Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel

Computational modeling of anoctamin 1 calcium-activated chloride channels as pacemaker channels in interstitial cells of Cajal

Interstitial cells of Cajal (ICC) act as pacemaker cells in the gastrointestinal tract by generating electrical slow waves to regulate rhythmic smooth muscle contractions. Intrinsic Ca(2+) oscillations in ICC appear to produce the slow waves by activating pacemaker currents, currently thought to be carried by the Ca(2+)-activated Cl(-) channel anoctamin 1 (Ano1). In this article we present a novel model of small intestinal ICC pacemaker activity that incorporates store-operated Ca(2+) entry and a new model of Ano1 current. A series of simulations were carried out with the ICC model to investigate current controversies about the reversal potential of the Ano1 Cl(-) current in ICC and to predict the characteristics of the other ion channels that are necessary to generate slow waves. The model results show that Ano1 is a plausible pacemaker channel when coupled to a store-operated Ca(2+) channel but suggest that small cyclical depolarizations may still occur in ICC in Ano1 knockout mice. The results predict that voltage-dependent Ca(2+) current is likely to be negligible during the slow wave plateau phase. The model shows that the Cl(-) equilibrium potential is an important modulator of slow wave morphology, highlighting the need for a better understanding of Cl(-) dynamics in ICC.

FINITE ELEMENT MODEL TO STUDY THE MECHANICS OF CALCIUM REGULATION IN OOCYTE

At various stages of fertilization specific spatial and temporal patterns of Ca 2+ are required for oocyte maturation. It is crucial to understand the mechanics of Ca 2+ regulation in cytosol of oocytes, in order to have better understanding of fertilization process. In this paper, a finite element model of cytosolic calcium regulation in oocyte has been developed for a two-dimensional unsteady state case. The model incorporates the important biophysical processes like diffusion, reaction, leak from endoplasmic recticulum (ER), efflux from cytosol to ER via sarco-ER calcium adenosine triphosphate (SERCA) pumps, buffers and sodium calcium exchanger. Appropriate boundary conditions have been framed. The effect of source, buffer, sodium calcium exchanger, etc. on spatial and temporal patterns of calcium in oocyte have been studied with the help of numerical results.

From free to effective diffusion coefficients in fluorescence correlation spectroscopy experiments

Diffusion is one of the main transport processes that occur inside cells determining the spatial and time distribution of relevant action molecules. In most cases these molecules not only diffuse but also interact with others as they get transported. When these interactions occur faster than diffusion the resulting transport can be characterized by "effective diffusion coefficients" that depend on both the reaction rates and the "free" diffusion coefficients. Fluorescence correlation spectroscopy (FCS) gives information on effective rather than free diffusion coefficients under this condition. In the present paper we investigate what coefficients can be drawn from FCS experiments for a wide range of values of the ratio of reaction to diffusion time scales, using different fitting functions. We find that the effective coefficients can be inferred with relatively small errors even when the condition of fast reactions does not exactly hold. Since the diffusion time scale depends on the size of the observation volume and the reaction time scale depends on concentrations, we also discuss how by changing either one or the other property one can switch between the two limits and extract more information on the system under study.

Permeant calcium ion feed-through regulation of single inositol 1,4,5-trisphosphate receptor channel gating

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) Ca(2+) release channel plays a central role in the generation and modulation of intracellular Ca(2+) signals, and is intricately regulated by multiple mechanisms including cytoplasmic ligand (InsP(3), free Ca(2+), free ATP(4-)) binding, posttranslational modifications, and interactions with cytoplasmic and endoplasmic reticulum (ER) luminal proteins. However, regulation of InsP(3)R channel activity by free Ca(2+) in the ER lumen ([Ca(2+)](ER)) remains poorly understood because of limitations of Ca(2+) flux measurements and imaging techniques. Here, we used nuclear patch-clamp experiments in excised luminal-side-out configuration with perfusion solution exchange to study the effects of [Ca(2+)](ER) on homotetrameric rat type 3 InsP(3)R channel activity. In optimal [Ca(2+)](i) and subsaturating [InsP(3)], jumps of [Ca(2+)](ER) from 70 nM to 300 µM reduced channel activity significantly. This inhibition was abrogated by saturating InsP(3) but restored when [Ca(2+)](ER) was raised to 1.1 mM. In suboptimal [Ca(2+)](i), jumps of [Ca(2+)](ER) (70 nM to 300 µM) enhanced channel activity. Thus, [Ca(2+)](ER) effects on channel activity exhibited a biphasic dependence on [Ca(2+)](i). In addition, the effect of high [Ca(2+)](ER) was attenuated when a voltage was applied to oppose Ca(2+) flux through the channel. These observations can be accounted for by Ca(2+) flux driven through the open InsP(3)R channel by [Ca(2+)](ER), raising local [Ca(2+)](i) around the channel to regulate its activity through its cytoplasmic regulatory Ca(2+)-binding sites. Importantly, [Ca(2+)](ER) regulation of InsP(3)R channel activity depended on cytoplasmic Ca(2+)-buffering conditions: it was more pronounced when [Ca(2+)](i) was weakly buffered but completely abolished in strong Ca(2+)-buffering conditions. With strong cytoplasmic buffering and Ca(2+) flux sufficiently reduced by applied voltage, both activation and inhibition of InsP(3)R channel gating by physiological levels of [Ca(2+)](ER) were completely abolished. Collectively, these results rule out Ca(2+) regulation of channel activity by direct binding to the luminal aspect of the channel.

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

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

Combined computational and experimental approaches to understanding the Ca(2+) regulatory network in neurons

Ca(2+) is a ubiquitous signaling ion that regulates a variety of neuronal functions by binding to and altering the state of effector proteins. Spatial relationships and temporal dynamics of Ca(2+) elevations determine many cellular responses of neurons to chemical and electrical stimulation. There is a wealth of information regarding the properties and distribution of Ca(2+) channels, pumps, exchangers, and buffers that participate in Ca(2+) regulation. At the same time, new imaging techniques permit characterization of evoked Ca(2+) signals with increasing spatial and temporal resolution. However, understanding the mechanistic link between functional properties of Ca(2+) handling proteins and the stimulus-evoked Ca(2+) signals they orchestrate requires consideration of the way Ca(2+) handling mechanisms operate together as a system in native cells. A wide array of biophysical modeling approaches is available for studying this problem and can be used in a variety of ways. Models can be useful to explain the behavior of complex systems, to evaluate the role of individual Ca(2+) handling mechanisms, to extract valuable parameters, and to generate predictions that can be validated experimentally. In this review, we discuss recent advances in understanding the underlying mechanisms of Ca(2+) signaling in neurons via mathematical modeling. We emphasize the value of developing realistic models based on experimentally validated descriptions of Ca(2+) transport and buffering that can be tested and refined through new experiments to develop increasingly accurate biophysical descriptions of Ca(2+) signaling in neurons.

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

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

Lobe specific Ca2+-calmodulin nano-domain in neuronal spines: a single molecule level analysis

Calmodulin (CaM) is a ubiquitous Ca²⁺ buffer and second messenger that affects cellular function as diverse as cardiac excitability, synaptic plasticity, and gene transcription. In CA1 pyramidal neurons, CaM regulates two opposing Ca²⁺-dependent processes that underlie memory formation: long-term potentiation (LTP) and long-term depression (LTD). Induction of LTP and LTD require activation of Ca²⁺-CaM-dependent enzymes: Ca²⁺/CaM-dependent kinase II (CaMKII) and calcineurin, respectively. Yet, it remains unclear as to how Ca²⁺ and CaM produce these two opposing effects, LTP and LTD. CaM binds 4 Ca²⁺ ions: two in its N-terminal lobe and two in its C-terminal lobe. Experimental studies have shown that the N- and C-terminal lobes of CaM have different binding kinetics toward Ca²⁺ and its downstream targets. This may suggest that each lobe of CaM differentially responds to Ca²⁺ signal patterns. Here, we use a novel event-driven particle-based Monte Carlo simulation and statistical point pattern analysis to explore the spatial and temporal dynamics of lobe-specific Ca²⁺-CaM interaction at the single molecule level. We show that the N-lobe of CaM, but not the C-lobe, exhibits a nano-scale domain of activation that is highly sensitive to the location of Ca²⁺ channels, and to the microscopic injection rate of Ca²⁺ ions. We also demonstrate that Ca²⁺ saturation takes place via two different pathways depending on the Ca²⁺ injection rate, one dominated by the N-terminal lobe, and the other one by the C-terminal lobe. Taken together, these results suggest that the two lobes of CaM function as distinct Ca²⁺ sensors that can differentially transduce Ca²⁺ influx to downstream targets. We discuss a possible role of the N-terminal lobe-specific Ca²⁺-CaM nano-domain in CaMKII activation required for the induction of synaptic plasticity.

Calmodulin is a versatile Ca²⁺ signal mediator and a buffer in a wide variety of body organs including the heart and brain. In the brain, calmodulin regulates intracellular molecular processes that change the strength of connectivity between neurons, thus contributing to various brain functions including memory formation. The exact molecular mechanism as to how calmodulin regulates these processes is not yet known. Interestingly, in other excitable tissues, including the heart, each of two lobes of calmodulin responds differentially toward Ca²⁺ influx and toward its target molecules (e.g., ion channels). This way, calmodulin precisely controls the Ca²⁺ dynamics of the cell. We wish to test if a similar mechanism may be operational in neurons so that two lobes of calmodulin interact differentially with Ca²⁺ ions to activate different downstream molecules that control the strength of connections between neurons. We constructed a detailed simulation of calmodulin that allows us to keep track of its interactions with Ca²⁺ ions and target proteins at the single molecule level. The simulation predicts that two lobes of calmodulin respond differentially to Ca²⁺ influx both in space and in time. This work opens a door to future experimental testing of the lobe-specific control of neural function by calmodulin.

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.

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.

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.

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.

Ryanodine receptor allosteric coupling and the dynamics of calcium sparks

Puffs and sparks are localized intracellular Ca(2+) elevations that arise from the cooperative activity of Ca(2+)-regulated inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or the sarcoplasmic reticulum. While the synchronous gating of Ca(2+)-regulated Ca(2+) channels can be mediated entirely though the buffered diffusion of intracellular Ca(2+), interprotein allosteric interactions also contribute to the dynamics of ryanodine receptor (RyR) gating and Ca(2+) sparks. In this article, Markov chain models of Ca(2+) release sites are used to investigate how the statistics of Ca(2+) spark generation and termination are related to the coupling of RyRs via local [Ca(2+)] changes and allosteric interactions. Allosteric interactions are included in a manner that promotes the synchronous gating of channels by stabilizing neighboring closed-closed and/or open-open channel pairs. When the strength of Ca(2+)-mediated channel coupling is systematically varied (e.g., by changing the Ca(2+) buffer concentration), simulations that include synchronizing allosteric interactions often exhibit more robust Ca(2+) sparks; however, for some Ca(2+) coupling strengths the sparks are less robust. We find no evidence that the distribution of spark durations can be used to distinguish between allosteric interactions that stabilize closed channel pairs, open channel pairs, or both in a balanced fashion. On the other hand, the changes in spark duration, interspark interval, and frequency observed when allosteric interactions that stabilize closed channel pairs are gradually removed from simulations are qualitatively different than the changes observed when open or both closed and open channel pairs are stabilized. Thus, our simulations clarify how changes in spark statistics due to pharmacological washout of the accessory proteins mediating allosteric coupling may indicate the type of synchronizing allosteric interactions exhibited by physically coupled RyRs. We also investigate the validity of a mean-field reduction applicable to the dynamics of a ryanodine receptor cluster coupled via local [Ca(2+)] and allosteric interactions. In addition to facilitating parameter studies of the effect of allosteric coupling on spark statistics, the derivation of the mean-field model establishes the correct functional form for cooperativity factors representing the coupled gating of RyRs. This mean-field formulation is well suited for use in computationally efficient whole cell simulations of excitation-contraction coupling.

EONS: an online synaptic modeling platform

Chemical synapses, although representing the smallest unit of communication between two neurons in the nervous system constitute a complex ensemble of mechanisms. Understanding these mechanisms and the way synaptic transmission occurs is critical for our comprehension of CNS functions in general and learning and memory in particular. Here we describe a modeling platform called EONS (Elementary Object of Neural System) accessible online, which allows neuroscientists throughout the world to study qualitatively, but also quantitatively the relative contributions of diverse mechanisms underlying synaptic efficacy: the relevance of each and every elements that comprise a synapse, the interactions between these components and their subcellular distribution, as well as the influence of synaptic geometry (presynaptic terminal, cleft and postsynaptic density).

Quasi-steady approximation for ion channel currents

Currents through ion channels are determined (among other parameters) by the concentration difference across the membrane containing the channel and the diffusive transport of the conducted ion toward the channel and away from it. Calculation of the current requires solving the diffusion equation around the channel. Here, we provide a quasi-steady approximation for the current and the local concentrations at the channel together with formulas linking the current and local concentrations at the channel to bulk concentrations and diffusion properties of the compartments.

Determining calcium concentration in heterogeneous model systems using multiple indicators

Intracellular free calcium concentrations ([Ca2+]i) are assessed by measuring indicator fluorescence in entire cells or subcellular regions using fluorescence microscopy. [Ca2+]i is calculated using equations which link fluorescence intensities (or intensity ratios) to calcium concentrations [G. Grynkiewicz, M. Poenie, R.Y. Tsien, A new generation of Ca2+ indicators with greatly improved fluorescence properties, J. Biol. Chem. 260 (1985) 3440-3450]. However, if calcium ions are heterogeneously distributed within a region of interest, then the observed average fluorescence intensity may not reflect average [Ca2+]i. We assessed potential calcium determination errors in mathematical and experimental models consisting of 'low' and 'high' calcium compartments, using indicators with different affinity for calcium. [Ca2+] calculated using average fluorescence intensity was lower than the actual mean concentrations. Low affinity indicators reported higher (more accurate) values than their high affinity counterparts. To estimate compartment dimensions and respective [Ca2+], we extended the standard approach by using different indicator responses to the same [Ca2+]. While two indicators were sufficient to provide a partial characterization of two-compartment model systems, the use of three or more indicators offered full description of the model provided compartmental [Ca2+] were within the indicator sensitivity ranges. These results show that uneven calcium distribution causes underestimation of actual [Ca2+], and offers novel approaches to estimating calcium heterogeneity.

Ca2+ from one or two channels controls fusion of a single vesicle at the frog neuromuscular junction

Neurotransmitter release is triggered by the cooperative action of approximately five Ca2+ ions entering the presynaptic terminal through Ca2+ channels. Depending on the organization of the active zone (AZ), influx through one or many channels may be needed to cause fusion of a vesicle. Using a combination of experiments and modeling, we examined the number of channels that contribute Ca2+ for fusion of a single vesicle in a frog neuromuscular AZ. We compared Ca2+ influx to neurotransmitter release by measuring presynaptic action potential-evoked (AP-evoked) Ca2+ transients simultaneously with postsynaptic potentials. Ca2+ influx was manipulated by changing extracellular [Ca2+] (Ca(ext)) to alter the flux per channel or by reducing the number of open Ca2+ channels with omega-conotoxin GVIA (omega-CTX). When Ca(ext) was reduced, the exponent of the power relationship relating release to Ca2+ influx was 4.16 +/- 0.62 (SD; n = 4), consistent with a biochemical cooperativity of approximately 5. In contrast, reducing influx with omega-CTX yielded a power relationship of 1.7 +/- 0.44 (n = 5) for Ca(ext) of 1.8 mM and 2.12 +/- 0.44 for Ca(ext) of 0.45 mM (n = 5). Using geometrically realistic Monte Carlo simulations, we tracked Ca2+ ions as they entered through each channel and diffused in the terminal. Experimental and modeling data were consistent with two to six channel openings per AZ per AP; the Ca2+ that causes fusion of a single vesicle originates from one or two channels. Channel cooperativity depends mainly on the physical relationship between channels and vesicles and is insensitive to changes in the non-geometrical parameters of our model.

Calcium microdomains in regulated exocytosis

Katz and co-workers showed that Ca(2+) triggers exocytosis. The existence of sub-micrometer domains of greater than 100 microM [Ca(2+)](i) was postulated on theoretical grounds. Using a modified, low-affinity aequorin, Llinas et al. were the first to demonstrate the existence of Ca(2+) 'microdomains' in squid presynaptic terminals. Over the past several years, it has become clear that individual Ca(2+) nano- and microdomains forming around the mouth of voltage-gated Ca(2+) channels ascertain the tight coupling of fast synaptic vesicle release to membrane depolarization by action potentials. Recent work has established different geometric arrangements of vesicles and Ca(2+) channels at different central synapses and pointed out the role of Ca(2+) syntillas - localized, store operated Ca(2+) signals - in facilitation and spontaneous release. The coupling between Ca(2+) increase and evoked exocytosis is more sluggish in peripheral terminals and neuroendocrine cells, where channels are less clustered and Ca(2+) comes from different sources, including Ca(2+) influx via the plasma membrane and the mobilization of Ca(2+) from intracellular stores. Finally, also non- (electrically) excitable cells display highly localized Ca(2+) signaling domains. We discuss in particular the organization of structural microdomains of Bergmann glia, specialized astrocytes of the cerebellum that have only recently been considered as secretory cells. Glial microdomains are the spatial substrate for functionally segregated Ca(2+) signals upon metabotropic activation. Our review emphasizes the large diversity of different geometric arrangements of vesicles and Ca(2+) sources, leading to a wide spectrum of Ca(2+) signals triggering release.

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.

Simple data-driven models of intracellular calcium dynamics with predictive power

Biology is complex. However, it is not clear how much of this complexity must necessarily translate into complicated mathematical models of biological processes. Simple models can be appealing to physicists but are usually deceiving for biologists. Complicated models, on the other hand, depend on too many parameters whose values are frequently unknown. Therefore, complicated models, although in principle more realistic, can lead to erroneous results if they are sensitive to these unknown parameter values. Intracellular calcium signals provide an example of utmost biological importance in which the issue of "simple vs complex" can be explored. In this paper we show that simple models describing the dynamics of intracellular calcium can be directly inferred from experimental data, without no a priori information on unknown parameters. A similar approach can be followed to study other reaction-diffusion systems. In spite of their simplicity, these models can provide quantitative information on some of the processes that shape calcium signals, such as the calcium current that underlies an experimental observation. This shows that simple models of biological systems are not limited to qualitative descriptions.

The asymptotic behavior of solutions of the buffered bistable system

In this paper, we study a model for calcium buffering with bistable nonlinearity. We present some results on the stability of equilibrium states and show that there exists a threshold phenomenon in our model. In comparing with the model without buffers, we see that stationary buffers cannot destroy the asymptotic stability of the associated equilibrium states and the threshold phenomenon. Moreover, we also investigate the propagation property of solutions with initial data being a disturbance of one of the stable states which is confined to a half-line. We show that the more stable state will eventually dominate the whole dynamics and that the speed of this propagation (or invading process) is positive.

The effect of residual on the stochastic gating of -regulated channel models

Single-channel models of intracellular Ca(2+) channels such as the inositol 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by a constant background [Ca(2+)] as opposed to a dynamic [Ca(2+)] representing the formation and collapse of a localized Ca(2+) domain. This assumption neglects the fact that Ca(2+) released by open intracellular Ca(2+) channels may influence subsequent gating through the processes of Ca(2+)-activation or -inactivation. We study the effect of such "residual Ca(2+)" from previous channel opening on the stochastic gating of minimal and realistic single-channel models coupled to a restricted cytoplasmic compartment. Using Monte Carlo simulation as well as analytical and numerical solution of a system of advection-reaction equations for the probability density of the domain [Ca(2+)] conditioned on the state of the channel, we determine how the steady-state open probability (p(open)) of single-channel models of Ca(2+)-regulated Ca(2+) channels depends on the time constant for Ca(2+) domain formation and collapse. As expected, p(open) for a minimal model including Ca(2+) activation increases as the domain time constant becomes large compared to the open and closed dwell times of the channel, that is, on average the channel is activated by residual Ca(2+) from previous openings. Interestingly, p(open) for a channel model that is inactivated by Ca(2+) also increases as a function of the domain time constant when the maximum domain [Ca(2+)] is fixed, because slow formation of the Ca(2+) domain attenuates Ca(2+)-mediated inactivation. Conversely, when the source amplitude of the channel is fixed, increasing the domain time constant leads to elevated domain [Ca(2+)] and decreased open probability. Consistent with these observations, a realistic De Young-Keizer-like IP(3)R model responds to residual Ca(2+) with a steady-state open probability that is a monotonic function of the domain time constant, though minimal models that include both Ca(2+)-activation and -inactivation show more complex behavior. We show how the probability density approach described here can be generalized for arbitrarily complex channel models and for any value of the domain time constant. In addition, we present a comparatively simple numerical procedure for estimating p(open) for models of Ca(2+)-regulated Ca(2+) channels in the limit of a very fast or very slow Ca(2+) domain. When the ordinary differential equation for the [Ca(2+)] in a restricted cytoplasmic compartment is replaced by a partial differential equation for the buffered diffusion of intracellular Ca(2+) in a homogeneous isotropic cytosol, we find the dependence of p(open) on the buffer time constant is qualitatively similar to the above-mentioned results.

Consequences of molecular-level Ca2+ channel and synaptic vesicle colocalization for the Ca2+ microdomain and neurotransmitter exocytosis: a monte carlo study

Morphological and biochemical studies indicate association between voltage-gated Ca2+ channels and the vesicle docking complex at vertebrate presynaptic active zones, which constrain the separation between some Ca2+ channels and vesicles to 20 nm or less. To address the effect of the precise geometrical relationship among the vesicles, the Ca2+ channel, and the proteins of the release machinery on neurotransmitter release, we developed a Monte Carlo simulation of Ca2+ diffusion and buffering with nanometer resolution. We find that the presence of a vesicle as a diffusion barrier alters the shape of the Ca2+ microdomain of a single Ca2+ channel around the vesicle. This effect is maximal in the vicinity of the vesicle and depends critically on the vesicle's distance from the plasmalemma. Ca2+-sensor(s) for release would be exposed to markedly different [Ca2+], varying by up to 13-fold, depending on their position around the vesicle. As a result, the precise position of Ca2+-sensor(s) with respect to the vesicle and the channel can be critical to determining the release probability. Variation in the position of Ca2+-sensor molecule(s) and their accessibility could be an important source of heterogeneity in vesicle release probability.

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

We simulate currents and concentration profiles generated by Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol through IP(3) receptor channel clusters. Clusters are described as conducting pores in the lumenal membrane with a diameter from 6 nm to 36 nm. The endoplasmic reticulum is modeled as a disc with a radius of 1-12 microm and an inner height of 28 nm. We adapt the dependence of the currents on the trans Ca(2+) concentration (intralumenal) measured in lipid bilayer experiments to the cellular geometry. Simulated currents are compared with signal mass measurements in Xenopus oocytes. We find that release currents depend linearly on the concentration of free Ca(2+) in the lumen. The release current is approximately proportional to the square root of the number of open channels in a cluster. Cytosolic concentrations at the location of the cluster range from 25 microM to 170 microM. Concentration increase due to puffs in a distance of a few micrometers from the puff site is found to be in the nanomolar range. Release currents decay biexponentially with timescales of <1 s and a few seconds. Concentration profiles decay with timescales of 0.125-0.250 s upon termination of release.

Analysing cardiac excitation–contraction coupling with mathematical models of local control

Cardiac excitation-contraction (E-C) coupling describes the process that links sarcolemmal Ca2+ influx via L-type Ca2+ channels to Ca2+ release from the sarcoplasmic reticulum via ryanodine receptors (RyRs). This process has proven difficult to study experimentally, and complete descriptions of how the cell couples surface membrane and intracellular signal transduction proteins to achieve both stable and sensitive intracellular calcium release are still lacking. Mathematical models provide a framework to test our understanding of how this is achieved. While no single model is yet capable of describing all features of cardiac E-C coupling, models of increasing complexity are revealing unexpected subtlety in the process. In particular, modelling has established a general failure of 'common-pool' models and has emphasized the requirement for 'local control' so that microscopic sub-cellular domains can separate local behaviour from the whole-cell average (common-pool) behaviour. The micro-architecture of the narrow diadic cleft in which the local control takes place is a key factor in determining local Ca2+ dynamics. There is still considerable uncertainty about the number of Ca2+ ions required to open RyRs within the cleft and various gating models have been proposed, many of which are in reasonable agreement with available experimental data. However, not all models exhibit a realistic voltage dependence of E-C coupling gain. Furthermore, it is unclear which model features are essential to producing reasonable gain properties. Thus, despite the success of local-control models in explaining many features of cardiac E-C coupling, more work will be needed to provide a sound theoretical basis of cardiac E-C coupling.

Responses of Ca2+-binding proteins to localized, transient changes in intracellular [Ca2+]

In smooth muscle cells, various transient, localized [Ca(2+)] changes have been observed that are thought to regulate cell function without necessarily inducing contraction. Although a great deal of effort has been put into detecting these transients and elucidating the mechanisms involved in their generation, the extent to which these transient Ca(2+) signals interact with intracellular Ca(2+)-binding molecules remains relatively unknown. To understand how the spatial and temporal characteristics of an intracellular Ca(2+) signal influence its interaction with Ca(2+)-binding proteins, mathematical models of Ca(2+) diffusion and regulation in smooth muscle cells were used to study Ca(2+) binding to prototypical proteins with one or two Ca(2+)-binding sites. Simulations with the models: (1) demonstrate the extent to which the rate constants for Ca(2+)-binding to proteins and the spatial and temporal characteristics of different Ca(2+) transients influence the magnitude and time course of the responses of these proteins to the transients; (2) predict significant differences in the responses of proteins with one or two Ca(2+)-binding sites to individual Ca(2+) transients and to trains of transients; (3) demonstrate how the kinetic characteristics determine the fidelity with which the responses of Ca(2+)-sensitive molecules reflect the magnitude and time course of transient Ca(2+) signals. Overall, this work demonstrates the clear need for complete information about the kinetics of Ca(2+) binding for determining how well Ca(2+)-binding molecules respond to different types of Ca(2+) signals. These results have important implications when considering the possible modulation of Ca(2+)- and Ca(2+)/calmodulin-dependent proteins by localized intracellular Ca(2+) transients in smooth muscle cells and, more generally, in other cell types.

Buffers and oscillations in intracellular Ca2+ dynamics

I model the behavior of intracellular Ca(2+) release with high buffer concentrations. The model uses a spatially discrete array of channel clusters. The channel subunit dynamics is a stochastic representation of the DeYoung-Keizer model. The calculations show that the concentration profile of fast buffer around an open channel is more localized than that of slow buffers. Slow buffers allow for release of larger amounts of Ca(2+) from the endoplasmic reticulum and hence bind more Ca(2+) than fast buffers with the same dissociation constant and concentration. I find oscillation-like behavior for high slow buffer concentration and low Ca(2+) content of the endoplasmic reticulum. High concentration of slow buffer leads to oscillation-like behavior by repetitive wave nucleation for high Ca(2+) content of the endoplasmic reticulum. Localization of Ca(2+) release by slow buffer, as used in experiments, can be reproduced by the modeling approach.

Dynamical description of sinoatrial node pacemaking: improved mathematical model for primary pacemaker cell

We developed an improved mathematical model for a single primary pacemaker cell of the rabbit sinoatrial node. Original features of our model include 1) incorporation of the sustained inward current (I(st)) recently identified in primary pacemaker cells, 2) reformulation of voltage- and Ca(2+)-dependent inactivation of the L-type Ca(2+) channel current (I(Ca,L)), 3) new expressions for activation kinetics of the rapidly activating delayed rectifier K(+) channel current (I(Kr)), and 4) incorporation of the subsarcolemmal space as a diffusion barrier for Ca(2+). We compared the simulated dynamics of our model with those of previous models, as well as with experimental data, and examined whether the models could accurately simulate the effects of modulating sarcolemmal ionic currents or intracellular Ca(2+) dynamics on pacemaker activity. Our model represents significant improvements over the previous models, because it can 1) simulate whole cell voltage-clamp data for I(Ca,L), I(Kr), and I(st); 2) reproduce the waveshapes of spontaneous action potentials and ionic currents during action potential clamp recordings; and 3) mimic the effects of channel blockers or Ca(2+) buffers on pacemaker activity more accurately than the previous models.

Slow time evolution of two-time-scale reaction-diffusion systems: the physical origin of nondiffusive transport

We study, from a mesoscopic point of view, the slow time-scale dynamics of a mixture of chemicals in which there is a chemical reaction that occurs much faster than all other processes, including diffusion. For a simple paradigmatic model reaction, it is possible to find a reduced set of dynamical equations analytically. This procedure, which yields the same mean field equations as the macroscopic approach described by Strier and Dawson [J. Chem. Phys, 112, 825 (2000)], clarifies the physical origin of some of the terms that appear in the reduced reaction-diffusion equations, such as "negative density dependent cross diffusion terms," whose actual meaning is hard to assess within the macroscopic framework. We also present a two-time-scale reactive lattice gas automaton with which it is possible to check the validity of the analytical results and the conditions under which the reduced description holds. Using this lattice gas we also show how the differential interaction with immobile species can give rise to the formation of stable Turing patterns in a system where all the other chemicals diffuse approximately at the same rate.

Functional triads consisting of ryanodine receptors, Ca(2+) channels, and Ca(2+)-activated K(+) channels in bullfrog sympathetic neurons. Plastic modulation of action potential

Fluorescent ryanodine revealed the distribution of ryanodine receptors in the submembrane cytoplasm (less than a few micrometers) of cultured bullfrog sympathetic ganglion cells. Rises in cytosolic Ca(2+) ([Ca(2+)](i)) elicited by single or repetitive action potentials (APs) propagated at a high speed (150 microm/s) in constant amplitude and rate of rise in the cytoplasm bearing ryanodine receptors, and then in the slower, waning manner in the deeper region. Ryanodine (10 microM), a ryanodine receptor blocker (and/or a half opener), or thapsigargin (1-2 microM), a Ca(2+)-pump blocker, or omega-conotoxin GVIA (omega-CgTx, 1 microM), a N-type Ca(2+) channel blocker, blocked the fast propagation, but did not affect the slower spread. Ca(2+) entry thus triggered the regenerative activation of Ca(2+)-induced Ca(2+) release (CICR) in the submembrane region, followed by buffered Ca(2+) diffusion in the deeper cytoplasm. Computer simulation assuming Ca(2+) release in the submembrane region reproduced the Ca(2+) dynamics. Ryanodine or thapsigargin decreased the rate of spike repolarization of an AP to 80%, but not in the presence of iberiotoxin (IbTx, 100 nM), a BK-type Ca(2+)-activated K(+) channel blocker, or omega-CgTx, both of which decreased the rate to 50%. The spike repolarization rate and the amplitude of a single AP-induced rise in [Ca(2+)](i) gradually decreased to a plateau during repetition of APs at 50 Hz, but reduced less in the presence of ryanodine or thapsigargin. The amplitude of each of the [Ca(2+)](i) rise correlated well with the reduction in the IbTx-sensitive component of spike repolarization. The apamin-sensitive SK-type Ca(2+)-activated K(+) current, underlying the afterhyperpolarization of APs, increased during repetitive APs, decayed faster than the accompanying rise in [Ca(2+)](i), and was suppressed by CICR blockers. Thus, ryanodine receptors form a functional triad with N-type Ca(2+) channels and BK channels, and a loose coupling with SK channels in bullfrog sympathetic neurons, plastically modulating AP.

The probability of quantal secretion near a single calcium channel of an active zone

A Monte Carlo analysis has been made of calcium dynamics and quantal secretion at microdomains in which the calcium reaches very high concentrations over distances of <50 nm from a channel and for which calcium dynamics are dominated by diffusion. The kinetics of calcium ions in microdomains due to either the spontaneous or evoked opening of a calcium channel, both of which are stochastic events, are described in the presence of endogenous fixed and mobile buffers. Fluctuations in the number of calcium ions within 50 nm of a channel are considerable, with the standard deviation about half the mean. Within 10 nm of a channel these numbers of ions can give rise to calcium concentrations of the order of 100 microM. The temporal changes in free calcium and calcium bound to different affinity indicators in the volume of an entire varicosity or bouton following the opening of a single channel are also determined. A Monte Carlo analysis is also presented of how the dynamics of calcium ions at active zones, after the arrival of an action potential and the stochastic opening of a calcium channel, determine the probability of exocytosis from docked vesicles near the channel. The synaptic vesicles in active zones are found docked in a complex with their calcium-sensor associated proteins and a voltage-sensitive calcium channel, forming a secretory unit. The probability of quantal secretion from an isolated secretory unit has been determined for different distances of an open calcium channel from the calcium sensor within an individual unit: a threefold decrease in the probability of secretion of a quantum occurs with a doubling of the distance from 25 to 50 nm. The Monte Carlo analysis also shows that the probability of secretion of a quantum is most sensitive to the size of the single-channel current compared with its sensitivity to either the binding rates of the sites on the calcium-sensor protein or to the number of these sites that must bind a calcium ion to trigger exocytosis of a vesicle.

Modeling study of the effects of overlapping Ca2+ microdomains on neurotransmitter release

Although single-channel Ca2+ microdomains are capable of gating neurotransmitter release in some instances, it is likely that in many cases the microdomains from several open channels overlap to activate vesicle fusion. We describe a mathematical model in which transmitter release is gated by single or overlapping Ca2+ microdomains produced by the opening of nearby Ca2+ channels. This model accounts for the presence of a mobile Ca2+ buffer, provided either that the buffer is unsaturable or that it is saturated near an open channel with Ca2+ binding kinetics that are rapid relative to Ca2+ diffusion. We show that the release time course is unaffected by the location of the channels (at least for distances up to 50 nm), but paired-pulse facilitation is greater when the channels are farther from the release sites. We then develop formulas relating the fractional release following selective or random channel blockage to the cooperative relationship between release and the presynaptic Ca2+ current. These formulas are used with the transmitter release model to study the dependence of this form of cooperativity, which we call Ca2+ current cooperativity, on mobile buffers and on the local geometry of Ca2+ channels. We find that Ca2+ current cooperativity increases with the number of channels per release site, but is considerably less than the number of channels, the theoretical upper bound. In the presence of a saturating mobile buffer the Ca2+ current cooperativity is greater, and it increases more rapidly with the number of channels. Finally, Ca2+ current cooperativity is an increasing function of channel distance, particularly in the presence of saturating mobile buffer.

Usefulness and limitations of linear approximations to the understanding of Ca++ signals

Simple approximations to some limiting cases of Ca++ signalling provide insight into the complex problems of buffered diffusion and of Ca++ homeostasis in the presence of buffers. Three cases are presented, where the influence of Ca++ buffers can readily be understood in the limit of small signals: the return of global cellular [Ca++] following a short stimulus in a 'Single Compartment', buffered diffusion along a cylindrical axon in the 'Rapid Buffer Approximation', and nonequilibrium microdomains of elevated [Ca++] in the immediate vicinity of open Ca++ channels.

Competitive calcium binding: implications for dendritic calcium signaling

Action potentials evoke calcium transients in dendrites of neocortical pyramidal neurons with time constants of < 100 ms at physiological temperature. This time period may not be sufficient for inflowing calcium ions to equilibrate with all present Ca2+-binding molecules. We therefore explored nonequilibrium dynamics of Ca2+ binding to numerous Ca2+ reaction partners within a dendritelike compartment using numerical simulations. After a brief Ca2+ influx, the reaction partner with the fastest Ca2+ binding kinetics initially binds more Ca2+ than predicted from chemical equilibrium, while companion reaction partners bind less. This difference is consolidated and may result in bypassing of slow reaction partners if a Ca2+ clearance mechanism is active. On the other hand, slower reaction partners effectively bind Ca2+ during repetitive calcium current pulses or during slower Ca2+ influx. Nonequilibrium Ca2+ distribution can further be enhanced through strategic placement of the reaction partners within the compartment. Using the Ca2+ buffer EGTA as a competitor of fluo-3, we demonstrate competitive Ca2+ binding within dendrites experimentally. Nonequilibrium calcium dynamics is proposed as a potential mechanism for differential and conditional activation of intradendritic targets.

A simple numerical model of calcium spark formation and detection in cardiac myocytes

The elementary events of excitation-contraction coupling in heart muscle are Ca2+ sparks, which arise from one or more ryanodine receptors in the sarcoplasmic reticulum (SR). Here a simple numerical model is constructed to explore Ca2+ spark formation, detection, and interpretation in cardiac myocytes. This model includes Ca2+ release, cytosolic diffusion, resequestration by SR Ca2+-ATPases, and the association and dissociation of Ca2+ with endogenous Ca2+-binding sites and a diffusible indicator dye (fluo-3). Simulations in a homogeneous, isotropic cytosol reproduce the brightness and the time course of a typical cardiac Ca2+ spark, but underestimate its spatial size (approximately 1.1 micron vs. approximately 2.0 micron). Back-calculating [Ca2+]i by assuming equilibrium with indicator fails to provide a good estimate of the free Ca2+ concentration even when using blur-free fluorescence data. A parameter sensitivity study reveals that the mobility, kinetics, and concentration of the indicator are essential determinants of the shape of Ca2+ sparks, whereas the stationary buffers and pumps are less influential. Using a geometrically more complex version of the model, we show that the asymmetric shape of Ca2+ sparks is better explained by anisotropic diffusion of Ca2+ ions and indicator dye rather than by subsarcomeric inhomogeneities of the Ca2+ buffer and transport system. In addition, we examine the contribution of off-center confocal sampling to the variance of spark statistics.

Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load

We construct a detailed mathematical model for Ca2+ regulation in the ventricular myocyte that includes novel descriptions of subcellular mechanisms based on recent experimental findings: 1) the Keizer-Levine model for the ryanodine receptor (RyR), which displays adaptation at elevated Ca2+; 2) a model for the L-type Ca2+ channel that inactivates by mode switching; and 3) a restricted subspace into which the RyRs and L-type Ca2+ channels empty and interact via Ca2+. We add membrane currents from the Luo-Rudy Phase II ventricular cell model to our description of Ca2+ handling to formulate a new model for ventricular action potentials and Ca2+ regulation. The model can simulate Ca2+ transients during an action potential similar to those seen experimentally. The subspace [Ca2+] rises more rapidly and reaches a higher level (10-30 microM) than the bulk myoplasmic Ca2+ (peak [Ca2+]i approximately 1 microM). Termination of sarcoplasmic reticulum (SR) Ca2+ release is predominately due to emptying of the SR, but is influenced by RyR adaptation. Because force generation is roughly proportional to peak myoplasmic Ca2+, we use [Ca2+]i in the model to explore the effects of pacing rate on force generation. The model reproduces transitions seen in force generation due to changes in pacing that cannot be simulated by previous models. Simulation of such complex phenomena requires an interplay of both RyR adaptation and the degree of SR Ca2+ loading. This model, therefore, shows improved behavior over existing models that lack detailed descriptions of subcellular Ca2+ regulatory mechanisms.

Modulation of stimulus-secretion coupling in single rat gonadotrophs

1. Exocytosis and intracellular [Ca2+] were determined simultaneously in single anterior pituitary gonadotrophs from ovariectomized female rats. Dispersed cells were cultured for 2-4 days with or without 0.2 nM oestradiol-17 beta (E2) before use. Cells were stimulated with either gonadotrophin releasing hormone (GnRH) or by membrane depolarization. Exocytosis was determined from the change in membrane capacitance (Cm) using the perforated-patch whole-cell recording technique. Intracellular [Ca2+] was measured using fura-2 fluorescence. 2. The exocytotic response to 1 nM GnRH was characterized by a wide spectrum of responses, ranging from exocytotic bursts to relatively slow, graded increases that were dependent on the evoked intracellular Ca2+ pattern. A kinetic model is presented that incorporates the observed steep dependence of exocytosis on measured intracellular [Ca2+]; simulated exocytosis reasonably approximated observed exocytotic responses, both kinetically and quantitatively. The model also suggests that the modulatory effects of E2 are brought about either by a change in the Ca2+ sensitivity of exocytosis or by a preferential clustering of docked-secretory granules close to sites of Ca2+ release. The results suggest that in gonadotrophs an oscillatory Ca2+ signal is sensed by the exocytotic apparatus in a modified form of digital encoding. 3. Exocytosis in E2-treated cells was 3-fold greater than in non-treated cells for GnRH-evoked secretion, and 38% greater for depolarization; however, there was no effect of E2 on the intracellular Ca2+ response to either stimulus. The results show that maximum expression of the effect of E2 on exocytosis requires activation of GnRH-dependent pathways.

Linearized buffered Ca2+ diffusion in microdomains and its implications for calculation of [Ca2+] at the mouth of a calcium channel

Immobile and mobile calcium buffers shape the calcium signal close to a channel by reducing and localizing the transient calcium increase to physiological compartments. In this paper, we focus on the impact of mobile buffers in shaping steady-state calcium gradients in the vicinity of an open channel, i.e. within its "calcium microdomain." We present a linear approximation of the combined reaction-diffusion problem, which can be solved explicitly and accounts for an arbitrary number of calcium buffers, either endogenous or added exogenously. It is valid for small saturation levels of the present buffers and shows that within a few hundred nanometers from the channel, standing calcium gradients develop in hundreds of microseconds after channel opening. It is shown that every buffer can be assigned a uniquely defined length-constant as a measure of its capability to buffer calcium close to the channel. The length-constant clarifies intuitively the significance of buffer binding and unbinding kinetics for understanding local calcium signals. Hence, we examine the parameters shaping these steady-state gradients. The model can be used to check the expected influence of single channel calcium microdomains on physiological processes such as excitation-secretion coupling or excitation-contraction coupling and to explore the differential effect of kinetic buffer parameters on the shape of these microdomains.

Predicted changes in concentrations of free and bound ATP and ADP during intracellular Ca2+ signaling

High Ca2+ concentrations can develop near Ca2+ sources during intracellular signaling and might lead to localized regulation of Ca2+-dependent processes. By shifting the amount of Ca2+ and other cations associated with ATP, local high Ca2+ concentrations might also alter the substrate available for membrane-associated and cytoplasmic enzymes. To study this, simultaneous equations were solved over a range of Ca2+ and Mg2+ concentrations to determine the general effects of Ca2+ on the concentrations of free and Ca2+- and Mg2+-bound forms of ATP. To obtain a more specific picture of the changes that might occur in smooth muscle cells, mathematical models of Ca2+ diffusion and regulation were used to predict the magnitude and time course of near-membrane Ca2+ transients and their effects on the free and bound forms of ATP near the membrane. The results of this work indicate that changes in free Ca2+ concentration over the range of 50 nM-100 microM would result in significant changes in free ATP concentration, MgATP concentration, and the CaATP-to-MgATP concentration ratio.

Linearized buffered Ca2+ diffusion in microdomains and its implications for calculation of [Ca2+] at the mouth of a calcium channel

Immobile and mobile calcium buffers shape the calcium signal close to a channel by reducing and localizing the transient calcium increase to physiological compartments. In this paper, we focus on the impact of mobile buffers in shaping steady-state calcium gradients in the vicinity of an open channel, i.e. within its "calcium microdomain." We present a linear approximation of the combined reaction-diffusion problem, which can be solved explicitly and accounts for an arbitrary number of calcium buffers, either endogenous or added exogenously. It is valid for small saturation levels of the present buffers and shows that within a few hundred nanometers from the channel, standing calcium gradients develop in hundreds of microseconds after channel opening. It is shown that every buffer can be assigned a uniquely defined length-constant as a measure of its capability to buffer calcium close to the channel. The length-constant clarifies intuitively the significance of buffer binding and unbinding kinetics for understanding local calcium signals. Hence, we examine the parameters shaping these steady-state gradients. The model can be used to check the expected influence of single channel calcium microdomains on physiological processes such as excitation-secretion coupling or excitation-contraction coupling and to explore the differential effect of kinetic buffer parameters on the shape of these microdomains.

A general computational framework for modeling cellular structure and function

The "Virtual Cell" provides a general system for testing cell biological mechanisms and creates a framework for encapsulating the burgeoning knowledge base comprising the distribution and dynamics of intracellular biochemical processes. It approaches the problem by associating biochemical and electrophysiological data describing individual reactions with experimental microscopic image data describing their subcellular localizations. Individual processes are collected within a physical and computational infrastructure that accommodates any molecular mechanism expressible as rate equations or membrane fluxes. An illustration of the method is provided by a dynamic simulation of IP3-mediated Ca2+ release from endoplasmic reticulum in a neuronal cell. The results can be directly compared to experimental observations and provide insight into the role of experimentally inaccessible components of the overall mechanism.