Optimal ion channel clustering for intracellular calcium signaling

[Experimental and computational approach of calcium signaling]

In many cell types, specific and robust signalling relies on a high level of spatiotemporal organization of Ca(2+) dynamics. In response to external stimulation, Ca(2+) signals ranging from a small increase of a few tens of nanomolar concentrations at the mouth of an inositol 1, 4, 5-trisphosphate receptor to the periodic propagation of waves invading an organ or a tissue, can be observed. Here, we review our combined experimental and computational approach of Ca(2+) dynamics, which has been mainly carried out on liver hepatocytes. We focus in particular on the understanding of the relationship between elementary Ca(2+) increases, Ca(2+) oscillations and intra- or intercellular Ca(2+) waves. The physiological impact of such signalling on liver function is also discussed.

Intercellular calcium waves in glial cells with bistable dynamics

A two-dimensional model is proposed for intercellular calcium (Ca(2 +)) waves with Ca(2 +)-induced IP(3) regeneration and the diffusion of IP(3) through gap junctions. Many experimental observations in glial cells, i.e. responding to local mechanical stimulation, glutamate application, mechanical stimulation followed by ACh application, and glutamate followed by mechanical stimulation, are reproduced and classified by the model. We show that a glial cell model with bistable dynamics, i.e. a Ca(2 +) oscillation state coexisting with a fixed point, can cause a prolonged plateau of Ca(2 +) signals in the cells nearby the stimulated cell when the cell network responds to the local mechanical stimulation.

Brownian diffusion of ion channels in different membrane patch geometries

We asymptotically calculate the spatially averaged mean first passage time (MFPT) of a diffusing channel protein in a finite membrane patch containing a small absorbing anchor site. Different two-dimensional membrane geometries are considered including a circular, a square-shaped, a rectangular, and a cylindrical domain. The asymptotic expressions are found to be in excellent agreement with results from Monte Carlo simulations if the radius of the diffusing protein is sufficiently small. For a larger radius, a simple correction to the asymptotic expressions is proposed. We show that the average MFPT for a circle and a square-shaped domain of the same area are approximately equal as long as the anchor site is close to the center of the domain. We also discuss how the average MFPT depends on the aspect ratio of a rectangular and a cylindrical domain. Among such domains with a fixed area, a minimal MFPT is obtained for the square-shaped domain.

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

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

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensen's inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

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.

Intrinsic variability of latency to first-spike

The assessment of the variability of neuronal spike timing is fundamental to gain understanding of latency coding. Based on recent mathematical results, we investigate theoretically the impact of channel noise on latency variability. For large numbers of ion channels, we derive the asymptotic distribution of latency, together with an explicit expression for its variance. Consequences in terms of information processing are studied with Fisher information in the Morris-Lecar model. A competition between sensitivity to input and precision is responsible for favoring two distinct regimes of latencies.

Design and analysis of molecular relay channels: an information theoretic approach

In this paper, we consider a molecular relay channel in which signal molecules are transmitted by a sender of communication, propagated, amplified, removed in the channel, and sensed by the receiver of communication that decodes the signal molecules. To understand and characterize the communication capacity of the molecular relay channel, we develop an information communication model that consists of the transmitter, channel, and receiver. Mutual information is used to quantify the amount of information that is transfered from the transmitter through the channel to the receiver. The method employed and results presented in this paper may help elucidate design principles of biological systems as well as help in the design and engineering of synthetic biological systems from the perspective of information theory.

IP3-induced cytosolic and nuclear Ca2+ signals in HL-1 atrial myocytes: possible role of IP3 receptor subtypes

HL-1 cells are the adult cardiac cell lines available that continuously divide while maintaining an atrial phenotype. Here we examined the expression and localization of inositol 1,4,5-trisphosphate receptor (IP(3)R) subtypes, and investigated how pattern of IP(3)-induced subcellular local Ca(2+) signaling is encoded by multiple IP(3)R subtypes in HL-1 cells. The type 1 IP(3)R (IP(3)R1) was expressed in the perinucleus with a diffuse pattern and the type 2 IP(3)R (IP(3)R2) was expressed in the cytosol with a punctate distribution. Extracellular ATP (1 mM) elicited transient intracellular Ca(2+) releases accompanied by a Ca(2+) oscillation, which was eliminated by the blocker of IP(3)Rs, 2-APB, and attenuated by ryanodine. Direct introduction of IP(3) into the permeabilized cells induced Ca(2+) transients with Ca(2+) oscillations at [Symbol: see text] 20 muM of IP(3), which was removed by the inhibition of IP(3)Rs using 2-APB and heparin. IP(3)-induced local Ca(2+) transients contained two distinct time courses: a rapid oscillation and a monophasic Ca(2+) transient. The magnitude of Ca(2+) oscillation was significantly larger in the cytosol than in the nucleus, while the monophasic Ca(2+) transient was more pronounced in the nucleus. These results provide evidence for the molecular and functional expression of IP(3)R1 and IP(3)R2 in HL-1 cells, and suggest that such distinct local Ca(2+) signaling may be correlated with the punctate distribution of IP(3)R2s in the cytosol and the diffuse localization of IP(3)R1 in the peri-nucleus.

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.

Importance of cell variability for calcium signaling in rat airway myocytes

Calcium signaling controls several essential physiological functions in different cell types. Hence, it is not surprising that different aspects of Ca(2+) dynamics are in the focus of in-depth and extensive investigations. Efforts concentrate on the development of proper theoretical models that would provide a unified description of Ca(2+) signaling. Remarkably, experimentally recorded Ca(2+) signals exhibit a rather large diversity, which can be observed irrespective of the cell type, measuring techniques, or the nature of the signal. Our goal in the present study therefore is to present a theoretical explanation for the variability observed in experiments, whereby we focus on caffeine-induced Ca(2+) responses in isolated airway myocytes. By employing a stochastic model, we first test whether the observed variability can be attributed to intrinsic fluctuations that are a common feature of biochemical reactions that govern Ca(2+) signalization. We find that stochastic effects, within ranges that correspond to actual conditions in the cell, are far too modest to explain the large diversity observed in experimental data. Foremost, we reveal that only cell variability in theoretical modeling can appropriately describe the observed diversity in single-cell responses.

Ca(2+) puffs originate from preestablished stable clusters of inositol trisphosphate receptors

Intracellular calcium ion (Ca(2+)) signaling crucially depends on the clustered organization of inositol trisphosphate receptors (IP(3)Rs) in the endoplasmic reticulum (ER) membrane. These ligand-gated ion channels liberate Ca(2+) to generate local signals known as Ca(2+) puffs. We tested the hypothesis that IP(3) itself elicits rapid clustering of IP(3)Rs by using flash photolysis of caged IP(3) in conjunction with high-resolution Ca(2+) imaging to monitor the activity and localization of individual IP(3)Rs within intact mammalian cells. Our results indicate that Ca(2+) puffs arising with latencies as short as 100 to 200 ms after photorelease of IP(3) already involve at least four IP(3)R channels, and that this number does not subsequently grow. Moreover, single active IP(3)Rs show limited mobility, and stochastic simulations suggest that aggregation of IP(3)Rs at puff sites by a diffusional trapping mechanism would require many seconds. We thus conclude that puff sites represent preestablished, stable clusters of IP(3)Rs and that functional IP(3)Rs are not readily diffusible within the ER membrane.

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.

Localization and socialization: experimental insights into the functional architecture of IP3 receptors

Inositol 1,4,5-trisphosphate (IP(3))-evoked Ca(2+) signals display great spatiotemporal malleability. This malleability depends on diversity in both the cellular organization and in situ functionality of IP(3) receptors (IP(3)Rs) that regulate Ca(2+) release from the endoplasmic reticulum (ER). Recent experimental data imply that these considerations are not independent, such that-as with other ion channels-the local organization of IP(3)Rs impacts their functionality, and reciprocally IP(3)R activity impacts their organization within native ER membranes. Here, we (i) review experimental data that lead to our understanding of the "functional architecture" of IP(3)Rs within the ER, (ii) propose an updated terminology to span the organizational hierarchy of IP(3)Rs observed in intact cells, and (iii) speculate on the physiological significance of IP(3)R socialization in Ca(2+) dynamics, and consequently the emerging need for modeling studies to move beyond gridded, planar, and static simulations of IP(3)R clustering even over short experimental timescales.

Reduction of calcium release site models via fast/slow analysis and iterative aggregation/disaggregation

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. In order to overcome the state-space explosion that occurs in such compositionally defined calcium release site models, we have implemented several automated procedures for model reduction using fast/slow analysis. After categorizing rate constants in the single channel model as either fast or slow, groups of states in the expanded release site model that are connected by fast transitions are lumped, and transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. For small problems these conditional probability distributions can be numerically calculated from the full model without approximation. For large problems the conditional probability distributions can be approximated without the construction of the full model by assuming rapid mixing of states connected by fast transitions. Alternatively, iterative aggregation/disaggregation may be employed to obtain reduced calcium release site models in a memory-efficient fashion. Benchmarking of several different iterative aggregation/disaggregation-based fast/slow reduction schemes establishes the effectiveness of automated calcium release site reduction utilizing the Koury-McAllister-Stewart method.

Targeting and clustering of IP3 receptors: key determinants of spatially organized Ca2+ signals

Inositol 1,4,5-trisphosphate receptors (IP3R) are intracellular Ca2+ channels that are almost ubiquitously expressed in animal cells. The spatiotemporal complexity of the Ca2+ signals evoked by IP3R underlies their versatility in cellular signaling. Here we review the mechanisms that contribute to the subcellular targeting of IP3R and the dynamic interplay between IP3R that underpin their ability to generate complex intracellular Ca2+ signals.

Effect of Ca(v)beta subunits on structural organization of Ca(v)1.2 calcium channels

BACKGROUND

Voltage-gated Ca(v)1.2 calcium channels play a crucial role in Ca(2+) signaling. The pore-forming alpha(1C) subunit is regulated by accessory Ca(v)beta subunits, cytoplasmic proteins of various size encoded by four different genes (Ca(v)beta(1)-beta(4)) and expressed in a tissue-specific manner.

METHODS AND RESULTS

Here we investigated the effect of three major Ca(v)beta types, beta(1b), beta(2d) and beta(3), on the structure of Ca(v)1.2 in the plasma membrane of live cells. Total internal reflection fluorescence microscopy showed that the tendency of Ca(v)1.2 to form clusters depends on the type of the Ca(v)beta subunit present. The highest density of Ca(v)1.2 clusters in the plasma membrane and the smallest cluster size were observed with neuronal/cardiac beta(1b) present. Ca(v)1.2 channels containing beta(3), the predominant Ca(v)beta subunit of vascular smooth muscle cells, were organized in a significantly smaller number of larger clusters. The inter- and intramolecular distances between alpha(1C) and Ca(v)beta in the plasma membrane of live cells were measured by three-color FRET microscopy. The results confirm that the proximity of Ca(v)1.2 channels in the plasma membrane depends on the Ca(v)beta type. The presence of different Ca(v)beta subunits does not result in significant differences in the intramolecular distance between the termini of alpha(1C), but significantly affects the distance between the termini of neighbor alpha(1C) subunits, which varies from 67 A with beta(1b) to 79 A with beta(3).

CONCLUSIONS

Thus, our results show that the structural organization of Ca(v)1.2 channels in the plasma membrane depends on the type of Ca(v)beta subunits present.

A computational analysis of localized Ca2+-dynamics generated by heterogeneous release sites

We investigate the role of heterogeneous expression of IP3R and RyR in generating diverse elementary Ca2+ signals. It has been shown empirically (Wojcikiewicz and Luo in Mol. Pharmacol. 53(4):656-662, 1998; Newton et al. in J. Biol. Chem. 269(46):28613-28619, 1994; Smedt et al. in Biochem. J. 322(Pt. 2):575-583, 1997) that tissues express various proportions of IP3 and RyR isoforms and this expression is dynamically regulated (Parrington et al. in Dev. Biol. 203(2):451-461, 1998; Fissore et al. in Biol. Reprod. 60(1):49-57, 1999; Tovey et al. in J. Cell Sci. 114(Pt. 22):3979-3989, 2001). Although many previous theoretical studies have investigated the dynamics of localized calcium release sites (Swillens et al. in Proc. Natl. Acad. Sci. U.S.A. 96(24):13750-13755, 1999; Shuai and Jung in Proc. Natl. Acad. Sci. U.S.A. 100(2):506-510, 2003a; Shuai and Jung in Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 67(3 Pt. 1):031905, 2003b; Thul and Falcke in Biophys. J. 86(5):2660-2673, 2004; DeRemigio and Smith in Cell Calcium 38(2):73-86, 2005; Nguyen et al. in Bull. Math. Biol. 67(3):393-432, 2005), so far all such studies focused on release sites consisting of identical channel types. We have extended an existing mathematical model (Nguyen et al. in Bull. Math. Biol. 67(3):393-432, 2005) to release sites with two (or more) receptor types, each with its distinct channel kinetics. Mathematically, the release site is represented by a transition probability matrix for a collection of nonidentical stochastically gating channels coupled through a shared Ca2+ domain. We demonstrate that under certain conditions a previously defined mean-field approximation of the coupling strength does not accurately reproduce the release site dynamics. We develop a novel approximation and establish that its performance in these instances is superior. We use this mathematical framework to study the effect of heterogeneity in the Ca2+-regulation of two colocalized channel types on the release site dynamics. We consider release sites consisting of channels with both Ca2+-activation and inactivation ("four-state channels") and channels with Ca2+-activation only ("two-state channels") and show that for the appropriate parameter values, synchronous channel openings within a release site with any proportion of two-state to four-state channels are possible, however, the larger the proportion of two-state channels, the more sensitive the dynamics are to the exact spatial positioning of the channels and the distance between channels. Specifically, the clustering of even a small number of two-state channels interferes with puff/spark termination and increases puff durations or leads to a tonic response.

Global noise and oscillations in clustered excitable media

We study the effects of global noise on waves in heterogeneous, spatially clustered, reaction-diffusion systems with possible applications to calcium signaling. We first discuss how clustering of the excitability determines the dynamics by shifting bifurcation points and creating new oscillatory solutions. We then consider the specific situation, where intrinsic noise, due to the smallness of the excitable patches, destroys the global oscillatory state. We show that additional small global fluctuations, however, can partially restore temporal and spatial coherence of the oscillatory signal.

What is the role of astrocyte calcium in neurophysiology?

Astrocytes comprise approximately half of the volume of the adult mammalian brain and are the primary neuronal structural and trophic supportive elements. Astrocytes are organized into distinct nonoverlapping domains and extend elaborate and dense fine processes that interact intimately with synapses and cerebrovasculature. The recognition in the mid 1990s that astrocytes undergo elevations in intracellular calcium concentration following activation of G protein-coupled receptors by synaptically released neurotransmitters demonstrated not only that astrocytes display a form of excitability but also that astrocytes may be active participants in brain information processing. The roles that astrocytic calcium elevations play in neurophysiology and especially in modulation of neuronal activity have been intensely researched in recent years. This review will summarize the current understanding of the function of astrocytic calcium signaling in neurophysiological processes and discuss areas where the role of astrocytes remains controversial and will therefore benefit from further study.

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

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

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

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

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

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

The clustering of inositol 1,4,5-trisphosphate (IP(3)) receptors is triggered by IP(3) binding and facilitated by depletion of the Ca(2+) store

The inositol 1,4,5-trisphosphate receptors (IP(3)Rs) form clusters following agonist stimulation, but its mechanism remains controversial. In this study, we visualized the clustering of green fluorescent protein (GFP)-tagged type 3 IP(3)R (GFP-IP(3)R3) in cultured living cells using confocal microscopy. Stimulation with ATP evoked GFP-IP(3)R3 clustering not only in cells with replete Ca(2+)-stores but also in cells with depleted Ca(2+) stores. Thapsigargin (ThG) and ionomycin failed to mimic the ATP-induced cluster formation despite the continuous elevation of intracellular Ca(2+) concentration ([Ca(2+)](i)). Application of IP(3) caused GFP-IP(3)R3 clustering in permeabilized cells, and the response was completely inhibited by heparin, a competitive inhibitor of IP(3)R. Experiments using LIBRAv, an IP(3) biosensor, showed that ATP significantly stimulated IP(3) generation even in store-depleted cells. We also found that pretreatment with ThG accelerated or enhanced the ATP-induced clustering in both the presence and absence of extracellular Ca(2+). When permeabilized cells were stimulated with the threshold of IP(3), the GFP-IP(3)R3 clustering clearly occurred in Ca(2+)-free medium but not in Ca(2+)-containing medium. These results strongly support the hypothesis that the agonist-induced clustering of IP(3)R is triggered by IP(3) binding, rather than [Ca(2+)](i) elevation. Although depletion of the Ca(2+) store by itself does not cause the clustering, it may increase the sensitivity of IP(3)R to cluster formation, leading to facilitation of IP(3)-triggered clustering.

Stochastic aspects of oscillatory Ca2+ dynamics in hepatocytes

Signal-induced Ca(2+) oscillations have been observed in many cell types and play a primary role in cell physiology. Although it is the regular character of these oscillations that first catches the attention, a closer look at time series of Ca(2+) increases reveals that the fluctuations on the period during individual spike trains are far from negligible. Here, we perform a statistical analysis of the regularity of Ca(2+) oscillations in norepinephrine-stimulated hepatocytes and find that the coefficient of variation lies between 10% and 15%. Stochastic simulations based on Gillespie's algorithm and considering realistic numbers of Ca(2+) ions and inositol trisphosphate (InsP(3)) receptors account for this variability if the receptors are assumed to be grouped in clusters of a few tens of channels. Given the relatively small number of clusters ( approximately 200), the model predicts the existence of repetitive spikes induced by fluctuations (stochastic resonance). Oscillations of this type are found in hepatocytes at subthreshold concentrations of norepinephrine. We next predict with the model that the isoforms of the InsP(3) receptor can affect the variability of the oscillations. In contrast, possible accompanying InsP(3) oscillations have no impact on the robustness of signal-induced repetitive Ca(2+) spikes.

How does intracellular Ca2+ oscillate: by chance or by the clock?

Ca2+ oscillations have been considered to obey deterministic dynamics for almost two decades. We show for four cell types that Ca2+ oscillations are instead a sequence of random spikes. The standard deviation of the interspike intervals (ISIs) of individual spike trains is similar to the average ISI; it increases approximately linearly with the average ISI; and consecutive ISIs are uncorrelated. Decreasing the effective diffusion coefficient of free Ca2+ using Ca2+ buffers increases the average ISI and the standard deviation in agreement with the idea that individual spikes are caused by random wave nucleation. Array-enhanced coherence resonance leads to regular Ca2+ oscillations with small standard deviation of ISIs.

ATP and adenosine trigger the interaction of plasma membrane IP3 receptors with protein kinase A in oviductal ciliated cells

We have demonstrated that adenosine did not produce any change of intracellular free Ca2+ concentration ([Ca2+]i) in oviductal ciliated cells; however, it increased the ATP-induced Ca2+ influx through the activation of protein kinase A (PKA). Uncaging of IP3 and cAMP triggered a larger Ca2+ influx than did IP3 alone. Furthermore, the IP3 effect was abolished by Xestospongin C, an IP3 receptor blocker. Whole-cell recordings demonstrated the presence of an ATP-induced Ca2+ current, and the addition of adenosine increased the peak of this current. This effect was not observed in the presence of H-89, a PKA inhibitor. Using excised macro-patches of plasma membrane, IP3 generated a current, which was higher in the presence of the catalytic PKA subunit and this current was blocked by Xestospongin C. We show here that activation of plasma membrane IP3 receptors directly triggers Ca2+ influx in response to ATP and that these receptors are modulated by adenosine-activated PKA.

Noise-induced dynamics in the mixed-feedback-loop network motif

In this paper, we present a stochastic model for the mixed-feedback loop (MFL), a motif found in integrated cellular networks of transcription regulation and protein-protein interaction. Previous bifurcation analysis indicates that this motif can serve as a bistable switch or a clock. We investigate how extrinsic and intrinsic noise affects its dynamic behaviors systematically. We find that this motif can exploit noise to enrich its dynamical performance. When the MFL is in the bistable region, under fluctuation of extrinsic noise, the MFL system can switch from one steady state to the other and meanwhile one protein's production is amplified for more than three orders of magnitude. Further, from an engineering perspective, this noise-based switch and amplifier for gene expression is very easy to control. Without extrinsic noise, spontaneous transition between states occurs as the consequence of intrinsic noise. Such a switch is controlled by the parameters and system size. On the other hand, intrinsic noise can induce sustained stochastic oscillation when the corresponding deterministic system does not oscillate. Such stochastic oscillation shows the best performance at an optimal noise level, indicating the occurrence of intrinsic noise stochastic resonance which can contribute to the robustness of this oscillator. When considering the effects of extrinsic noise near bifurcation points, a similar phenomenon of extrinsic noise stochastic resonance is unveiled.

Internal noise-sustained circadian rhythms in a Drosophila model

Circadian rhythmic processes, mainly regulated by gene expression at the molecular level, have inherent stochasticity. Their robustness or resistance to internal noise has been extensively investigated by most of the previous studies. This work focuses on the constructive roles of internal noise in a reduced Drosophila model, which incorporates negative and positive feedback loops, each with a time delay. It is shown that internal noise sustains reliable oscillations with periods close to 24 h in a region of parameter space, where the deterministic kinetics would evolve to a stable steady state. The amplitudes of noise-sustained oscillations are significantly affected by the variation of internal noise level, and the best performance of the oscillations could be found at an optimal noise intensity, indicating the occurrence of intrinsic coherence resonance. In the oscillatory region of the deterministic model, the coherence of noisy circadian oscillations is suppressed by internal noise, while the period remains nearly constant over a large range of noise intensity, demonstrating robustness of the Drosophila model for circadian rhythms to intrinsic noise. In addition, the effects of time delay in the positive feedback on the oscillations are also investigated. It is found that the time delay could efficiently tune the performance of the noise-sustained oscillations. These results might aid understanding of the exploitation of intracellular noise in biochemical and genetic regulatory systems.

Distribution of inositol-1,4,5-trisphosphate receptor isotypes and ryanodine receptor isotypes during maturation of the rat hippocampus

Activation of inositol-1,4,5-trisphosphate receptors (InsP(3)Rs) and ryanodine receptors (RyRs) can lead to the release of Ca(2+) from intracellular stores and propagating Ca(2+) waves. Previous studies of these proteins in neurons have focused on their distribution in adult tissue, whereas, recent functional studies have examined neural tissue extracted from prenatal and young postnatal animals. In this study we examined the distribution of InsP(3)R isotypes 1-3 and RyR isotypes 1-3 in rat hippocampus during postnatal maturation, with a focus on InsP(3)R1 because it is most prominent in the hippocampus. InsP(3)R1 was observed in pyramidal cells and granule cells, InsP(3)R2 immunoreactivity was observed in perivascular astrocytes and endothelial cells, and InsP(3)R3 immunoreactivity was detected in axon terminals located in stratum pyramidale of CA1 and microvessels in stratum radiatum. RyR1 immunolabeling was enriched in CA1, RyR2 was most intense in CA3 and the dentate gyrus, and RyR3 immunolabeling was detected in all subfields of the hippocampus, but was most intense in stratum lacunosum-moleculare. During maturation from 2 to 10 weeks of age there was a shift in InsP(3)R1 immunoreactivity from a high density in the proximal apical dendrites to a uniform distribution along the dendrites. Independent of age, InsP(3)R1 immunoreactivity was observed to form clusters within the primary apical dendrite and at dendritic bifurcations of pyramidal neurons. As CA1 pyramidal neurons matured, InsP(3)R1 was often co-localized with the Ca(2+) binding protein calbindin D-28k. In contrast, InsP(3)R1 immunolabel was never co-localized with calbindin D-28k immunopositive interneurons located outside of stratum pyramidale or with parvalbumin, typically found in hippocampal basket cells, suggesting that InsP(3)R1s do not play a role in internal Ca(2+) release in these interneurons. These findings should help to interpret past functional studies and inform future studies examining the characteristics and consequences of InsP(3)R-mediated internal Ca(2+) release and Ca(2+) waves in hippocampal neurons.

Reversible interactions between smooth domains of the endoplasmic reticulum and mitochondria are regulated by physiological cytosolic Ca2+ levels

The 3F3A monoclonal antibody to autocrine motility factor receptor (AMFR) labels mitochondria-associated smooth endoplasmic reticulum (ER) tubules. siRNA down-regulation of AMFR expression reduces mitochondria-associated 3F3A labelling. The 3F3A-labelled ER domain does not overlap with reticulon-labelled ER tubules, the nuclear membrane or perinuclear ER markers and only partially overlaps with the translocon component Sec61alpha. Upon overexpression of FLAG-tagged AMFR, 3F3A labelling is mitochondria associated, excluded from the perinuclear ER and co-distributes with reticulon. 3F3A labelling therefore defines a distinct mitochondria-associated ER domain. Elevation of free cytosolic Ca(2+) levels with ionomycin promotes dissociation of 3F3A-labelled tubules from mitochondria and, judged by electron microscopy, disrupts close contacts (<50 nm) between smooth ER tubules and mitochondria. The ER tubule-mitochondria association is similarly disrupted upon thapsigargin-induced release of ER Ca(2+) stores or purinergic receptor stimulation by ATP. The inositol (1,4,5)-trisphosphate [Ins(1,4,5)P(3)] receptor (IP3R) colocalises to 3F3A-labelled mitochondria-associated ER tubules, and conditions that induce ER tubule-mitochondria dissociation disrupt continuity between 3F3A- and IP3R-labelled ER domains. RAS-transformed NIH-3T3 cells have increased basal cytosolic Ca(2+) levels and show dissociation of the 3F3A-labelled, but not IP3R-labelled, ER from mitochondria. Our data indicate that regulation of the ER-mitochondria association by free cytosolic Ca(2+) is a characteristic of smooth ER domains and that multiple mechanisms regulate the interaction between these organelles.

A theoretical study on activation of transcription factor modulated by intracellular Ca2+ oscillations

This work presents both deterministic and stochastic models of genetic expression modulated by intracellular calcium (Ca2+) oscillations, based on macroscopic differential equations and chemical Langevin equations, respectively. In deterministic case, the oscillations of intracellular Ca2+ decrease the effective Ca2+ threshold for the activation of transcriptional activator (TF-A). The average activation of TF-A increases with the increase of the average amplitude of intracellular Ca2+ oscillations, but decreases with the increase of the period of intracellular Ca2+ oscillations, which are qualitatively consistent with the experimental results on the gene expression in lymphocytes. In stochastic case, it is found that a large internal fluctuation of the biochemical reaction can enhance gene expression efficiency specifically at a low level of external stimulations or at a small rate of TF-A dimer phosphorylation activated by Ca2+, which reduces the threshold of the average intracellular Ca2+ concentration for gene expression.

Selective stimulation of astrocyte calcium in situ does not affect neuronal excitatory synaptic activity

Astrocytes are considered the third component of the synapse, responding to neurotransmitter release from synaptic terminals and releasing gliotransmitters--including glutamate--in a Ca(2+)-dependent manner to affect neuronal synaptic activity. Many studies reporting astrocyte-driven neuronal activity have evoked astrocyte Ca(2+) increases by application of endogenous ligands that directly activate neuronal receptors, making astrocyte contribution to neuronal effect(s) difficult to determine. We have made transgenic mice that express a Gq-coupled receptor only in astrocytes to evoke astrocyte Ca(2+) increases using an agonist that does not bind endogenous receptors in brain. By recording from CA1 pyramidal cells in acute hippocampal slices from these mice, we demonstrate that widespread Ca(2+) elevations in 80%-90% of stratum radiatum astrocytes do not increase neuronal Ca(2+), produce neuronal slow inward currents, or affect excitatory synaptic activity. Our findings call into question the developing consensus that Ca(2+)-dependent glutamate release by astrocytes directly affects neuronal synaptic activity in situ.

A kinetic model of single and clustered IP3 receptors in the absence of Ca2+ feedback

Ca2+ liberation through inositol 1,4,5-trisphosphate receptor (IP3R) channels generates complex patterns of spatiotemporal cellular Ca2+ signals owing to the biphasic modulation of channel gating by Ca2+ itself. These processes have been extensively studied in Xenopus oocytes, where imaging studies have revealed local Ca2+ signals ("puffs") arising from clusters of IP3R, and patch-clamp studies on isolated oocyte nuclei have yielded extensive data on IP3R gating kinetics. To bridge these two levels of experimental data, we developed an IP3R model and applied stochastic simulation and transition matrix theory to predict the behavior of individual and clustered IP3R channels. The channel model consists of four identical, independent subunits, each of which has an IP3-binding site together with one activating and one inactivating Ca2+-binding site. The channel opens when at least three subunits undergo a conformational change to an "active" state after binding IP3 and Ca2+. The model successfully reproduces patch-clamp data; including the dependence of open probability, mean open duration, and mean closed duration on [IP3] and [Ca2+]. Notably, the biexponential distribution of open-time duration and the dependence of mean open time on [Ca2+] are explained by populations of openings involving either three or four active subunits. As a first step toward applying the single IP3R model to describe cellular responses, we then simulated measurements of puff latency after step increases of [IP3]. Assuming that stochastic opening of a single IP3R at basal cytosolic [Ca2+] and any given [IP3] has a high probability of rapidly triggering neighboring channels by calcium-induced calcium release to evoke a puff, optimal correspondence with experimental data of puff latencies after photorelease of IP3 was obtained when the cluster contained a total of 40-70 IP3Rs.

Hybrid stochastic and deterministic simulations of calcium blips

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

Simulations of intracellular calcium release dynamics in response to a high-intensity, ultrashort electric pulse

Numerical simulations for electrically induced, intracellular calcium release from the endoplasmic reticulum are reported. A two-step model is used for self-consistency. Distributed electrical circuit representation coupled with the Smoluchowski equation yields the ER membrane nanoporation for calcium outflow based on a numerical simulation. This is combined with the continuum Li-Rinzel model and drift diffusion for calcium dynamics. Our results are shown to be in agreement with reported calcium release data. A modest increase (rough doubling) of the cellular calcium is predicted in the absence of extra-cellular calcium. In particular, the applied field of 15 kV/cm with 60 ns pulse duration makes for a strong comparison. No oscillations are predicted and the net recovery period of about 5 min are both in agreement with published experimental results. A quantitative explanation for the lack of such oscillatory behavior, based on the density dependent calcium fluxes, is also provided.

Structural-diversity-enhanced cellular ability to detect subthreshold extracellular signals

We study the influence of the structural diversity of cells on subthreshold agonist signals in coupled hepatocytes systems. The variance of the cellular structural parameter sigma is employed to characterize the structural diversity. It is found that structural diversity enhances the cellular ability to detect extracellular weak signals through intracellular Ca2+ oscillations and the regularity of Ca2+ spikes undergoes a maximum with a variation of sigma, indicating the occurrence of structural-diversity-induced coherence resonance. Furthermore, the effects of the level of subthreshold stimulus and junctional coupling strength on the behavior of Ca2+ dynamics are also considered. Analysis indicates that these phenomena have inherent relevance to both the bifurcation feature of a single cell and intercellular interaction through junctional coupling. Our findings may exhibit that structural diversity plays a constructive role in biological systems.

The dynamics of luminal depletion and the stochastic gating of Ca2+-activated Ca2+ channels and release sites

Single channel models of intracellular calcium (Ca(2+)) channels such as the 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by constant background cytosolic [Ca(2+)]. This assumption neglects the fact that Ca(2+) released by open channels may influence subsequent gating through the processes of Ca(2+)-activation or inactivation. Similarly, the influence of the dynamics of luminal depletion on the stochastic gating of intracellular Ca(2+) channels is often neglected, in spite of the fact that the sarco/endoplasmic reticulum [Ca(2+)] near the luminal face of intracellular Ca(2+) channels influences the driving force for Ca(2+), the rate of Ca(2+) release, and the magnitude and time course of the consequent increase in cytosolic domain [Ca(2+)]. Here we analyze how the steady-state open probability of several minimal Ca(2+)-regulated Ca(2+) channel models depends on the conductance of the channel and the time constants for the relaxation of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] to the bulk [Ca(2+)] of both compartments. Our approach includes Monte Carlo simulation as well as numerical solution of a system of advection-reaction equations for the multivariate probability density of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] conditioned on each state of the stochastically gating channel. Both methods are subsequently used to study the role of luminal depletion in the dynamics of Ca(2+) puff/spark termination in release sites composed of Ca(2+) channels that are activated, but not inactivated, by cytosolic Ca(2+). The probability density approach shows that such minimal Ca(2+) release site models may exhibit puff/spark-like dynamics in either of two distinct parameter regimes. In one case, puffs/spark termination is due to the process of stochastic attrition and facilitated by rapid Ca(2+) domain collapse [cf. DeRemigio, H., Smith, G., 2005. The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain. Cell Calcium 38, 73-86]. In the second case, puff/spark termination is promoted by the local depletion of luminal Ca(2+).

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.

Excitation-induced dynamics of external pH pattern in Chara corallina cells and its dependence on external calcium concentration

The influence of cell excitation and external calcium level on the dynamics of light-induced pH bands along the length of Chara corallina cells is studied in the present paper. Generation of an action potential (AP) transiently quenched these pH patterns, which was more pronounced at 0.05-0.1 mM Ca2+ than at higher concentrations of Ca2+ (0.6-2 mM) in the medium. After transient smoothing of the pH bands, some alkaline peaks reemerged at slightly shifted positions in media with low Ca2+ concentrations, while at high Ca2+ concentrations, the alkaline spots reappeared exactly at their initial positions. This Ca2+ dependency has been revealed by both digital imaging and pH microelectrodes. The stabilizing effect of external Ca2+ on the locations of recovering alkaline peaks is supposedly due to formation of a physically heterogeneous environment around the cell owing to precipitation of CaCO3 in the alkaline zones at high Ca2+ during illumination. The elevation of local pH by dissolving CaCO3 facilitates the reappearance of alkaline spots at their initial locations after temporal suppression caused by cell excitation. At low Ca2+ concentrations, when the solubility product of CaCO3 is not attained, the alkaline peaks are not stabilized by CaCO3 dissolution and may appear at random locations.