optimal size range of 4-8 nm. This suggests that a nanometer-sized zeolite may be best for oscillations.

Intracellular calcium channels: inositol-1,4,5-trisphosphate receptors

The inositol-1,4,5-trisphosphate receptors (InsP3Rs) are the major intracellular Ca(2+)-release channels in cells. Activity of InsP3Rs is essential for elementary and global Ca(2+) events in the cell. There are three InsP3Rs isoforms that are present in mammalian cells. In this review we will focus primarily on InsP3R type 1. The InsP3R1 is a predominant isoform in neurons and it is the most extensively studied isoform. Combination of biophysical and structural methods revealed key mechanisms of InsP3R function and modulation. Cell biological and biochemical studies lead to identification of a large number of InsP3R-binding proteins. InsP3Rs are involved in the regulation of numerous physiological processes, including learning and memory, proliferation, differentiation, development and cell death. Malfunction of InsP3R1 play a role in a number of neurodegenerative disorders and other disease states. InsP3Rs represent a potentially valuable drug target for treatment of these disorders and for modulating activity of neurons and other cells. Future studies will provide better understanding of physiological functions of InsP3Rs in health and disease.

Towards the Physics of Calcium Signalling in Plants

Calcium is an abundant element with a wide variety of important roles within cells. Calcium ions are inter- and intra-cellular messengers that are involved in numerous signalling pathways. Fluctuating compartment-specific calcium ion concentrations can lead to localised and even plant-wide oscillations that can regulate downstream events. Understanding the mechanisms that give rise to these complex patterns that vary both in space and time can be challenging, even in cases for which individual components have been identified. Taking a systems biology approach, mathematical and computational techniques can be employed to produce models that recapitulate experimental observations and capture our current understanding of the system. Useful models make novel predictions that can be investigated and falsified experimentally. This review brings together recent work on the modelling of calcium signalling in plants, from the scale of ion channels through to plant-wide responses to external stimuli. Some in silico results that have informed later experiments are highlighted.

The emergence of subcellular pacemaker sites for calcium waves and oscillations

Calcium (Ca(2+)) waves generating oscillatory Ca(2+) signals are widely observed in biological cells. Experimental studies have shown that under certain conditions, initiation of Ca(2+) waves is random in space and time, while under other conditions, waves occur repetitively from preferred locations (pacemaker sites) from which they entrain the whole cell. In this study, we use computer simulations to investigate the self-organization of Ca(2+) sparks into pacemaker sites generating Ca(2+) oscillations. In both ventricular myocyte experiments and computer simulations of a heterogeneous Ca(2+) release unit (CRU) network model, we show that Ca(2+) waves occur randomly in space and time when the Ca(2+) level is low, but as the Ca(2+) level increases, waves occur repetitively from the same sites. Our analysis indicates that this transition to entrainment can be attributed to the fact that random Ca(2+) sparks self-organize into Ca(2+) oscillations differently at low and high Ca(2+) levels. At low Ca(2+), the whole cell Ca(2+) oscillation frequency of the coupled CRU system is much slower than that of an isolated single CRU. Compared to a single CRU, the distribution of interspike intervals (ISIs) of the coupled CRU network exhibits a greater variation, and its ISI distribution is asymmetric with respect to the peak, exhibiting a fat tail. At high Ca(2+), however, the coupled CRU network has a faster frequency and lesser ISI variation compared to an individual CRU. The ISI distribution of the coupled network no longer exhibits a fat tail and is well-approximated by a Gaussian distribution. This same Ca(2+) oscillation behaviour can also be achieved by varying the number of ryanodine receptors per CRU or the distance between CRUs. Using these results, we develop a theory for the entrainment of random oscillators which provides a unified explanation for the experimental observations underlying the emergence of pacemaker sites and Ca(2+) oscillations.

Maternal diabetes causes abnormal dynamic changes of endoplasmic reticulum during mouse oocyte maturation and early embryo development

BACKGROUND

The adverse effects of maternal diabetes on oocyte maturation and embryo development have been reported.

METHODS

In this study, we used time-lapse live cell imaging confocal microscopy to investigate the dynamic changes of ER and the effects of diabetes on the ER's structural dynamics during oocyte maturation, fertilization and early embryo development.

RESULTS

We report that the ER first became remodeled into a dense ring around the developing MI spindle, and then surrounded the spindle during migration to the cortex. ER reorganization during mouse early embryo development was characterized by striking localization around the pronuclei in the equatorial section, in addition to larger areas of fluorescence deeper within the cytoplasm. In contrast, in diabetic mice, the ER displayed a significantly higher percentage of homogeneous distribution patterns throughout the entire ooplasm during oocyte maturation and early embryo development. In addition, a higher frequency of large ER aggregations was detected in GV oocytes and two cell embryos from diabetic mice.

CONCLUSIONS

These results suggest that the diabetic condition adversely affects the ER distribution pattern during mouse oocyte maturation and early embryo development.

Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

The probability of triggering calcium puffs is linearly related to the number of inositol trisphosphate receptors in a cluster

Puffs are local Ca(2+) signals that arise by Ca(2+) liberation from the endoplasmic reticulum through concerted opening of tightly clustered inositol trisphosphate receptor/channels (IP(3)R). They serve both local signaling functions and trigger global Ca(2+) waves. The numbers of functional IP(3)R within clusters differ appreciably between different puff sites, and we investigated how the probability of puff occurrence varies with cluster size. We imaged puffs in SH-SY5Y cells using total internal fluorescence microscopy, and estimated cluster sizes from the magnitude of the largest puff observed at each site relative to the signal from a single channel. We find that the initial triggering rate of puffs following photorelease of IP(3), and the average frequency of subsequent repetitive puffs, vary about linearly with cluster size. These data accord well with stochastic simulations in which opening of any individual IP(3)R channel within a cluster triggers a puff via Ca(2+)-induced Ca(2+) release. An important consequence is that the signaling power of a puff site (average amount of Ca(2+) released per puff × puff frequency) varies about the square of cluster size, implying that large clusters contribute disproportionately to cellular signaling and, because of their higher puff frequency, preferentially act as pacemakers to initiate Ca(2+) waves.

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.

Influence fields: a quantitative framework for representation and analysis of active dendrites

Neuronal dendrites express numerous voltage-gated ion channels (VGICs), typically with spatial gradients in their densities and properties. Dendritic VGICs, their gradients, and their plasticity endow neurons with information processing capabilities that are higher than those of neurons with passive dendrites. Despite this, frameworks that incorporate dendritic VGICs and their plasticity into neurophysiological and learning theory models have been far and few. Here, we develop a generalized quantitative framework to analyze the extent of influence of a spatially localized VGIC conductance on different physiological properties along the entire stretch of a neuron. Employing this framework, we show that the extent of influence of a VGIC conductance is largely independent of the conductance magnitude but is heavily dependent on the specific physiological property and background conductances. Morphologically, our analyses demonstrate that the influences of different VGIC conductances located on an oblique dendrite are confined within that oblique dendrite, thus providing further credence to the postulate that dendritic branches act as independent computational units. Furthermore, distinguishing between active and passive propagation of signals within a neuron, we demonstrate that the influence of a VGIC conductance is spatially confined only when propagation is active. Finally, we reconstruct functional gradients from VGIC conductance gradients using influence fields and demonstrate that the cumulative contribution of VGIC conductances in adjacent compartments plays a critical role in determining physiological properties at a given location. We suggest that our framework provides a quantitative basis for unraveling the roles of dendritic VGICs and their plasticity in neural coding, learning, and homeostasis.

P2X7 receptor-mediated calcium dynamics in HEK293 cells: experimental characterization and modelling approach

The P2X7 receptor (P2X7R) induces ionotropic Ca²⁺ signalling in different cell types. It plays an important role in the immune response and in the nervous system. Here, the mechanisms underlying intracellular Ca²⁺ variations evoked by 3'-O-(4-benzoyl)benzoyl-ATP (BzATP), a potent agonist of the P2X7R, in transfected HEK293 cells, are investigated both experimentally and theoretically. We propose a minimal model of P2X7R that is capable of reproducing, qualitatively and quantitatively, the experimental data. This approach was also adopted for the P2X7R variant, which lacks the entire C-terminus tail (trP2X7R). Then we introduce a biophysical model describing the Ca²⁺ dynamics in HEK293. Our model gives an account of the ionotropic Ca²⁺ influx evoked by BzATP on the basis of the kinetics model of P2X7R. To explain the complex Ca²⁺ responses evoked by BzATP, the model predicted that an impairment in Ca²⁺ extrusion flux through the plasma membrane is a key factor for Ca²⁺ homeostasis in HEK293 cells.

Quantifying the uncertainty of spontaneous Ca2+ oscillations in astrocytes: particulars of Alzheimer's disease

The quantification of spontaneous calcium (Ca(2+)) oscillations (SCOs) in astrocytes presents a challenge because of the large irregularities in the amplitudes, durations, and initiation times of the underlying events. In this article, we use a stochastic context to account for such SCO variability, which is based on previous models for cellular Ca(2+) signaling. First, we found that passive Ca(2+) influx from the extracellular space determine the basal concentration of this ion in the cytosol. Second, we demonstrated the feasibility of estimating both the inositol 1,4,5-trisphosphate (IP(3)) production levels and the average number of IP(3) receptor channels in the somatic clusters from epifluorescent Ca(2+) imaging through the combination of a filtering strategy and a maximum-likelihood criterion. We estimated these two biophysical parameters using data from wild-type adult mice and age-matched transgenic mice overexpressing the 695-amino-acid isoform of human Alzheimer β-amyloid precursor protein. We found that, together with an increase in the passive Ca(2+) influx, a significant reduction in the sensitivity of G protein-coupled receptors might lie beneath the abnormalities in the astrocytic Ca(2+) signaling, as was observed in rodent models of Alzheimer's disease. This study provides new, to our knowledge, indices for a quantitative analysis of SCOs in normal and pathological astrocytes.

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.

Inositol 1,4,5-trisphosphate receptor subtype-specific regulation of calcium oscillations

Oscillatory fluctuations in the cytosolic concentration of free calcium ions (Ca²⁺) are considered a ubiquitous mechanism for controlling multiple cellular processes. Inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃R) are intracellular Ca²⁺ release channels that mediate Ca²⁺ release from endoplasmic reticulum (ER) Ca²⁺ stores. The three IP₃R subtypes described so far exhibit differential structural, biophysical, and biochemical properties. Subtype specific regulation of IP₃R by the endogenous modulators IP₃, Ca²⁺, protein kinases and associated proteins have been thoroughly examined. In this article we will review the contribution of each IP₃R subtype in shaping cytosolic Ca²⁺ oscillations.

Inositol (1,4,5)-trisphosphate receptor microarchitecture shapes Ca2+ puff kinetics

Inositol (1,4,5)-trisphosphate receptors (IP(3)Rs) release intracellular Ca(2+) as localized Ca(2+) signals (Ca(2+) puffs) that represent the activity of small numbers of clustered IP(3)Rs spaced throughout the endoplasmic reticulum. Although much emphasis has been placed on estimating the number of active Ca(2+) release channels supporting Ca(2+) puffs, less attention has been placed on understanding the role of cluster microarchitecture. This is important as recent data underscores the dynamic nature of IP(3)R transitions between heterogeneous cellular architectures and the differential behavior of IP(3)Rs socialized into clusters. Here, we applied a high-resolution model incorporating stochastically gating IP(3)Rs within a three-dimensional cytoplasmic space to demonstrate: 1), Ca(2+) puffs are supported by a broad range of clustered IP(3)R microarchitectures; 2), cluster ultrastructure shapes Ca(2+) puff characteristics; and 3), loosely corralled IP(3)R clusters (>200 nm interchannel separation) fail to coordinate Ca(2+) puffs, owing to inefficient triggering and impaired coupling due to reduced Ca(2+)-induced Ca(2+) release microwave velocity (<10 nm/s) throughout the channel array. Dynamic microarchitectural considerations may therefore influence Ca(2+) puff occurrence/properties in intact cells, contrasting with a more minimal role for channel number over the same simulated conditions in shaping local Ca(2+) dynamics.

Reduction of calcium release site models via moment fitting of phase-type distributions

Models of calcium (Ca(2 +)) release sites derived from continuous-time Markov chain (CTMC) models of intracellular Ca(2 +) channels exhibit collective gating reminiscent of the experimentally observed phenomenon of Ca(2 +) puffs and sparks. In order to overcome the state-space explosion that occurs in compositionally defined Ca(2 +) release site models, we have implemented an automated procedure for model reduction that replaces aggregated states of the full release site model with much simpler CTMCs that have similar within-group phase-type sojourn times and inter-group transitions. Error analysis based on comparison of full and reduced models validates the method when applied to release site models composed of 20 three-state channels that are both activated and inactivated by Ca(2 +). Although inspired by existing techniques for fitting moments of phase-type distributions, the automated reduction method for compositional Ca(2 +) release site models is unique in several respects and novel in this biophysical context.