Cellular diversity promotes intercellular Ca2+ wave propagation

Calcium ions are an important second messenger in living cells. Calcium signals in form of waves serve as a means of intercellular communication and thus represent a vibrant subject for experimental and theoretical investigations. Here we study the role of cellular variability on the occurrence of Ca2+ wave propagation in a net of diffusively coupled cells. Dynamics of individual cells is simulated by a mathematical model for Ca2+ oscillations. Structural diversity of cells is introduced via variations of the bifurcation parameters, which signify cell sensitivity for external stimulation. Remarkably, for sufficient values of variability Ca2+ waves emerge, which are mostly ordered for intermediate variability strength. We analyze the spatial profile via the autocorrelation function, which confirms a resonance-like response due to the cellular variability. Thus, the reported phenomenon is a novel observation of diversity-induced spatial coherence resonance in a tissue-like media.     

Purinergic signalling and intercellular Ca2+ wave propagation in the organ of Corti

Extracellular ATP is a key neuromodulator of visual and auditory sensory epithelia. In the rat cochlea, pharmacological dissection indicates that ATP, acting through a highly sensitive purinergic/IP(3)-mediated signaling pathway with (little or) no involvement of ryanodine receptors, is the principal paracrine mediator implicated in the propagation of calcium waves through supporting and epithelial cells. Measurement of sensitivity to UTP and other purinergic agonists implicate P2Y(2) and P2Y(4) as the main P2Y receptor isoforms involved in these responses. Ca2+ waves, elicited under highly reproducible conditions by carefully controlling dose (1 microM) and timing of focal agonist application (0.2s), extended over radial distance greater than 160 microm from the source, identical to those activated by damaging single outer hair cells. Altogether, these results indicate that intercellular calcium waves are a robust phenomenon that confers a significant ability for cell-cell communication in the mammalian cochlea. Further ongoing research will reveal the roles that such Ca2+ waves play in the inner ear.     

The effect of residual Ca2+ on the stochastic gating of Ca2+-regulated Ca2+ 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.     

A spatiotemporal stochastic process model for species spread

We use a spatiotemporal Markov process to model the spread of an ecological population through its environment over time. Available habitat is divided into sites, and a parametric function of spatial variables is used to model the probability that one site is colonized from another. This allows us both to make predictions about the future spread of a population, and to determine which are the important factors governing colonizations. The model evolves in discrete time, allowing the population distribution to change seasonally in accordance with breeding patterns. Discrete time formulations are natural for ecological populations, but are problematic due to difficulties of fitting and predicting over irregular time intervals. The model described here can accommodate years of missing data and can therefore fit and predict at irregular intervals. Two methods of approximating the likelihood are described and applied to ornithological survey data for the woodlark, Lullula arborea, from Thetford Forest in the U.K.     

Intercellular Ca2+ wave propagation through gap-junctional Ca2+ diffusion: a theoretical study

Intercellular regenerative calcium waves in systems such as the liver and the blowfly salivary gland have been hypothesized to spread through calcium-induced calcium release (CICR) and gap-junctional calcium diffusion. A simple mathematical model of this mechanism is developed. It includes CICR and calcium removal from the cytoplasm, cytoplasmic and gap-junctional calcium diffusion, and calcium buffering. For a piecewise linear approximation of the calcium kinetics, expressions in terms of the cellular parameters are derived for 1) the condition for the propagation of intercellular waves, and 2) the characteristic time of the delay of a wave encountered at the gap junctions. Intercellular propagation relies on the local excitation of CICR in the perijunctional space by gap-junctional calcium influx. This mechanism is compatible with low effective calcium diffusivity, and necessitates that CICR can be excited in every cell along the path of a wave. The gap-junctional calcium permeability required for intercellular waves in the model falls in the range of reported gap-junctional permeability values. The concentration of diffusive cytoplasmic calcium buffers and the maximal rate of CICR, in the case of inositol 1,4,5-trisphosphate (IP3) receptor calcium release channels set by the IP(3) concentration, are shown to be further determinants of wave behavior.     

Local Ca2+ influx through Ca2+ release-activated Ca2+ (CRAC) channels stimulates production of an intracellular messenger and an intercellular pro-inflammatory signal

Ca2+ entry through store-operated Ca2+ channels drives the production of the pro-inflammatory molecule leukotriene C4 (LTC4) from mast cells through a pathway involving Ca2+-dependent protein kinase C, mitogen-activated protein kinases ERK1/2, phospholipase A2, and 5-lipoxygenase. Here we examine whether local Ca2+ influx through store-operated Ca2+ release-activated Ca2+ (CRAC) channels in the plasma membrane stimulates this signaling pathway. Manipulating the amplitude and spatial extent of Ca2+ entry by altering chemical and electrical gradients for Ca2+ influx or changing the Ca2+ buffering of the cytoplasm all impacted on protein kinase C and ERK activation, generation of arachidonic acid and LTC4 secretion, with little change in the bulk cytoplasmic Ca2+ rise. Similar bulk cytoplasmic Ca2+ concentrations were achieved when CRAC channels were activated in 0.25 mm external Ca2+ versus 2 mm Ca2+ and 100 nm La3+, an inhibitor of CRAC channels. However, despite similar bulk cytoplasmic Ca2+, protein kinase C activation and LTC4 secretion were larger in 2 mm Ca2+ and La3+ than in 0.25 mm Ca2+, consistent with the central involvement of a subplasmalemmal Ca2+ rise. The nonreceptor tyrosine kinase Syk coupled CRAC channel opening to protein kinase C and ERK activation. Recombinant TRPC3 channels also activated protein kinase C, suggesting that subplasmalemmal Ca2+ rather than a microdomain exclusive to CRAC channels is the trigger. Hence a subplasmalemmal Ca2+ increase in mast cells is highly versatile in that it triggers cytoplasmic responses through generation of intracellular messengers as well as long distance changes through increased secretion of paracrine signals.     

Temperature dependence of Ca2+ wave properties in cardiomyocytes: implications for the mechanism of autocatalytic Ca2+ release in wave propagation.

Digital imaging microscopy of fluo-3 fluorescence was used to study the velocity and shape of intracellular Ca2+ waves in isolated rat cardiomyocytes as a function of temperature. Decreasing the temperature from 37 to 17 degrees C reduced the longitudinal wave velocity by a factor of 1.8 and remarkably slowed the decay of [Ca2+]i in the trailing flank of a wave. Using image analysis, rise times, and half-maximum decay times of local Ca2+ transients, which characterize the processes of local Ca2+ release and removal, were determined as a function of temperature. Apparent activation energies for wave front propagation, local Ca2+ release, and local Ca2+ removal were derived from Arrhenius plots and amounted to -23, -28, and -46 kJ/mol, respectively. The high activation energy of Ca2+ removal, which arises from the activity of the sarcoplasmic reticulum (SR) Ca2+ ATPase, relative to those of longitudinal wave propagation and local Ca2+ release excludes the hypothetical mechanism of regenerative "spontaneous Ca2+ release," in which Ca2+ that has been taken up from the approaching wavefront triggers Ca2+ release at a luminal site of the SR. It is consistent, however, with the hypothesis that Ca2+ wave propagation is based on Ca(2+)-induced Ca2+ release where Ca2+ triggers release on the cytosolic face of the SR.     

A non-equilibrium thermodynamics model of reconstituted Ca2+-ATPase

A non-equilibrium thermodynamics (NET) model describing the action of completely coupled or `slipping' reconstituted Ca<sup>2+</sup>-ATPase is presented. Variation of the coupling stoichiometries with the magnitude of the electrochemical gradients, as the ATPase hydrolyzes ATP, is an indication of molecular slip. However, the Ca<sup>2+</sup> and H<sup>+</sup> membrane-leak conductances may also be a function of their respective gradients. Such non-ohmic leak typically yields `flow-force' relationships that are similar to those that are obtained when the pump slips; hence, caution needs to be exercised when interpreting data of Ca²⁺-ATPase-mediated fluxes that display a non-linear dependence on the electrochemical proton (Δμ˜_H) and/or calcium gradients (Δμ˜_Ca). To address this issue, three experimentally verifiable relationships differentiating between membrane leak and enzymic slip were derived. First, by measuring Δμ˜_H as a function of the rate of ATP hydrolysis by the enzyme. Second, by measuring the overall `efficiency' of the pump as a function of Δμ˜_H. Third, by measuring the proton ejection rate by the pump as a function of its ATP hydrolysis rate. Received: 19 June 1997 / Accepted: 3 December 1997     

A kinetic model for Ca2+ efflux mediated by the Ca2+ + Mg2+-activated ATPase of sarcoplasmic reticulum.

We present a model for Ca2+ efflux from vesicles of sarcoplasmic reticulum (SR). It is proposed that efflux is mediated by the Ca2+ + Mg2+-activated ATPase that is responsible for Ca2+ uptake in this system. In the normal ATPase cycle of the ATPase, phosphorylation of the ATPase is followed by a conformational change in which the Ca2+-binding sites change from being outward-facing and of high affinity to being inward-facing and of low affinity. To mediate Ca2+ efflux, it is proposed that the ATPase can adopt a conformation in which the Ca2+-binding sites are of low affinity but still outward-facing. It is shown that experimental data on the rates of Ca2+ efflux can be simulated in terms of this model, with Ca2+-binding-site affinities previously proposed to explain ATPase activity [Gould, East, Froud, McWhirter, Stefanova & Lee (1986) Biochem. J. 237, 217-227]. Effects of Mg2+ and adenine nucleotides on efflux rates are explained. It is suggested that Ca2+ efflux from SR mediated by the ATPase could be important in excitation-contraction coupling in skeletal muscle.     

A model of intracellular Ca2+ oscillations based on the activity of the intermediate-conductance Ca2+-activated K+ channels

Intracellular Ca2+ oscillations are observed in a large number of non-excitable cells. While most appear to reflect an intermittent Ca2+ release from intracellular stores, in some instances intracellular Ca2+ oscillations strongly depend on Ca2+ influx, and are coupled to oscillations of the membrane potential, suggesting that a plasma membrane-based mechanism may be involved. We have developed a theoretical model for the latter type of intracellular Ca2+ oscillations based on the Ca2+-dependent modulation of the intermediate-conductance, Ca2+-activated K+ (IKCa) channel. The functioning of this model relies on the Ca2+-dependent activation, and the much slower Ca2+-dependent rundown of this channel. We have shown that Ca2+-dependent activation of the IKCa channels, the consequent membrane hyperpolarization and the resulting increase in Ca2+ influx may confer the positive feedback mechanism required for the ascending phase of the oscillation. The much slower Ca2+-dependent rundown process will conversely halt this positive loop, and establish the descending phase of the intracellular Ca2+ oscillation. We found that this simple model gives rise to intracellular Ca2+ oscillations when using physiologically reasonable parameters, suggesting that IKCa channels could participate in the generation of intracellular Ca2+ oscillations.     

A minimal model for decoding of time-limited Ca2+ oscillations

Calcium oscillations regulate several cellular processes by activating particular proteins. Most theoretical studies focused on the idealized situation of infinitely long oscillations. Here we analyze information transfer by time-limited calcium spike trains. We show that proteins can be selectively activated in a resonance-like manner by time-limited spike trains of different frequencies, while infinitely long oscillations do not show this resonance phenomenon. We found that proteins are activated more specifically by shorter oscillatory signals with narrower spikes.     

Ca2+-calmodulin-dependent facilitation and Ca2+ inactivation of Ca2+ release-activated Ca2+ channels

In non-excitable cells, one major route for Ca2+ influx is through store-operated Ca2+ channels in the plasma membrane. These channels are activated by the emptying of intracellular Ca2+ stores, and in some cell types store-operated influx occurs through Ca2+ release-activated Ca2+ (CRAC) channels. Here, we report that intracellular Ca2+ modulates CRAC channel activity through both positive and negative feedback steps in RBL-1 cells. Under conditions in which cytoplasmic Ca2+ concentration can fluctuate freely, we find that store-operated Ca2+ entry is impaired either following overexpression of a dominant negative calmodulin mutant or following whole-cell dialysis with a calmodulin inhibitory peptide. The peptide had no inhibitory effect when intracellular Ca2+ was buffered strongly at low levels. Hence, Ca2+-calmodulin is not required for the activation of CRAC channels per se but is an important regulator under physiological conditions. We also find that the plasma membrane Ca2+ATPase is the dominant Ca2+ efflux pathway in these cells. Although the activity of the Ca2+ pump is regulated by calmodulin, the store-operated Ca2+ entry is more sensitive to inhibition by the calmodulin mutant than by Ca2+ extrusion. Hence, these two plasmalemmal Ca2+ transport systems may differ in their sensitivities to endogenous calmodulin. Following the activation of Ca2+ entry, the rise in intracellular Ca2+ subsequently feeds back to further inhibit Ca2+ influx. This slow inactivation can be activated by a relatively brief Ca2+ influx (30-60 s); it reverses slowly and is not altered by overexpression of the calmodulin mutant. Hence, the same messenger, intracellular Ca2+, can both facilitate and inactivate Ca2+ entry through store-operated CRAC channels and through different mechanisms.     

Divergence of Ca2+ selectivity and equilibrium Ca2+ blockade in a Ca2+ release-activated Ca2+ channel

Prevailing models postulate that high Ca²⁺ selectivity of Ca²⁺ release-activated Ca²⁺ (CRAC) channels arises from tight Ca²⁺ binding to a high affinity site within the pore, thereby blocking monovalent ion flux. Here, we examined the contribution of high affinity Ca²⁺ binding for Ca²⁺ selectivity in recombinant Orai3 channels, which function as highly Ca²⁺-selective channels when gated by the endoplasmic reticulum Ca²⁺ sensor STIM1 or as poorly Ca²⁺-selective channels when activated by the small molecule 2-aminoethoxydiphenyl borate (2-APB). Extracellular Ca²⁺ blocked Na⁺ currents in both gating modes with a similar inhibition constant (K_i; ∼25 µM). Thus, equilibrium binding as set by the K_i of Ca²⁺ blockade cannot explain the differing Ca²⁺ selectivity of the two gating modes. Unlike STIM1-gated channels, Ca²⁺ blockade in 2-APB–gated channels depended on the extracellular Na⁺ concentration and exhibited an anomalously steep voltage dependence, consistent with enhanced Na⁺ pore occupancy. Moreover, the second-order rate constants of Ca²⁺ blockade were eightfold faster in 2-APB–gated channels than in STIM1-gated channels. A four-barrier, three–binding site Eyring model indicated that lowering the entry and exit energy barriers for Ca²⁺ and Na⁺ to simulate the faster rate constants of 2-APB–gated channels qualitatively reproduces their low Ca²⁺ selectivity, suggesting that ion entry and exit rates strongly affect Ca²⁺ selectivity. Noise analysis indicated that the unitary Na⁺ conductance of 2-APB–gated channels is fourfold larger than that of STIM1-gated channels, but both modes of gating show a high open probability (P_o; ∼0.7). The increase in current noise during channel activation was consistent with stepwise recruitment of closed channels to a high P_o state in both cases, suggesting that the underlying gating mechanisms are operationally similar in the two gating modes. These results suggest that both high affinity Ca²⁺ binding and kinetic factors contribute to high Ca²⁺ selectivity in CRAC channels.     

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+).     

On the role of stochastic channel behavior in intracellular Ca2+ dynamics

I present a stochastic model for intracellular Ca(2+) oscillations. The model starts from stochastic binding and dissociation of Ca(2+) to binding sites on a single subunit of the IP(3)-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.     

A fast Ca2+-induced Ca2+-release mechanism in Dictyostelium discoideum

In vertebrate cells calcium-induced calcium release (CICR) is thought to be responsible for rapid cytosolic Ca(2+) elevations despite the occurrence of strong Ca(2+) buffering within the cytosol. In Dictyostelium, a CICR mechanism has not been reported. While analyzing Ca(2+) regulation in a vesicular fraction of Dictyostelium rich in Ca(2+)-flux activity, containing contractile vacuoles (CV) as the main component of acidic Ca(2+) stores and ER, we detected a rapid Ca(2+) change upon addition of Ca(2+) (CIC). CIC was three times larger in active stores accumulating Ca(2+) than before Ca(2+) uptake and in inactivated stores. Ca(2+) release was demonstrated with the calmodulin antagonist W7 that inhibits the V-type H(+)ATPase activity and Ca(2+) uptake of acidic Ca(2+) stores. W7 caused a rapid and large increase of extravesicular Ca(2+) ([Ca(2+)](e)), much faster and larger than thapsigargin (Tg), a Ca(2+)-uptake inhibitor of the ER. W7 treatment blocked CIC indicating that a large part of CIC is due to Ca(2+) release. The height of CIC depended on the filling state of the Ca(2+) stores. CIC was virtually unchanged in the iplA(-) strain that lacks a putative IP(3) or ryanodine receptor thought to be located at the endoplasmic reticulum. By contrast, CIC was reduced in two mutants, HGR8 and lvsA(-), that are impaired in acidic Ca(2+)-store function. Purified Ca(2+) stores enriched in CV still displayed CIC, indicating that CV are a source of Ca(2+)-induced Ca(2+) release. CIC-defective mutants were altered in their oscillatory properties. The irregularity of the HGR8 oscillation suggests that the principal oscillator is affected in this mutant.     

Mechanism of Ca2+ wave propagation in pancreatic acinar cells.

An increase in cytosolic Ca2+ often begins as a Ca2+ wave, and this wave is thought to result from sequential activation of Ca(2+)-sensitive Ca2+ stores across the cell. We tested that hypothesis in pancreatic acinar cells, and since Ca2+ waves may regulate acinar Cl- secretion, we examined whether such waves also are important for amylase secretion. Ca2+ wave speed and direction was determined in individual cells within rat pancreatic acini using confocal line scanning microscopy. Both acetylcholine (ACh) and cholecystokinin-8 induced rapid Ca2+ waves which usually travelled in an apical-to-basal direction. Both caffeine and ryanodine, at concentrations that inhibit Ca(2+)-induced Ca2+ release (CICR), markedly slowed the speed of these waves. Amylase secretion was increased over 3-fold in response to ACh stimulation, and this increase was preserved in the presence of ryanodine. These results indicate that 1) stimulation of either muscarinic or cholecystokinin-8 receptors induces apical-to-basal Ca2+ waves in pancreatic acinar cells, 2) the speed of such waves is dependent upon mobilization of caffeine- and ryanodine-sensitive Ca2+ stores, and 3) ACh-induced amylase secretion is not inhibited by ryanodine. These observations provide direct evidence that Ca(2+)-induced Ca2+ release is important for propagation of cytosolic Ca2+ waves in pancreatic acinar cells.     

Cardiac cellular coupling and the spread of early instabilities in intracellular Ca2+

Recent experimental and modeling studies demonstrate the fine spatial scale, complex nature, and independent contribution of Ca²⁺ dynamics as a proarrhythmic factor in the heart. The mechanism of progression of cell-level Ca²⁺ instabilities, known as alternans, to tissue-level arrhythmias is not well understood. Because gap junction coupling dictates cardiac syncytial properties, we set out to elucidate its role in the spatiotemporal evolution of Ca²⁺ instabilities. We experimentally perturbed cellular coupling in cardiac syncytium in vitro. Coupling was quantified by fluorescence recovery after photobleaching, and related to function, including subtle fine-scale Ca²⁺ alternans, captured by optical mapping. Conduction velocity and threshold for alternans monotonically increased with coupling. Lower coupling enhanced Ca²⁺ alternans amplitude, but the spatial spread of early (<2 Hz) alternation was the greatest under intermediate (not low) coupling. This nonmonotonic relationship was closely matched by the percent of samples exhibiting large-scale alternans at higher pacing rates. Computer modeling corroborated these experimental findings for strong but not weak electromechanical (voltage-Ca²⁺) coupling, and offered mechanistic insight. In conclusion, using experimental and modeling approaches, we reveal a general mechanism for the spatial spread of subtle cellular Ca²⁺ alternans that relies on a combination of gap-junctional and voltage-Ca²⁺ coupling.