Release currents of IP(3) receptor channel clusters and concentration profiles

Luminal transport through intact endoplasmic reticulum limits the magnitude of localized Ca2+ signals

The endoplasmic reticulum (ER) forms an interconnected network of tubules stretching throughout the cell. Understanding how ER functionality relies on its structural organization is crucial for elucidating cellular vulnerability to ER perturbations, which have been implicated in several neuronal pathologies. One of the key functions of the ER is enabling Ca[Formula: see text] signaling by storing large quantities of this ion and releasing it into the cytoplasm in a spatiotemporally controlled manner. Through a combination of physical modeling and live-cell imaging, we demonstrate that alterations in ER shape significantly impact its ability to support efficient local Ca[Formula: see text] releases, due to hindered transport of luminal content within the ER. Our model reveals that rapid Ca[Formula: see text] release necessitates mobile luminal buffer proteins with moderate binding strength, moving through a well-connected network of ER tubules. These findings provide insight into the functional advantages of normal ER architecture, emphasizing its importance as a kinetically efficient intracellular Ca[Formula: see text] delivery system.

An integrate-and-fire approach to Ca2+ signaling. Part II: Cumulative refractoriness

Inositol 1,4,5-trisphosphate-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. The agonist concentration stimulating Ca2+ signals is encoded in the frequency of a Ca2+ concentration spike sequence. When a cell is stimulated, the interspike intervals (ISIs) often show a distinct transient during which they gradually increase, a system property we refer to as cumulative refractoriness. We extend a previously published stochastic model to include the Ca2+ concentration in the intracellular Ca2+ store as a slow adaptation variable. This model can reproduce both stationary and transient statistics of experimentally observed ISI sequences. We derive approximate expressions for the mean and coefficient of variation of the stationary ISIs. We also consider the response to the onset of a constant stimulus and estimate the length of the transient and the strength of the adaptation of the ISI. We show that the adaptation sets the coefficient of variation in agreement with current ideas derived from experiments. Moreover, we explain why, despite a pronounced transient behavior, ISI correlations can be weak, as often observed in experiments. Finally, we fit our model to reproduce the transient statistics of experimentally observed ISI sequences in stimulated HEK cells. The fitted model is able to qualitatively reproduce the relationship between the stationary interval correlations and the number of transient intervals, as well as the strength of the ISI adaptation. We also find positive correlations in the experimental sequence that cannot be explained by our model.

T cell Ca2+ microdomains through the lens of computational modeling

Cellular Ca2+ signaling is highly organized in time and space. Locally restricted and short-lived regions of Ca2+ increase, called Ca2+ microdomains, constitute building blocks that are differentially arranged to create cellular Ca2+ signatures controlling physiological responses. Here, we focus on Ca2+ microdomains occurring in restricted cytosolic spaces between the plasma membrane and the endoplasmic reticulum, called endoplasmic reticulum-plasma membrane junctions. In T cells, these microdomains have been finely characterized. Enough quantitative data are thus available to develop detailed computational models of junctional Ca2+ dynamics. Simulations are able to predict the characteristics of Ca2+ increases at the level of single channels and in junctions of different spatial configurations, in response to various signaling molecules. Thanks to the synergy between experimental observations and computational modeling, a unified description of the molecular mechanisms that create Ca2+ microdomains in the first seconds of T cell stimulation is emerging.

An efficient reduced-lattice model of IP3R for probing Ca2+ dynamics

Numerous cellular processes are regulated by Ca2+ signals, and the endoplasmic reticulum (ER) membrane's inositol triphosphate receptor (IP3R) is critical for modulating intracellular Ca2+ dynamics. The IP3Rs are seen to be clustered in a variety of cell types. The combination of IP3Rs clustering and IP3Rs-mediated Ca2+-induced Ca2+ release results in the hierarchical organization of the Ca2+ signals, which challenges the numerical simulation given the multiple spatial and temporal scales that must be covered. The previous methods rather ignore the spatial feature of IP3Rs or fail to coordinate the conflicts between the real biological relevance and the computational cost. In this work, a general and efficient reduced-lattice model is presented for the simulation of IP3Rs-mediated multiscale Ca2+ dynamics. The model highlights biological details that make up the majority of the calcium events, including IP3Rs clustering and calcium domains, and it reduces the complexity by approximating the minor details. The model's extensibility provides fresh insights into the function of IP3Rs in producing global Ca2+ events and supports the research under more physiological circumstances. Our work contributes to a novel toolkit for modeling multiscale Ca2+ dynamics and advances knowledge of Ca2+ signals.

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.

Adhesion to laminin-1 and collagen IV induces the formation of Ca2+ microdomains that sensitize mouse T cells for activation

During an immune response, T cells migrate from blood vessel walls into inflamed tissues by migrating across the endothelium and through extracellular matrix (ECM). Integrins facilitate T cell binding to endothelial cells and ECM proteins. Here, we report that Ca²⁺ microdomains observed in the absence of T cell receptor (TCR)/CD3 stimulation are initial signaling events triggered by adhesion to ECM proteins that increase the sensitivity of primary murine T cells to activation. Adhesion to the ECM proteins collagen IV and laminin-1 increased the number of Ca²⁺ microdomains in a manner dependent on the kinase FAK, phospholipase C (PLC), and all three inositol 1,4,5-trisphosphate receptor (IP₃R) subtypes and promoted the nuclear translocation of the transcription factor NFAT-1. Mathematical modeling predicted that the formation of adhesion-dependent Ca²⁺ microdomains required the concerted activity of two to six IP₃Rs and ORAI1 channels to achieve the increase in the Ca²⁺ concentration in the ER-plasma membrane junction that was observed experimentally and that required SOCE. Further, adhesion-dependent Ca²⁺ microdomains were important for the magnitude of the TCR-induced activation of T cells on collagen IV as assessed by the global Ca²⁺ response and NFAT-1 nuclear translocation. Thus, adhesion to collagen IV and laminin-1 sensitizes T cells through a mechanism involving the formation of Ca²⁺ microdomains, and blocking this low-level sensitization decreases T cell activation upon TCR engagement.

Diffusive exit rates through pores in membrane-enclosed structures

The function of many membrane-enclosed intracellular structures relies on release of diffusing particles that exit through narrow pores or channels in the membrane. The rate of release varies with pore size, density, and length of the channel. We propose a simple approximate model, validated with stochastic simulations, for estimating the effective release rate from cylinders, and other simple-shaped domains, as a function of channel parameters. The results demonstrate that, for very small pores, a low density of channels scattered over the boundary is sufficient to achieve substantial rates of particle release. Furthermore, we show that increasing the length of passive channels will both reduce release rates and lead to a less steep dependence on channel density. Our results are compared to previously-measured local calcium release rates from tubules of the endoplasmic reticulum, providing an estimate of the relevant channel density responsible for the observed calcium efflux.

Termination of Ca2+ puffs during IP3-evoked global Ca2+ signals

We previously described that cell-wide cytosolic Ca²⁺ transients evoked by inositol trisphosphate (IP₃) are generated by two modes of Ca²⁺ liberation from the ER; 'punctate' release via an initial flurry of transient Ca²⁺ puffs from local clusters of IP₃ receptors, succeeded by a spatially and temporally 'diffuse' Ca²⁺ liberation. Those findings were derived using statistical fluctuation analysis to monitor puff activity which is otherwise masked as global Ca²⁺ levels rise. Here, we devised imaging approaches to resolve individual puffs during global Ca²⁺ elevations to better investigate the mechanisms terminating the puff flurry. We find that puffs contribute about 40% (∼90 attomoles) of the total Ca²⁺ liberation, largely while the global Ca²⁺ signal rises halfway to its peak. The major factor terminating punctate Ca²⁺ release is an abrupt decline in puff frequency. Although the amplitudes of large puffs fall during the flurry, the amplitudes of more numerous small puffs remain steady, so overall puff amplitudes decline only modestly (∼30%). The Ca²⁺ flux through individual IP₃ receptor/channels does not measurably decline during the flurry, or when puff activity is depressed by pharmacological lowering of Ca²⁺ levels in the ER lumen, indicating that the termination of punctate release is not a simple consequence of reduced driving force for Ca²⁺ liberation. We propose instead that the gating of IP₃ receptors at puff sites is modulated such that their openings become suppressed as the bulk [Ca²⁺] in the ER lumen falls during global Ca²⁺ signals.

Three-Dimensional Model of Sub-Plasmalemmal Ca2+ Microdomains Evoked by the Interplay Between ORAI1 and InsP3 Receptors

Ca²⁺ signaling plays an essential role in T cell activation, which is a key step to start an adaptive immune response. During the transition from a quiescent to a fully activated state, Ca²⁺ microdomains characterized by reduced spatial and temporal extents are observed in the junctions between the plasma membrane (PM) and the endoplasmic reticulum (ER). Such Ca²⁺ responses can also occur in response to T cell adhesion to other cells or extracellular matrix proteins in otherwise unstimulated T cells. These non-TCR/CD3-dependent Ca²⁺ microdomains rely on d-myo-inositol 1,4,5-trisphosphate (IP₃) signaling and subsequent store operated Ca²⁺ entry (SOCE) via the ORAI/STIM system. The detailed molecular mechanism of adhesion-dependent Ca²⁺ microdomain formation remains to be fully elucidated. We used mathematical modeling to investigate the spatiotemporal characteristics of T cell Ca²⁺ microdomains and their molecular regulators. We developed a reaction-diffusion model using COMSOL Multiphysics to describe the evolution of cytosolic and ER Ca²⁺ concentrations in a three-dimensional ER-PM junction. Equations are based on a previously proposed realistic description of the junction, which is extended to take into account IP₃ receptors (IP₃R) that are located next to the junction. The first model only considered the ORAI channels and the SERCA pumps. Taking into account the existence of preformed clusters of ORAI1 and STIM2, ORAI1 slightly opens in conditions of a full ER. These simulated Ca²⁺ microdomains are too small as compared to those observed in unstimulated T cells. When considering the opening of the IP₃Rs located near the junction, the local depletion of ER Ca²⁺ allows for larger Ca²⁺ fluxes through the ORAI1 channels and hence larger local Ca²⁺ concentrations. Computational results moreover show that Ca²⁺ diffusion in the ER has a major impact on the Ca²⁺ changes in the junction, by affecting the local Ca²⁺ gradients in the sub-PM ER. Besides pointing out the likely involvement of the spontaneous openings of IP₃Rs in the activation of SOCE in conditions of T cell adhesion prior to full activation, the model provides a tool to investigate how Ca²⁺ microdomains extent and interact in response to T cell receptor activation.

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Calcium (Ca²⁺) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP₃)-induced Ca²⁺ release. These Ca²⁺ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca²⁺, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca²⁺ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca²⁺ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca²⁺ transients in different dynamical situations. We test two different models of the IP₃ receptor (IP₃R). The model with nonlinear concentration response of binding of activating Ca²⁺ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca²⁺-dependent inhibition of the IP₃R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca²⁺ and IP₃ production sensitizing IP₃R for Ca²⁺-induced Ca²⁺ release. We also model ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca²⁺ transients. Finally, we propose possible ways of recovering Ca²⁺ transients under ataxia.

Mechanisms and functions of calcium microdomains produced by ORAI channels, d-myo-inositol 1,4,5-trisphosphate receptors, or ryanodine receptors

With the discovery of local Ca²⁺ signals in the 1990s the concept of 'elementary Ca²⁺ signals' and 'fundamental Ca²⁺ signals' was developed. While 'elementary Ca²⁺signals' relate to optical signals gained by activity of small clusters of Ca²⁺channels, 'fundamental signals' describe such optical signals that arise from opening of single Ca²⁺channels. In this review, we discuss (i) concepts of local Ca²⁺ signals and Ca²⁺ microdomains, (ii) molecular mechanisms underlying Ca²⁺ microdomains, (iii) functions of Ca²⁺ microdomains, and (iv) mathematical modelling of Ca²⁺ microdomains. We focus on Ca²⁺ microdomains produced by ORAI channels, D-myo-inositol 1,4,5-trisphosphate receptors, or ryanodine receptors. In summary, research on local Ca²⁺ signals in different cell models aims to better understand how cells use the Ca²⁺ toolkit to produce Ca²⁺ microdomains as relevant signals for specific cellular responses, but also how local Ca²⁺ signals as building blocks merge into global Ca²⁺ signaling.

Bicarbonate permeation through anion channels: its role in health and disease

Many anion channels, frequently referred as Cl^- channels, are permeable to different anions in addition to Cl^-. As the second-most abundant anion in the human body, HCO₃^- permeation via anion channels has many important physiological roles. In addition to its classical role as an intracellular pH regulator, HCO₃^- also controls the activity and stability of dissolved proteins in bodily fluids such as saliva, pancreatic juice, intestinal fluid, and airway surface liquid. Moreover, HCO₃^- permeation through these channels affects membrane potentials that are the driving forces for transmembrane transport of solutes and water in epithelia and affect neuronal excitability in nervous tissue. Consequently, aberrant HCO₃^- transport via anion channels causes a number of human diseases in respiratory, gastrointestinal, genitourinary, and neuronal systems. Notably, recent studies have shown that the HCO₃^- permeabilities of several anion channels are not fixed and can be altered by cellular stimuli, findings which may have both physiological and pathophysiological significance. In this review, we summarize recent progress in understanding the molecular mechanisms and the physiological roles of HCO₃^- permeation through anion channels. We hope that the present discussions can stimulate further research into this very important topic, which will provide the basis for human disorders associated with aberrant HCO₃^- transport.

Fundamentals of Cellular Calcium Signaling: A Primer

Ionized calcium (Ca²⁺) is the most versatile cellular messenger. All cells use Ca²⁺ signals to regulate their activities in response to extrinsic and intrinsic stimuli. Alterations in cellular Ca²⁺ signaling and/or Ca²⁺ homeostasis can subvert physiological processes into driving pathological outcomes. Imaging of living cells over the past decades has demonstrated that Ca²⁺ signals encode information in their frequency, kinetics, amplitude, and spatial extent. These parameters alter depending on the type and intensity of stimulation, and cellular context. Moreover, it is evident that different cell types produce widely varying Ca²⁺ signals, with properties that suit their physiological functions. This primer discusses basic principles and mechanisms underlying cellular Ca²⁺ signaling and Ca²⁺ homeostasis. Consequently, we have cited some historical articles in addition to more recent findings. A brief summary of the core features of cellular Ca²⁺ signaling is provided, with particular focus on Ca²⁺ stores and Ca²⁺ transport across cellular membranes, as well as mechanisms by which Ca²⁺ signals activate downstream effector systems.

Extension of Rapid Buffering Approximation to Ca2+ Buffers with Two Binding Sites

Fundamental cell processes such as synaptic neurotransmitter release, endocrine hormone secretion, and myocyte contraction are controlled by highly localized calcium (Ca²⁺) signals resulting from brief openings of trans-membrane Ca²⁺ channels. On short temporal and spatial scales, the corresponding local Ca²⁺ nanodomains formed in the vicinity of a single or several open Ca²⁺ channels can be effectively approximated by quasi-stationary solutions. The rapid buffering approximation (RBA) is one of the most powerful of such approximations, and is based on the assumption of instantaneous equilibration of the bimolecular Ca²⁺ buffering reaction, combined with the conservation condition for the total Ca²⁺ and buffer molecule numbers. Previously, RBA has been generalized to an arbitrary arrangement of Ca²⁺ channels on a flat membrane, in the presence of any number of simple Ca²⁺ buffers with one-to-one Ca²⁺ binding stoichiometry. However, many biological buffers have multiple binding sites. For example, buffers and sensors phylogenetically related to calmodulin consist of two Ca²⁺-binding domains (lobes), with each domain binding two Ca²⁺ ions in a cooperative manner. Here we consider an extension of RBA to such buffers with two interdependent Ca²⁺ binding sites. We show that in the presence of such buffers, RBA solution is given by the solution to a cubic equation, analogous to the quadratic equation describing RBA in the case of a simple, one-to-one Ca²⁺ buffer. We examine in detail the dependence of RBA accuracy on buffering parameters, to reveal conditions under which RBA provides sufficient precision.

Coding and decoding of oscillatory Ca2+ signals

About 30 years after their first observation, Ca²⁺ oscillations are now recognised as a universal mechanism of signal transduction. These oscillations are driven by periodic cycles of release and uptake of Ca²⁺ between the cytoplasm and the endoplasmic reticulum. Their frequency often increases with the level of stimulation, which can be decoded by some molecules. However, it is becoming increasingly evident that the widespread core oscillatory mechanism is modulated in many ways, depending on the cell type and on the physiological conditions. Interplay with inositol 1,4,5-trisphosphate metabolism and with other Ca²⁺ stores as the extracellular medium or mitochondria can much affect the properties of these oscillations. In many cases, these finely tuned characteristics of Ca²⁺ oscillations impact the physiological response that is triggered by the signal. Moreover, oscillations are intrinsically irregular. This randomness can also be exploited by the cell. In this review, we discuss evidences of these additional manifestations of the versatility of Ca²⁺ signalling.

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.

Padé Approximation of a Stationary Single-Channel Ca2+ Nanodomain

We consider the stationary solution for the Ca²⁺ concentration near a point Ca²⁺ source describing a single-channel Ca²⁺ nanodomain, in the presence of a single mobile buffer with one-to-one Ca²⁺ binding stoichiometry. Previously, a number of Ca²⁺ nanodomains approximations have been developed, for instance the excess buffer approximation (EBA), the rapid buffering approximation (RBA), and the linear approximation (LIN), each valid for appropriate buffering conditions. Apart from providing a simple method of estimating Ca²⁺ and buffer concentrations without resorting to computationally expensive numerical solution of reaction-diffusion equations, such approximations proved useful in revealing the dependence of nanodomain Ca²⁺ distribution on crucial parameters such as buffer mobility and its Ca²⁺ binding properties. In this study, we present a different form of analytic approximation, which is based on matching the short-range Taylor series of the nanodomain concentration with the long-range asymptotic series expressed in inverse powers of distance from channel location. Namely, we use a "dual" Padé rational function approximation to simultaneously match terms in the short- and the long-range series, and we show that this provides an accurate approximation to the nanodomain Ca²⁺ and buffer concentrations. We compare this approximation with the previously obtained approximations and show that it yields a better estimate of the free buffer concentration for a wide range of buffering conditions. The drawback of our method is that it has a complex algebraic form for any order higher than the lowest bilinear order, and cannot be readily extended to multiple Ca²⁺ channels. However, it may be possible to extend the Padé method to estimate Ca²⁺ nanodomains in the presence of cooperative Ca²⁺ buffers with two Ca²⁺ binding sites, the case that existing methods do not address.

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 multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles

Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca²⁺ content within tens of seconds. Our concept for a multiscale mathematical model of Ca²⁺ -induced Ca²⁺ release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca²⁺-concentration profiles using three previously published RyR-channel Markov schemes.

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.

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.

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.

Role of calcium signaling in epithelial bicarbonate secretion

Transepithelial bicarbonate secretion plays a key role in the maintenance of fluid and protein secretion from epithelial cells and the protection of the epithelial cell surface from various pathogens. Epithelial bicarbonate secretion is mainly under the control of cAMP and calcium signaling. While the physiological roles and molecular mechanisms of cAMP-induced bicarbonate secretion are relatively well defined, those induced by calcium signaling remain poorly understood in most epithelia. The present review summarizes the current status of knowledge on the role of calcium signaling in epithelial bicarbonate secretion. Specifically, this review introduces how cytosolic calcium signaling can increase bicarbonate secretion by regulating membrane transport proteins and how it synergizes with cAMP-induced mechanisms in epithelial cells. In addition, tissue-specific variations in the pancreas, salivary glands, intestines, bile ducts, and airways are discussed. We hope that the present report will stimulate further research into this important topic. These studies will provide the basis for future medicines for a wide spectrum of epithelial disorders including cystic fibrosis, Sjögren's syndrome, and chronic pancreatitis.

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.

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

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

Canonical transient receptor potential 3 channels regulate mitochondrial calcium uptake

Mitochondrial Ca(2+) homeostasis is fundamental to regulation of mitochondrial membrane potential, ATP production, and cellular Ca(2+) homeostasis. It has been known for decades that isolated mitochondria can take up Ca(2+) from the extramitochondrial solution, but the molecular identity of the Ca(2+) channels involved in this action is largely unknown. Here, we show that a fraction of canonical transient receptor potential 3 (TRPC3) channels is localized to mitochondria, a significant fraction of mitochondrial Ca(2+) uptake that relies on extramitochondrial Ca(2+) concentration is TRPC3-dependent, and the up- and down-regulation of TRPC3 expression in the cell influences the mitochondrial membrane potential. Our findings suggest that TRPC3 channels contribute to mitochondrial Ca(2+) uptake. We anticipate our observations may provide insights into the mechanisms of mitochondrial Ca(2+) uptake and advance understanding of the physiological role of TRPC3.

Dynamic modulation of ANO1/TMEM16A HCO3(-) permeability by Ca2+/calmodulin

Anoctamin 1 (ANO1)/transmembrane protein 16A (TMEM16A) is a calcium-activated anion channel that may play a role in HCO(3)(-) secretion in epithelial cells. Here, we report that the anion selectivity of ANO1 is dynamically regulated by the Ca(2+)/calmodulin complex. Whole-cell current measurements in HEK 293T cells indicated that ANO1 becomes highly permeable to HCO(3)(-) at high [Ca(2+)](i). Interestingly, this result was not observed in excised patches, indicating the involvement of cytosolic factors in this process. Further studies revealed that the direct association between ANO1 and calmodulin at high [Ca(2+)](i) is responsible for changes in anion permeability. Calmodulin physically interacted with ANO1 in a [Ca(2+)](i)-dependent manner, and addition of recombinant calmodulin to the cytosolic side of excised patches reversibly increased P(HCO3)/P(Cl). In addition, the high [Ca(2+)](i)-induced increase in HCO(3)(-) permeability was reproduced in mouse submandibular gland acinar cells, in which ANO1 plays a critical role in fluid secretion. These results indicate that the HCO(3)(-) permeability of ANO1 can be dynamically modulated and that ANO1 may play an important role in cellular HCO(3)(-) transport, especially in transepithelial HCO(3)(-) secretion.

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh-Nagumo model, to get the buffered FitzHugh-Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant (K = K(a)) characterized by system excitability parameter a such that calcium buffers can be classified into two types: weak buffers (K ∈ (K(a), ∞)) and strong buffers (K ∈ (0, K(a))). We analytically show that the addition of weak buffers or strong buffers but with its total concentration b(0)(1) below some critical total concentration b(0,c)(1) into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity D1 of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to √D1 as D1 --> ∞. In contrast, the addition of strong buffers with the total concentration b(0)(1) > b(0,c)(1) into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity D1 of the added buffers is sufficiently large.

Multi-scale data-driven modeling and observation of calcium puffs

The spatiotemporal dynamics of elementary Ca(2+) release events, such as "blips" and "puffs" shapes the hierarchal Ca(2+) signaling in many cell types. Despite being the building blocks of Ca(2+) patterning, the mechanism responsible for the observed properties of puffs, especially their termination is incompletely understood. In this paper, we employ a data-driven approach to gain insights into the complex dynamics of blips and puffs. We use a model of inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) derived directly from single channel patch clamp data taken at 10 μM concentration of IP(3) to simulate calcium puffs. We first reproduce recent observations regarding puffs and blips and then investigate the mechanism of puff termination. Our model suggests that during a puff, IP(3)R s proceed around a loop through kinetic states from "rest" to "open" to "inhibited" and back to "rest". A puff terminates because of self-inhibition. Based on our simulations, we rule out the endoplasmic reticulum (ER) Ca(2+) depletion as a possible cause for puff termination. The data-driven approach also enables us to estimate the current through a single IP(3)R and the peak Ca(2+) concentration near the channel pore.

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.

Timescales of IP(3)-evoked Ca(2+) spikes emerge from Ca(2+) puffs only at the cellular level

The behavior of biological systems is determined by the properties of their component molecules, but the interactions are usually too complex to understand fully how molecular behavior generates cellular behavior. Ca(2+) signaling by inositol trisphosphate receptors (IP(3)R) offers an opportunity to understand this relationship because the cellular behavior is defined largely by Ca(2+)-mediated interactions between IP(3)R. Ca(2+) released by a cluster of IP(3)R (giving a local Ca(2+) puff) diffuses and ignites the behavior of neighboring clusters (to give repetitive global Ca(2+) spikes). We use total internal reflection fluorescence microscopy of two mammalian cell lines to define the temporal relationships between Ca(2+) puffs (interpuff intervals, IPI) and Ca(2+) spikes (interspike intervals) evoked by flash photolysis of caged IP(3). We find that IPI are much shorter than interspike intervals, that puff activity is stochastic with a recovery time that is much shorter than the refractory period of the cell, and that IPI are not periodic. We conclude that Ca(2+) spikes do not arise from oscillatory dynamics of IP(3)R clusters, but that repetitive Ca(2+) spiking with its longer timescales is an emergent property of the dynamics of the whole cluster array.

TEMPERATURE EFFECTS ON THE STOCHASTIC GATING OF THE IP3R CALCIUM RELEASE CHANNEL: A NUMERICAL SIMULATION STUDY

The importance of the kinetic study of endoplasmatic calcium ion channels in different intracellular processes is known today. Although there are few experimental reports on the temperature dependency of IP3R channel functions, we did not find any detailed theoretical study on this subject. For this purpose, we used a modified Gillespie algorithm to investigate the effect of temperature on the conditions affecting the open state of a single subunit of the De Young-Keizer (DYK) model. Population of the states was considered as the subject of fluctuation. Key features of the channel, such as bell-shaped dependency of open probability to the Calcium concentrations were modeled at different temperatures, too. The range of temperature variation was selected by regarding the experimental data on IP3R channel.

By increasing the temperature, we had the very slow time domains (t: 10-1 s ) and the much slower time domains (t: 100 s ) in addition to other time domains, which could be seen as new time categories in InsP3R studies, and so the results were reported in these time domains, as well.

We found out that increase in temperature declined the open probability in some concentrations of Ca 2+ and/or IP3. Also, by introducing the intensity graphs, broadening of the range of fluctuations and lowering of the order of frequency of fluctuations for the population of each state were observed due to the temperature increments.

The temperature effects on the activation and inactivation states of the channel were studied in the framework of the reaction paths. We did not find similar paths at different time domains; several paths observed which were totally different all together. These time-dependent reaction paths are also depending on the Ca 2+ and/or the IP3 concentrations. So, one can predict the most probable reaction paths at different concentrations and temperatures and also determine which kind of the path it is; a path for closing the channel or a path to open it.

Finally, the temperature effects on the calcium inhibited states were studied. We found out that calcium ion inhibitions were shifted to lower calcium concentration by increasing the temperature. The results suggests that inhibiting role of calcium is not only [ Ca 2+] and/or [IP3] dependent, but also temperature dependent.

Fundamental properties of Ca2+ signals

BACKGROUND

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

SCOPE OF REVIEW

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

MAJOR CONCLUSIONS

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

GENERAL SIGNIFICANCE

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

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.

Model of cell signal transduction in a three-dimensional domain

Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.

Derivation of Ca2+ signals from puff properties reveals that pathway function is robust against cell variability but sensitive for control

Ca(2+) is a universal second messenger in eukaryotic cells transmitting information through sequences of concentration spikes. A prominent mechanism to generate these spikes involves Ca(2+) release from the endoplasmic reticulum Ca(2+) store via inositol 1,4,5-trisphosphate (IP(3))-sensitive channels. Puffs are elemental events of IP(3)-induced Ca(2+) release through single clusters of channels. Intracellular Ca(2+) dynamics are a stochastic system, but a complete stochastic theory has not been developed yet. We formulate the theory in terms of interpuff interval and puff duration distributions because, unlike the properties of individual channels, they can be measured in vivo. Our theory reproduces the typical spectrum of Ca(2+) signals like puffs, spiking, and bursting in analytically treatable test cases as well as in more realistic simulations. We find conditions for spiking and calculate interspike interval (ISI) distributions. Signal form, average ISI and ISI distributions depend sensitively on the details of cluster properties and their spatial arrangement. In contrast to that, the relation between the average and the standard deviation of ISIs does not depend on cluster properties and cluster arrangement and is robust with respect to cell variability. It is controlled by the global feedback processes in the Ca(2+) signaling pathway (e.g., via IP(3)-3-kinase or endoplasmic reticulum depletion). That relation is essential for pathway function because it ensures frequency encoding despite the randomness of ISIs and determines the maximal spike train information content. Hence, we find a division of tasks between global feedbacks and local cluster properties that guarantees robustness of function while maintaining sensitivity of control of the average ISI.

Unitary Ca(2+) current through recombinant type 3 InsP(3) receptor channels under physiological ionic conditions

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) channel, localized primarily in the endoplasmic reticulum (ER) membrane, releases Ca(2+) into the cytoplasm upon binding InsP(3), generating and modulating intracellular Ca(2+) signals that regulate numerous physiological processes. Together with the number of channels activated and the open probability of the active channels, the size of the unitary Ca(2+) current (i(Ca)) passing through an open InsP(3)R channel determines the amount of Ca(2+) released from the ER store, and thus the amplitude and the spatial and temporal nature of Ca(2+) signals generated in response to extracellular stimuli. Despite its significance, i(Ca) for InsP(3)R channels in physiological ionic conditions has not been directly measured. Here, we report the first measurement of i(Ca) through an InsP(3)R channel in its native membrane environment under physiological ionic conditions. Nuclear patch clamp electrophysiology with rapid perfusion solution exchanges was used to study the conductance properties of recombinant homotetrameric rat type 3 InsP(3)R channels. Within physiological ranges of free Ca(2+) concentrations in the ER lumen ([Ca(2+)](ER)), free cytoplasmic [Ca(2+)] ([Ca(2+)](i)), and symmetric free [Mg(2+)] ([Mg(2+)](f)), the i(Ca)-[Ca(2+)](ER) relation was linear, with no detectable dependence on [Mg(2+)](f). i(Ca) was 0.15 +/- 0.01 pA for a filled ER store with 500 microM [Ca(2+)](ER). The i(Ca)-[Ca(2+)](ER) relation suggests that Ca(2+) released by an InsP(3)R channel raises [Ca(2+)](i) near the open channel to approximately 13-70 microM, depending on [Ca(2+)](ER). These measurements have implications for the activities of nearby InsP(3)-liganded InsP(3)R channels, and they confirm that Ca(2+) released by an open InsP(3)R channel is sufficient to activate neighboring channels at appropriate distances away, promoting Ca(2+)-induced Ca(2+) release.

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.

Computational modelling of the initiation and development of spontaneous intracellular Ca2+ waves in ventricular myocytes

Intracellular Ca(2+) dynamics provides excitation-contraction coupling in cardiac myocytes. Under pathological conditions, spontaneous Ca(2+) release events can lead to intracellular Ca(2+) travelling waves, which can break, giving transitory or persistent intracellular re-entrant Ca(2+) scroll waves. Intracellular Ca(2+) waves can trigger cellular delayed after-depolarizations of membrane potential, which if they occur in a cluster of a few hundred neighbouring myocytes may lead to cardiac arrhythmia. Quantitative prediction of the initiation and propagation of intracellular Ca(2+) waves requires the dynamics of Ca(2+)-induced Ca(2+) release, and the intracellular spatial distribution of Ca(2+) release units (CRUs). The spatial distribution of ryanodine receptor clusters within a few sarcomeres was reconstructed directly from confocal imaging measurements. It was then embedded into a three-dimensional ventricular cell model, with a resting membrane potential and simple stochastic Ca(2+)-induced Ca(2+) release dynamics. Isotropic global Ca(2+) wave propagation can be produced within the anisotropic intracellular architecture, by isotropic local Ca(2+) diffusion, and the branching Z-disc structure providing inter Z-disc pathways for Ca(2+) propagation. The branching Z-disc provides a broader spatial distribution of ryanodine receptor clusters across Z-discs, which reduces the likelihood of wave initiation by spontaneous Ca(2+) releases. Intracellular Ca(2+) dynamics during catecholaminergic polymorphic ventricular tachycardia (CPVT) was simulated phenomenologically by increasing the Ca(2+) sensitivity factor of the CRU, which results in an increased rate of Ca(2+) release events. Flecainide has been shown to prevent arrhythmias in a murine model of CPVT and in patients. The modelled actions of flecainide on the time course of Ca(2+) release events prevented the initiation of Ca(2+) waves.

Exocytosis, dependent on Ca2+ release from Ca2+ stores, is regulated by Ca2+ microdomains

The relationship between the cellular Ca2+ signal and secretory vesicle fusion (exocytosis) is a key determinant of the regulation of the kinetics and magnitude of the secretory response. Here, we have investigated secretion in cells where the exocytic response is controlled by Ca2+ release from intracellular Ca2+ stores. Using live-cell two-photon microscopy that simultaneously records Ca2+signals and exocytic responses, we provide evidence that secretion is controlled by changes in Ca2+ concentration [Ca2+] in relatively large-volume microdomains. Our evidence includes: (1) long latencies (>2 seconds) between the rise in [Ca2+] and exocytosis, (2) observation of exocytosis all along the lumen and not clustered around Ca2+ release hot-spots, (3) high affinity (Kd=1.75 microM) Ca2+dependence of exocytosis, (4) significant reduction in exocytosis in the presence of cytosolic EGTA, (5) spatial exclusion of secretory granules from the cell membrane by the endoplasmic reticulum, and (6) inability of local Ca2+ responses to trigger exocytosis. These results strongly indicate that the control of exocytosis, triggered by Ca2+ release from stores, is through the regulation of cytosolic[Ca2+] within a microdomain.

Calcium signals driven by single channel noise

Usually, the occurrence of random cell behavior is appointed to small copy numbers of molecules involved in the stochastic process. Recently, we demonstrated for a variety of cell types that intracellular Ca2+ oscillations are sequences of random spikes despite the involvement of many molecules in spike generation. This randomness arises from the stochastic state transitions of individual Ca2+ release channels and does not average out due to the existence of steep concentration gradients. The system is hierarchical due to the structural levels channel--channel cluster--cell and a corresponding strength of coupling. Concentration gradients introduce microdomains which couple channels of a cluster strongly. But they couple clusters only weakly; too weak to establish deterministic behavior on cell level. Here, we present a multi-scale modelling concept for stochastic hierarchical systems. It simulates active molecules individually as Markov chains and their coupling by deterministic diffusion. Thus, we are able to follow the consequences of random single molecule state changes up to the signal on cell level. To demonstrate the potential of the method, we simulate a variety of experiments. Comparisons of simulated and experimental data of spontaneous oscillations in astrocytes emphasize the role of spatial concentration gradients in Ca2+ signalling. Analysis of extensive simulations indicates that frequency encoding described by the relation between average and standard deviation of interspike intervals is surprisingly robust. This robustness is a property of the random spiking mechanism and not a result of control.

Spatiotemporal organization of Ca dynamics: a modeling-based approach

Calcium is a ubiquitous second messenger that mediates vital physiological responses such as fertilization, secretion, gene expression, or apoptosis. Given this variety of processes mediated by Ca(2+), these signals are highly organized both in time and space to ensure reliability and specificity. This review deals with the spatiotemporal organization of the Ca(2+) signaling pathway in electrically nonexcitable cells in which InsP(3) receptors are by far the most important Ca(2+) channels. We focus on the aspects of this highly regulated dynamical system for which an interplay between experiments and modeling is particularly fruitful. In particular, the importance of the relative densities of the different InsP(3) receptor subtypes will be discussed on the basis of a modeling approach linking the steady-state behaviors of these channels in electrophysiological experiments with their behavior in a cellular environment. Also, the interplay between InsP(3) metabolism and Ca(2+) oscillations will be considered. Finally, we discuss the relationships between stochastic openings of the Ca(2+) releasing channels at the microscopic level and the coordinated, regular behavior observed at the whole cell level on the basis of a combined experimental and modeling approach.

Quantifying calcium fluxes underlying calcium puffs in Xenopus laevis oocytes

We determine the calcium fluxes through inositol 1,4,5-trisphosphate receptor/channels underlying calcium puffs of Xenopus laevis oocytes using a simplified version of the algorithm of Ventura et al. An analysis of 130 puffs obtained with Fluo-4 indicates that Ca2+ release comes from a region of width approximately 450 nm, that the release duration is peaked around 18 s and that the underlying Ca2+ currents range between 0.12 and 0.95 pA. All these parameters are independent of IP(3) concentration. We explore what distributions of channels that open during a puff, N(p), and what relations between current and number of open channels, I(N(p)), are compatible with our findings and with the distribution of puff-to-trigger amplitude ratio reported in Rose et al. To this end, we use simple "mean field" models in which all channels open and close simultaneously. We find that the variability among clusters plays an important role in shaping the observed puff amplitude distribution and that a model for which I(N(p)) approximately N(p) for small N(p) and I(N(p)) approximately N(p)(1/alpha) (alpha > 1) for large N(p), provides the best agreement. Simulations of more detailed models in which channels open and close stochastically show that this nonlinear behavior can be attributed to the limited time resolution of the observations and to the averaging procedure that is implicit in the mean-field models. These conclusions are also compatible with observations of approximately 400 puffs obtained using the dye Oregon green.

IP3 receptors: some lessons from DT40 cells

Inositol-1,4,5-trisphosphate receptors (IP3Rs) are intracellular Ca2+ channels that are regulated by IP3 and Ca2+ and are modulated by many additional signals. These properties allow them to initiate and, via Ca2+-induced Ca2+ release, regeneratively propagate Ca2+ signals evoked by receptors that stimulate formation of IP3. The ubiquitous expression of IP3R highlights their importance, but it also presents problems when attempting to resolve the behavior of defined IP3R. DT40 cells are a pre-B-lymphocyte cell line in which high rates of homologous recombination afford unrivalled opportunities to disrupt endogenous genes. DT40-knockout cells with both alleles of each of the three IP3R genes disrupted provide the only null-background for analysis of homogenous recombinant IP3R. We review the properties of DT40 cells and consider three areas where they have contributed to understanding IP3R behavior. Patch-clamp recording from the nuclear envelope and Ca2+ release from intracellular stores loaded with a low-affinity Ca2+ indicator address the mechanisms leading to activation of IP(3)R. We show that IP3 causes intracellular IP3R to cluster and re-tune their responses to IP3 and Ca2+, better equipping them to mediate regenerative Ca2+ signals. Finally, we show that DT40 cells reliably count very few IP3R into the plasma membrane, where they mediate about half the Ca2+ entry evoked by the B-cell antigen receptor.

Modeling of the modulation by buffers of Ca2+ release through clusters of IP3 receptors

Intracellular Ca(2+) release is a versatile second messenger system. It is modeled here by reaction-diffusion equations for the free Ca(2+) and Ca(2+) buffers, with spatially discrete clusters of stochastic IP(3) receptor channels (IP(3)Rs) controlling the release of Ca(2+) from the endoplasmic reticulum. IP(3)Rs are activated by a small rise of the cytosolic Ca(2+) concentration and inhibited by large concentrations. Buffering of cytosolic Ca(2+) shapes global Ca(2+) transients. Here we use a model to investigate the effect of buffers with slow and fast reaction rates on single release spikes. We find that, depending on their diffusion coefficient, fast buffers can either decouple clusters or delay inhibition. Slow buffers have little effect on Ca(2+) release, but affect the time course of the signals from the fluorescent Ca(2+) indicator mainly by competing for Ca(2+). At low [IP(3)], fast buffers suppress fluorescence signals, slow buffers increase the contrast between bulk signals and signals at open clusters, and large concentrations of buffers, either fast or slow, decouple clusters.

A model-based method for estimating Ca2+ release fluxes from linescan images in Xenopus oocytes

We propose a model-based method of interpreting linescan images observed in Xenopus oocytes with the use of Oregon Green-1 as a fluorescent dye. We use a detailed modeling formalism based on numerical simulations that incorporate physical barriers for local diffusion, and, by assuming a Gaussian distribution of release durations, we derive the distributions of release Ca(2+) amounts and currents, fluorescence amplitudes, and puff widths. We analyze a wide set of available data collected from 857 and 281 events observed in the animal and the vegetal hemispheres of the oocyte, respectively. A relatively small fraction of events appear to involve coupling of two or three adjacent clusters of Ca(2+) releasing channels. In the animal hemisphere, the distribution of release currents with a mean of 1.4 pA presents a maximum at 1.0 pA and a rather long tail extending up to 5 pA. The overall distribution of liberated Ca(2+) amounts exhibits a dominant peak at 120 fC, a smaller peak at 375 fC, and an average of 166 fC. Ca(2+) amounts and release fluxes in the vegetal hemisphere appear to be 3.6 and 1.6 times smaller than in the animal hemisphere, respectively. Predicted diameters of elemental release sites are approximately 1.0 microm in the animal and approximately 0.5 microm in the vegetal hemisphere, but the side-to-side separation between adjacent sites appears to be identical (approximately 0.4 microm). By fitting the model to individual puffs we can estimate the quantity of liberated calcium, the release current, the orientation of the scan line, and the dimension of the corresponding release site.

A simple sequential-binding model for calcium puffs

Calcium puffs describe the transient release of Ca(2+) ions into the cytosol, through small clusters of 1,4,5-inositol triphosphate (IP(3)) receptors, present on internal stores such as the endoplasmic reticulum. Statistical properties of puffs, such as puff amplitudes and durations, have been well characterized experimentally. We model calcium puffs using a simple, sequential-binding model for the IP(3) receptor in conjunction with a computationally inexpensive point-source approximation. We follow two different protocols, a sequential protocol and a renewal protocol. In the sequential protocol, puffs are generated successively by the same cluster; in the renewal protocol, the system is reset after each puff. In both cases for a single set of parameters our results are in excellent agreement with experimental results for puff amplitudes and durations but indicate puff-to-puff correlations for the sequential protocol, consistent with recent experimental findings [H. J. Rose, S. Dargan, J. W. Shuai, and I. Parker, Biophys. J. 91, 4024 (2006)]. The model is then used to test the consistency of the hypothesized steep Ca(2+) gradients around single channels with the experimentally observed features of puff durations and amplitudes. A three-dimensional implementation of our point-source model suggests that a peak Ca(2+) concentration of the order of 10 muM at the cluster site (not channel) is consistent with the statistical features of observed calcium puffs.

What can we learn from the irregularity of Ca2+ oscillations?

In most cells, Ca(2+) increases in response to external stimulation are organized in the form of oscillations and waves that sometimes propagate from one cell to another. Numerous experimental and theoretical studies reveal that this spatiotemporal organization contains a non-negligible level of stochasticity. In this study, we extend the previous work based on a statistical analysis of experimental Ca(2+) traces in isolated, hormone-stimulated hepatocytes and on stochastic simulations of Ca(2+) oscillations based on the Gillespie's algorithm. Comparison of the coefficients of variation in the periods of experimental and simulated Ca(2+) spikes provides information about the clustering and the specific subtypes of the Ca(2+) channels. In hepatocytes coupled by gap junctions, the global perfusion with a hormone leads to successive Ca(2+) responses, giving the appearance of an intercellular wave. Statistical analysis of experimental Ca(2+) oscillations in coupled hepatocytes confirms that this coordinated Ca(2+) spiking corresponds to a phase wave but suggests the existence of an additional coupling mechanism.