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.     

Effects of inositol 1,4,5-trisphosphate receptor-mediated intracellular stochastic calcium oscillations on activation of glycogen phosphorylase

In various cell types cytosolic calcium (Ca(2+)) is an important regulator. The possible role of Ca(2+) release from the inositol 1,4,5-trisphosphate (IP(3)) receptor channel in the regulation of the phosphorylation-dephosphorylation cycle process involved in glycogen degradation by glycogen phosphorylase have theoretically investigated by using the Li-Rinzel model for cytosolic Ca(2+) oscillations. For the case of deterministic cytosolic Ca(2+) oscillations, there exists an optimal frequency of cytosolic Ca(2+) oscillations at which the average fraction of active glycogen phosphorylase reaches a maximum value, and a mutation for the average fraction of active glycogen phosphorylase occurs at the higher bifurcation point of Ca(2+) oscillations. For the case of stochastic cytosolic Ca(2+) oscillations, the fraction of active phosphorylase is strongly affected by the number of IP(3) receptor channels and the level of IP(3) concentration. Small number of IP(3) receptor channels can potentiate the sensitivity of the activity of glycogen phosphorylase. The average frequency and amplitude of active phosphorylase stochastic oscillations are increased with the level of increasing IP(3) stimuli. The various distributions for the amplitude of active glycogen phosphorylase oscillations in parameters plane are discussed.     

Dynamic simulation of the effect of calcium-release activated calcium channel on cytoplasmic Ca2+ oscillation

A mathematical model is proposed to illustrate the activation of STIM1 (stromal interaction molecule 1) protein, the assembly and activation of calcium-release activated calcium (CRAC) channels in T cells. In combination with De Young-Keizer-Li-Rinzel model, we successfully reproduce a sustained Ca(2+) oscillation in cytoplasm. Our results reveal that Ca(2+) oscillation dynamics in cytoplasm can be significantly affected by the way how the Orai1 CRAC channel are assembled and activated. A low sustained Ca(2+) influx is observed through the CRAC channels across the plasma membrane. In particular, our model shows that a tetrameric channel complex can effectively regulate the total quantity of the channels and the ratio of the active channels to the total channels, and a period of Ca(2+) oscillation about 29 s is in agreement with published experimental data. The bifurcation analyses illustrate the different dynamic properties between our mixed Ca(2+) feedback model and the single positive or negative feedback models.     

Phantom bursting is highly sensitive to noise and unlikely to account for slow bursting in beta-cells: considerations in favor of metabolically driven oscillations

Pancreatic beta-cells show bursting electrical activity with a wide range of burst periods ranging from a few seconds, often seen in isolated cells, over tens of seconds (medium bursting), usually observed in intact islets, to several minutes. The phantom burster model [Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic beta-cells. Biophys. J. 79, 2880-2892] provided a framework, which covered this span, and gave an explanation of how to obtain medium bursting combining two processes operating on different time scales. However, single cells are subjected to stochastic fluctuations in plasma membrane currents, which are likely to disturb the bursting mechanism and transform medium bursters into spikers or very fast bursters. We present a polynomial, minimal, phantom burster model and show that noise modifies the plateau fraction and lowers the burst period dramatically in phantom bursters. It is therefore unlikely that slow bursting in single cells is driven by the slow phantom bursting mechanism, but could instead be driven by oscillations in glycolysis, which we show are stable to random ion channel fluctuations. Moreover, so-called compound bursting can be converted to apparent slow bursting by noise, which could explain why compound bursting and mixed Ca(2+) oscillations are seen mainly in intact islets.     

InsP3 signaling induces pulse-modulated Ca2+ signals in the nucleus of airway epithelial ciliated cells

The phenomenology of nuclear Ca(2+) dynamics has experienced important progress revealing the broad range of cellular processes that it regulates. Although several agonists can mobilize Ca(2+) from storage in the nuclear envelope (NE) to the intranuclear compartment (INC), the mechanisms of Ca(2+) signaling in the nucleus still remain uncertain. Here we report that the NE/INC complex can function as an inositol-1,4,5-trisphosphate (InsP(3))-controlled Ca(2+) oscillator. Thin optical sectioning combined with fluorescent labeling of Ca(2+) probes show in cultured airway epithelial ciliated cells that ATP can trigger periodic oscillations of Ca(2+) in the NE ([Ca(2+)](NE)) and corresponding pulses of Ca(2+) release to the INC. Identical results were obtained in InsP(3)-stimulated isolated nuclei of these cells. Our data show that [Ca(2+)](NE) oscillations and Ca(2+) release to the INC result from the interplay between the Ca(2+)/K(+) ion-exchange properties of the intralumenal polyanionic matrix of the NE and two Ca(2+)-sensitive ion channels-an InsP(3)-receptor-Ca(2+) channel and an apamin-sensitive K(+) channel. A similar Ca(2+) signaling system operating under the same functional protocol and molecular hardware controls Ca(2+) oscillations and release in/to the endoplasmic reticulum/cytosol and in/to the granule/cytosol complexes in airway and mast cells. These observations suggest that these intracellular organelles share a remarkably conserved mechanism of InsP(3)-controlled frequency-encoded Ca(2+) signaling.     

Three roads to islet bursting: emergent oscillations in coupled phantom bursters

Glucose-induced membrane potential and Ca(2+) oscillations in isolated pancreatic beta-cells occur over a wide range of frequencies, from >6/min (fast) to <1/min (slow). However, cells within intact islets generally oscillate with periods of 10-60 s (medium). The phantom bursting concept addresses how beta-cells can generate such a wide range of frequencies. Here, we explore an updated phantom bursting model to determine how heterogeneity in a single parameter can explain both the broad frequency range observed in single cells and the rarity of medium oscillations. We then incorporate the single-cell model into an islet model with parameter heterogeneity. We show that strongly coupled islets must be composed of predominantly medium oscillating single cells or a mixture of fast and slow cells to robustly produce medium oscillations. Surprisingly, we find that this constraint does not hold for moderate coupling, and that robustly medium oscillating islets can arise from populations of single cells that are essentially all slow or all fast. Thus, with coupled phantom bursters, medium oscillating islets can be constructed out of cells that are either all fast, all slow, or a combination of the two.     

A theoretical modeling for frequency modulation of Ca(2+) signal on activation of MAPK cascade

It is known that Ca(2+) signal regulates mitogen-activated protein kinase (MAPK) cascade by a central Ras protein in GTPase-cycle. Therefore, we construct an integrated theoretical model comprising Ca(2+) oscillations, GTPase-cycle and MAPK cascade modules sequentially. Meanwhile, based on multiple feedback regulations in MAPK cascade, three operation modes of this model are introduced. An extended version of this model is further built when spatial heterogeneity is involved. These models allow us to investigate the very interesting and broad question about the effects of Ca(2+) oscillations on the activation of MAPK cascade in both the homogeneous and heterogeneous systems. When the Li-Rinzel model is adopted to simulate endogenous Ca(2+) oscillations, our theoretical results illustrate that the appropriate operation mode of MAPK cascade is required for the negative correlation between a decreasing frequency of Ca(2+) oscillations and activation of MAPK cascade, which was found in the experiment (S. Kupzig et al. PNAS 102 (2005) 7577-7582). While a piecewise function is used to generate Ca(2+) signal to explore much larger range of periods of Ca(2+) oscillations, it is found that the negative correlation feature is independent on the operation mode of MAPK cascade. In this case, different operation modes only influence the strength of negative correlation between activation of MAPK cascade and periods of Ca(2+) oscillations. The quantitative results may be of great use in analyzing interaction of IP3-Ca(2+) and Ras-MAPK signaling pathways, and motivate the further experimental research.     

Regulation of cAMP dynamics by Ca2+ and G protein-coupled receptors in the pancreatic beta-cell: a computational approach

In this report we describe a mathematical model for the regulation of cAMP dynamics in pancreatic beta-cells. Incretin hormones such as glucagon-like peptide 1 (GLP-1) increase cAMP and augment insulin secretion in pancreatic beta-cells. Imaging experiments performed in MIN6 insulinoma cells expressing a genetically encoded cAMP biosensor and loaded with fura-2, a calcium indicator, showed that cAMP oscillations are differentially regulated by periodic changes in membrane potential and GLP-1. We modeled the interplay of intracellular calcium (Ca(2+)) and its interaction with calmodulin, G protein-coupled receptor activation, adenylyl cyclases (AC), and phosphodiesterases (PDE). Simulations with the model demonstrate that cAMP oscillations are coupled to cytoplasmic Ca(2+) oscillations in the beta-cell. Slow Ca(2+) oscillations (<1 min(-1)) produce low-frequency cAMP oscillations, and faster Ca(2+) oscillations (>3-4 min(-1)) entrain high-frequency, low-amplitude cAMP oscillations. The model predicts that GLP-1 receptor agonists induce cAMP oscillations in phase with cytoplasmic Ca(2+) oscillations. In contrast, observed antiphasic Ca(2+) and cAMP oscillations can be simulated following combined glucose and tetraethylammonium-induced changes in membrane potential. The model provides additional evidence for a pivotal role for Ca(2+)-dependent AC and PDE activation in coupling of Ca(2+) and cAMP signals. Our results reveal important differences in the effects of glucose/TEA and GLP-1 on cAMP dynamics in MIN6 beta-cells.     

Animal-vegetal axis patterning mechanisms in the early sea urchin embryo

During mouse fertilization the spermatozoon induces a series of low-frequency long-lasting Ca(2+) oscillations. It is generally accepted that these oscillations are due to Ca(2+) release through the inositol 1,4,5-trisphosphate (InsP(3)) receptor. However, InsP(3) microinjection does not mimic sperm-induced Ca(2+) oscillations, leading to the suggestion that the spermatozoon causes Ca(2+) release by sensitizing the InsP(3) receptor to basal levels of InsP(3). This contradicts recent evidence that the spermatozoon triggers Ca(2+) oscillations by introducing a phospholipase C or else an activator of phospholipase C. Here we show for the first time that sperm-induced Ca(2+) oscillations may be mimicked by the photolysis of caged InsP(3) in both mouse metaphase II eggs and germinal vesicle stage oocytes. Eggs, and also oocytes that had displayed spontaneous Ca(2+) oscillations, gave long-lasting Ca(2+) oscillations when fertilized or when caged InsP(3) was photolyzed. In contrast, oocytes that had shown no spontaneous Ca(2+) oscillations did not generate many oscillations when fertilized or following photolysis of caged InsP(3). Fertilization in eggs was most closely mimicked when InsP(3) was uncaged at relatively low amounts for extended periods. Here we observed an initial Ca(2+) transient with superimposed spikes, followed by a series of single transients with a low frequency; all characteristics of the Ca(2+) changes at fertilization. We therefore show that InsP(3) can mimic the distinctive pattern of Ca(2+) release in mammalian eggs at fertilization. It is proposed that a sperm Ca(2+)-releasing factor operates by generating a continuous small amount of InsP(3) over an extended period of time, consistent with the evidence for the involvement of a phospholipase C.     

Determinants for calmodulin binding on voltage-dependent Ca2+ channels

Calmodulin, bound to the alpha(1) subunit of the cardiac L-type calcium channel, is required for calcium-dependent inactivation of this channel. Several laboratories have suggested that the site of interaction of calmodulin with the channel is an IQ-like motif in the carboxyl-terminal region of the alpha(1) subunit. Mutations in this IQ motif are linked to L-type Ca(2+) current (I(Ca)) facilitation and inactivation. IQ peptides from L, P/Q, N, and R channels all bind Ca(2+)calmodulin but not Ca(2+)-free calmodulin. Another peptide representing a carboxyl-terminal sequence found only in L-type channels (designated the CB domain) binds Ca(2+)calmodulin and enhances Ca(2+)-dependent I(Ca) facilitation in cardiac myocytes, suggesting the CB domain is functionally important. Calmodulin blocks the binding of an antibody specific for the CB sequence to the skeletal muscle L-type Ca(2+) channel, suggesting that this is a calmodulin binding site on the intact protein. The binding of the IQ and CB peptides to calmodulin appears to be competitive, signifying that the two sequences represent either independent or alternative binding sites for calmodulin rather than both sequences contributing to a single binding site.     

Correlated oscillations in glucose consumption, oxygen consumption, and intracellular free Ca(2+) in single islets of Langerhans

Micron-sized sensors were used to monitor glucose and oxygen levels in the extracellular space of single islets of Langerhans in real-time. At 10 mM glucose, oscillations in intraislet glucose concentration were readily detected. Changes in glucose level correspond to changes in glucose consumption by glycolysis balanced by mass transport into the islet. Oscillations had a period of 3.1 +/- 0.2 min and amplitude of 0.8 +/- 0.1 mM glucose (n = 21). Superimposed on these oscillations were faster fluctuations in glucose level during the periods of low glucose consumption. Oxygen level oscillations that were out of phase with the glucose oscillations were also detected. Oscillations in both oxygen and glucose consumption were strongly dependent upon extracellular Ca(2+) and sensitive to nifedipine. Simultaneous measurements of glucose with intracellular Ca(2+) ([Ca(2+)](i)) revealed that decreases in [Ca(2+)](i) preceded increases in glucose consumption by 7.4 +/- 2.1 s during an oscillation (n = 9). Conversely, increases in [Ca(2+)](i) preceded increases in oxygen consumption by 1.5 +/- 0.2 s (n = 4). These results suggest that during oscillations, bursts of glycolysis begin after Ca(2+) has stopped entering the cell. Glycolysis stimulates further Ca(2+) entry, which in turn stimulates increases in respiration. The data during oscillation are in contrast to the time course of events during initial exposure to glucose. Under these conditions, a burst of oxygen consumption precedes the initial rise in [Ca(2+)](i). A model to explain these results is described.     

Comparison of simulated and measured calcium sparks in intact skeletal muscle fibers of the frog

Calcium sparks in frog intact skeletal muscle fibers were modeled as stereotypical events that arise from a constant efflux of Ca(2+) from a point source for a fixed period of time (e.g., 2.5 pA of Ca(2+) current for 4.6 ms; 18 degrees C). The model calculates the local changes in the concentrations of free Ca(2+) and of Ca(2+) bound to the major intrinsic myoplasmic Ca(2+) buffers (troponin, ATP, parvalbumin, and the SR Ca(2+) pump) and to the Ca(2+) indicator (fluo-3). A distinctive feature of the model is the inclusion of a binding reaction between fluo-3 and myoplasmic proteins, a process that strongly affects fluo-3's Ca(2+)-reaction kinetics, its apparent diffusion constant, and hence the morphology of sparks. DeltaF/F (the change in fluo-3's fluorescence divided by its resting fluorescence) was estimated from the calculated changes in fluo-3 convolved with the microscope point-spread function. To facilitate comparisons with measured sparks, noise and other sources of variability were included in a random repetitive fashion to generate a large number of simulated sparks that could be analyzed in the same way as the measured sparks. In the initial simulations, the binding of Ca(2+) to the two regulatory sites on troponin was assumed to follow identical and independent binding reactions. These simulations failed to accurately predict the falling phase of the measured sparks. A second set of simulations, which incorporated the idea of positive cooperativity in the binding of Ca(2+) to troponin, produced reasonable agreement with the measurements. Under the assumption that the single channel Ca(2+) current of a ryanodine receptor (RYR) is 0.5-2 pA, the results suggest that 1-5 active RYRs generate an average Ca(2+) spark in a frog intact muscle fiber.     

Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms

Intracellular Ca(2+) oscillations are commonly observed in a large number of cell types in response to stimulation by an extracellular agonist. In most cell types the mechanism of regular spiking is well understood and models based on Ca(2+)-induced Ca(2+) release (CICR) can account for many experimental observations. However, cells do not always exhibit simple Ca(2+) oscillations. In response to given agonists, some cells show more complex behaviour in the form of bursting, i.e. trains of Ca(2+) spikes separated by silent phases. Here we develop several theoretical models, based on physiologically plausible assumptions, that could account for complex intracellular Ca(2+) oscillations. The models are all based on one- or two-pool models based on CICR. We extend these models by (i) considering the inhibition of the Ca(2+)-release channel on a unique intracellular store at high cytosolic Ca(2+) concentrations, (ii) taking into account the Ca(2+)-activated degradation of inositol 1,4,5-trisphosphate (IP(3)), or (iii) considering explicity the evolution of the Ca(2+) concentration in two different pools, one sensitive and the other one insensitive to IP(3). Besides simple periodic oscillations, these three models can all account for more complex oscillatory behaviour in the form of bursting. Moreover, the model that takes the kinetics of IP(3) into account shows chaotic behaviour.     

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.     

Two adaptation processes in auditory hair cells together can provide an active amplifier

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves Ca(2+) ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce "self-tuned critical oscillations", i.e., maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the Ca(2+) dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.     

Mechanism of local and global Ca2+ sensing by calmodulin in complex with a Ca2+ channel

Calmodulin (CaM) in complex with Ca(2+) channels constitutes a prototype for Ca(2+) sensors that are intimately colocalized with Ca(2+) sources. The C-lobe of CaM senses local, large Ca(2+) oscillations due to Ca(2+) influx from the host channel, and the N-lobe senses global, albeit diminutive Ca(2+) changes arising from distant sources. Though biologically essential, the mechanism underlying global Ca(2+) sensing has remained unknown. Here, we advance a theory of how global selectivity arises, and we experimentally validate this proposal with methodologies enabling millisecond control of Ca(2+) oscillations seen by the CaM/channel complex. We find that global selectivity arises from rapid Ca(2+) release from CaM combined with greater affinity of the channel for Ca(2+)-free versus Ca(2+)-bound CaM. The emergence of complex decoding properties from the juxtaposition of common elements, and the techniques developed herein, promise generalization to numerous molecules residing near Ca(2+) sources.     

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+).