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

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.

Anti-cross-correlation between the adjacent open and closed durations of Markovian channels

We show that a non-Markovian behavior can appear in a type of Markovian multimeric channel. Such a channel consists of N independent subunits, and each subunit has at least one open state and more than one closed state. Suppose the open state of the channel is defined as M out of N subunits in the open state with N>M>0. We show that, although the gating dynamics for each subunit between open and closed states is Markovian, the channel can show a memory behavior of weak anti-cross-correlation between the adjacent open and closed durations. Our study indicates that a non-Markovian binary time series can be obtained from a linear superposition of some independent channel subunits with Markovian gating dynamics.

Changes in Ca2+ Removal Can Mask the Effects of Geometry During IP3R Mediated Ca2+ Signals

Calcium (Ca²⁺) signals are ubiquitous. Most intracellular Ca²⁺ signals involve the release of Ca²⁺ from the endoplasmic reticulum (ER) through Inositol 1,4,5-Trisphosphate Receptors (IP₃Rs). The non-uniform spatial organization of IP₃Rs and the fact that their individual openings are coupled via cytosolic Ca²⁺ are key factors for the variety of spatio-temporal distributions of the cytosolic [Ca²⁺] and the versatility of the signals. In this paper we combine experiments performed in untreated and in progesterone-treated Xenopus laevis oocytes and mathematical models to investigate how the interplay between geometry (the IP₃R spatial distribution) and dynamics (the processes that characterize the release, transport, and removal of cytosolic Ca²⁺) affects the resulting signals. Signal propagation looks more continuous and spatially uniform in treated (mature) than in untreated (immature) oocytes. This could be due to the different underlying IP₃R spatial distribution that has been observed in both cell types. The models, however, show that the rate of cytosolic Ca²⁺ removal, which is also different in both cell types, plays a key role affecting the coupling between Ca²⁺ release sites in such a way that the effect of the underlying IP₃R spatial distribution can be modified.

Deterministic Limit of Intracellular Calcium Spikes

In nonexcitable cells, global Ca^{2+} spikes emerge from the collective dynamics of clusters of Ca^{2+} channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin, while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

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.

Mapping Interpuff Interval Distribution to the Properties of Inositol Trisphosphate Receptors

Tightly clustered inositol trisphosphate receptors (IP₃Rs) control localized Ca²⁺ liberation from the endoplasmic reticulum to generate repetitive Ca²⁺ puffs. Distributions of the interpuff interval (IPI), i.e., the waiting time between successive puffs, are found to be well characterized by a probability density function involving only two parameters, λ and ξ, which represent the basal rate of puff generation and the recovery rate from refractoriness, respectively. However, how the two parameters depend on the kinetic parameters of single IP₃Rs in a cluster is still unclear. In this article, using a stochastic puff model and a single-channel data-based IP₃R model, we establish the dependencies of λ and ξ on two important IP₃R model parameters, IP₃ concentration ([IP₃]) and the recovery rate from Ca²⁺ inhibition (r_low). By varying [IP₃] and r_low in physiologically plausible ranges, we find that the ξ-λ plane is comprised of only two disjoint regions, a biologically impermissible region and a region where each parameter set (ξ, λ) can be caused by using two different combinations of [IP₃] and r_low. The two combinations utilize very different mechanisms to maintain the same IPI distribution, and the mechanistic difference provides a way of identifying IP₃R kinetic parameters by observing properties of the IPI.

Factors determining the recruitment of inositol trisphosphate receptor channels during calcium puffs

Puffs are localized, transient elevations in cytosolic Ca(2+) that serve both as the building blocks of global cellular Ca(2+) signals and as local signals in their own right. They arise from clustered inositol 1,4,5-trisphosphate receptor/channels (IP3Rs), whose openings are coordinated by Ca(2+)-induced Ca(2+) release (CICR). We utilized total internal reflection fluorescence imaging of Ca(2+) signals in neuroblastoma cells with single-channel resolution to elucidate the mechanisms determining the triggering, amplitudes, kinetics, and spatial spread of puffs. We find that any given channel in a cluster has a mean probability of ∼66% of opening following opening of an initial "trigger" channel, and the probability of puff triggering thus increases steeply with increasing number of channels in a cluster (cluster size). Mean puff amplitudes scale with cluster size, but individual amplitudes vary widely, even at sites of similar cluster size, displaying similar proportions of events involving any given number of the channels in the cluster. Stochastic variation in numbers of Ca(2+)-inhibited IP3Rs likely contributes to the variability of amplitudes of repeated puffs at a site but the amplitudes of successive puffs were uncorrelated, even though we observed statistical correlations between interpuff intervals and puff amplitudes. Initial puffs evoked following photorelease of IP3-which would not be subject to earlier Ca(2+)-inhibition-also showed wide variability, indicating that mechanisms such as stochastic variation in IP3 binding and channel recruitment by CICR further determine puff amplitudes. The mean termination time of puffs lengthened with increasing puff amplitude size, consistent with independent closings of channels after a given mean open time, but we found no correlation of termination time with cluster size independent of puff amplitude. The spatial extent of puffs increased with their amplitude, and puffs of similar size were of similar width, independent of cluster size.

Basis for a neuronal version of Grover's quantum algorithm

Grover's quantum (search) algorithm exploits principles of quantum information theory and computation to surpass the strong Church–Turing limit governing classical computers. The algorithm initializes a search field into superposed N (eigen)states to later execute nonclassical “subroutines” involving unitary phase shifts of measured states and to produce root-rate or quadratic gain in the algorithmic time (O(N^1/2)) needed to find some “target” solution m. Akin to this fast technological search algorithm, single eukaryotic cells, such as differentiated neurons, perform natural quadratic speed-up in the search for appropriate store-operated Ca²⁺ response regulation of, among other processes, protein and lipid biosynthesis, cell energetics, stress responses, cell fate and death, synaptic plasticity, and immunoprotection. Such speed-up in cellular decision making results from spatiotemporal dynamics of networked intracellular Ca²⁺-induced Ca²⁺ release and the search (or signaling) velocity of Ca²⁺ wave propagation. As chemical processes, such as the duration of Ca²⁺ mobilization, become rate-limiting over interstore distances, Ca²⁺ waves quadratically decrease interstore-travel time from slow saltatory to fast continuous gradients proportional to the square-root of the classical Ca²⁺ diffusion coefficient, D^1/2, matching the computing efficiency of Grover's quantum algorithm. In this Hypothesis and Theory article, I elaborate on these traits using a fire-diffuse-fire model of store-operated cytosolic Ca²⁺ signaling valid for glutamatergic neurons. Salient model features corresponding to Grover's quantum algorithm are parameterized to meet requirements for the Oracle Hadamard transform and Grover's iteration. A neuronal version of Grover's quantum algorithm figures to benefit signal coincidence detection and integration, bidirectional synaptic plasticity, and other vital cell functions by rapidly selecting, ordering, and/or counting optional response regulation choices.

Fluorescence fluctuations and equivalence classes of Ca²⁺ imaging experiments

release into the cytosol through inositol 1,4,5-trisphosphate receptors (IP₃Rs) plays a relevant role in numerous physiological processes. IP₃R-mediated signals involve -induced -release (CICR) whereby release through one open IP₃R induces the opening of other channels. IP₃Rs are apparently organized in clusters. The signals can remain localized (i.e., puffs) if CICR is limited to one cluster or become waves that propagate between clusters. puffs are the building blocks of waves. Thus, there is great interest in determining puff properties, especially in view of the current controversy on the spatial distribution of activatable IP₃Rs. puffs have been observed in intact cells with optical techniques proving that they are intrinsically stochastic. Obtaining a correct picture of their dynamics then entails being able to detect the whole range of puff sizes. puffs are observed using visible single-wavelength dyes, slow exogenous buffers (e.g., EGTA) to disrupt inter-cluster CICR and UV-photolyzable caged IP₃. Single-wavelength dyes increase their fluorescence upon calcium binding producing images that are strongly dependent on their kinetic, transport and photophysical properties. Determining the artifacts that the imaging setting introduces is particularly relevant when trying to analyze the smallest signals. In this paper we introduce a method to estimate the expected signal-to-noise ratio of imaging experiments that use single-wavelength dyes. The method is based on the Number and Brightness technique. It involves the performance of a series of experiments and their subsequent analysis in terms of a fluorescence fluctuation model with which the model parameters are quantified. Using the model, the expected signal-to-noise ratio is then computed. Equivalence classes between different experimental conditions that produce images with similar signal-to-noise ratios can then be established. The method may also be used to estimate the smallest signals that can reliably be observed with each setting.

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.

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.

Intracellular calcium signals display an avalanche-like behavior over multiple lengthscales

Many natural phenomena display “self-organized criticality” (SOC), (Bak et al., 1987). This refers to spatially extended systems for which patterns of activity characterized by different lengthscales can occur with a probability density that follows a power law with pattern size. Differently from power laws at phase transitions, systems displaying SOC do not need the tuning of an external parameter. Here we analyze intracellular calcium (Ca²⁺) signals, a key component of the signaling toolkit of almost any cell type. Ca²⁺ signals can either be spatially restricted (local) or propagate throughout the cell (global). Different models have suggested that the transition from local to global signals is similar to that of directed percolation. Directed percolation has been associated, in turn, to the appearance of SOC. In this paper we discuss these issues within the framework of simple models of Ca²⁺ signal propagation. We also analyze the size distribution of local signals (“puffs”) observed in immature Xenopus Laevis oocytes. The puff amplitude distribution obtained from observed local signals is not Gaussian with a noticeable fraction of large size events. The experimental distribution of puff areas in the spatio-temporal record of the image has a long tail that is approximately log-normal. The distribution can also be fitted with a power law relationship albeit with a smaller goodness of fit. The power law behavior is encountered within a simple model that includes some coupling among individual signals for a wide range of parameter values. An analysis of the model shows that a global elevation of the Ca²⁺ concentration plays a major role in determining whether the puff size distribution is long-tailed or not. This suggests that Ca²⁺-clearing from the cytosol is key to determine whether IP₃-mediated Ca²⁺ signals can display a SOC-like behavior or not.

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.

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.

Custom-made modification of a commercial confocal microscope to photolyze caged compounds using the conventional illumination module and its application to the observation of Inositol 1,4,5-trisphosphate-mediated calcium signals

The flash photolysis of "caged" compounds is a powerful experimental technique for producing rapid changes in concentrations of bioactive signaling molecules. These caged compounds are inactive and become active when illuminated with ultraviolet light. This paper describes an inexpensive adaptation of an Olympus confocal microscope that uses as source of ultraviolet light the mercury lamp that comes with the microscope for conventional fluorescence microscopy. The ultraviolet illumination from the lamp (350 - 400 nm) enters through an optical fiber that is coupled to a nonconventional port of the microscope. The modification allows to perform the photolysis of caged compounds over wide areas (∼ 200 μm) and obtain confocal fluorescence images simultaneously. By controlling the ultraviolet illumination exposure time and intensity it is possible to regulate the amount of photolyzed compounds. In the paper we characterize the properties of the system and show its capabilities with experiments done in aqueous solution and in Xenopus Laevis oocytes. The latter demonstrate its applicability for the study of Inositol 1,4,5-trisphosphate-mediated intracellular calcium signals.

Calcium oscillations

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

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

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

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.

Puff-wave transition in an inhomogeneous model for calcium signals

In many cell types, calcium ion channels on the endoplasmic reticulum membrane occur in a clustered distribution. The channels generate either localized puffs, each comprising channels of only one cluster, or global calcium waves. In this work we model the calcium system as a two-dimensional lattice of active elements distributed regularly in an otherwise passive space. We address an important feature of the puff-wave transition, which is the difference in lifetime of puffs at a few hundred milliseconds and long-lived global waves with periods of several seconds. We show that such a lifetime difference between puffs and waves can be understood with strongly reduced ordinary differential equations modified by a time-scale factor that takes into account the coupling strength of active and passive regions determined by the Ca2+ diffusion coefficient. Furthermore, we show that the point model can also describe very well the dependence of Ca2+ oscillation characteristics on the cluster-cluster distance in the case of large diffusivity.

Intra-cluster percolation of calcium signals

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

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.

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.

Imaging the quantal substructure of single IP3R channel activity during Ca2+ puffs in intact mammalian cells

The spatiotemporal patterning of Ca(2+) signals regulates numerous cellular functions, and is determined by the functional properties and spatial clustering of inositol trisphosphate receptor (IP(3)R) Ca(2+) release channels in the endoplasmic reticulum membrane. However, studies at the single-channel level have been hampered because IP(3)Rs are inaccessible to patch-clamp recording in intact cells, and because excised organelle and bilayer reconstitution systems disrupt the Ca(2+)-induced Ca(2+) release (CICR) process that mediates channel-channel coordination. We introduce here the use of total internal reflection fluorescence microscopy to image single-channel Ca(2+) flux through individual and clustered IP(3)Rs in intact mammalian cells. This enables a quantal dissection of the local calcium puffs that constitute building blocks of cellular Ca(2+) signals, revealing stochastic recruitment of, on average, approximately 6 active IP(3)Rs clustered within <500 nm. Channel openings are rapidly ( approximately 10 ms) recruited by opening of an initial trigger channel, and a similarly rapid inhibitory process terminates puffs despite local [Ca(2+)] elevation that would otherwise sustain Ca(2+)-induced Ca(2+) release indefinitely. Minimally invasive, nano-scale Ca(2+) imaging provides a powerful tool for the functional study of intracellular Ca(2+) release channels while maintaining the native architecture and dynamic interactions essential for discrete and selective cell signaling.

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

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

Calcium Dynamics: Spatio‐Temporal Organization from the Subcellular to the Organ Level

Many essential physiological processes are controlled by calcium. To ensure reliability and specificity, calcium signals are highly organized in time and space in the form of oscillations and waves. Interesting findings have been obtained at various scales, ranging from the stochastic opening of a single calcium channel to the intercellular calcium wave spreading through an entire organ. A detailed understanding of calcium dynamics thus requires a link between observations at different scales. It appears that some regulations such as calcium-induced calcium release or PLC activation by calcium, as well as the weak diffusibility of calcium ions play a role at all levels of organization in most cell types. To comprehend how calcium waves spread from one cell to another, specific gap-junctional coupling and paracrine signaling must also be taken into account. On the basis of a pluridisciplinar approach ranging from physics to physiology, a unified description of calcium dynamics is emerging, which could help understanding how such a small ion can mediate so many vital functions in living systems.

Ca2+ microdomains in smooth muscle

In smooth muscle, Ca(2+) controls diverse activities including cell division, contraction and cell death. Of particular significance in enabling Ca(2+) to perform these multiple functions is the cell's ability to localize Ca(2+) signals to certain regions by creating high local concentrations of Ca(2+) (microdomains), which differ from the cytoplasmic average. Microdomains arise from Ca(2+) influx across the plasma membrane or release from the sarcoplasmic reticulum (SR) Ca(2+) store. A single Ca(2+) channel can create a microdomain of several micromolar near (approximately 200 nm) the channel. This concentration declines quickly with peak rates of several thousand micromolar per second when influx ends. The high [Ca(2+)] and the rapid rates of decline target Ca(2+) signals to effectors in the microdomain with rapid kinetics and enable the selective activation of cellular processes. Several elements within the cell combine to enable microdomains to develop. These include the brief open time of ion channels, localization of Ca(2+) by buffering, the clustering of ion channels to certain regions of the cell and the presence of membrane barriers, which restrict the free diffusion of Ca(2+). In this review, the generation of microdomains arising from Ca(2+) influx across the plasma membrane and the release of the ion from the SR Ca(2+) store will be discussed and the contribution of mitochondria and the Golgi apparatus as well as endogenous modulators (e.g. cADPR and channel binding proteins) will be considered.