Modeling spiral Ca2+ waves in single cardiac cells: role of the spatial heterogeneity created by the nucleus

Spiral wave initiation in excitable media

Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Spiral wave initiation in excitable media

Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Ryanodine receptor gating controls generation of diastolic calcium waves in cardiac myocytes

Calcium waves can form and propagate at low frequencies of spontaneous calcium sparks if the calcium dependence of spark frequency is sufficiently steep, or the number of open RyRs is sufficiently large.

The role of cardiac ryanodine receptor (RyR) gating in the initiation and propagation of calcium waves was investigated using a mathematical model comprising a stochastic description of RyR gating and a deterministic description of calcium diffusion and sequestration. We used a one-dimensional array of equidistantly spaced RyR clusters, representing the confocal scanning line, to simulate the formation of calcium sparks. Our model provided an excellent description of the calcium dependence of the frequency of diastolic calcium sparks and of the increased tendency for the production of calcium waves after a decrease in cytosolic calcium buffering. We developed a hypothesis relating changes in the propensity to form calcium waves to changes of RyR gating and tested it by simulation. With a realistic RyR gating model, increased ability of RyR to be activated by Ca²⁺ strongly increased the propensity for generation of calcium waves at low (0.05–0.1-µM) calcium concentrations but only slightly at high (0.2–0.4-µM) calcium concentrations. Changes in RyR gating altered calcium wave formation by changing the calcium sensitivity of spontaneous calcium spark activation and/or the average number of open RyRs in spontaneous calcium sparks. Gating changes that did not affect RyR activation by Ca²⁺ had only a weak effect on the propensity to form calcium waves, even if they strongly increased calcium spark frequency. Calcium waves induced by modulating the properties of the RyR activation site could be suppressed by inhibiting the spontaneous opening of the RyR. These data can explain the increased tendency for production of calcium waves under conditions when RyR gating is altered in cardiac diseases.

An integrated formulation of anisotropic force-calcium relations driving spatio-temporal contractions of cardiac myocytes

Isolated cardiac myocytes exhibit spontaneous patterns of rhythmic contraction, driven by intracellular calcium waves. In order to study the coupling between spatio-temporal calcium dynamics and cell contraction in large deformation regimes, a new strain-energy function, describing the influence of sarcomere length on the calcium-dependent generation of active intracellular stresses, is proposed. This strain-energy function includes anisotropic passive and active contributions that were first validated separately from experimental stress-strain curves and stress-sarcomere length curves, respectively. An extended validation of this formulation was then conducted by considering this strain-energy function as the core of an integrated mechano-chemical three-dimensional model of cardiac myocyte contraction, where autocatalytic intracellular calcium dynamics were described by a representative two-variable model able to generate realistic intracellular calcium waves similar to those observed experimentally. Finite-element simulations of the three-dimensional cell model, conducted for different intracellular locations of triggering calcium sparks, explained very satisfactorily, both qualitatively and quantitatively, the contraction patterns of cardiac myocytes observed by time-lapse videomicroscopy. This integrative approach of the mechano-chemical couplings driving cardiac myocyte contraction provides a comprehensive framework for analysing active stress regulation and associated mechano-transduction processes that contribute to the efficiency of cardiac cell contractility in both physiological and pathological contexts.

Calcium sparks

The calcium ion (Ca(2+)) is the simplest and most versatile intracellular messenger known. The discovery of Ca(2+) sparks and a related family of elementary Ca(2+) signaling events has revealed fundamental principles of the Ca(2+) signaling system. A newly appreciated "digital" subsystem consisting of brief, high Ca(2+) concentration over short distances (nanometers to microns) comingles with an "analog" global Ca(2+) signaling subsystem. Over the past 15 years, much has been learned about the theoretical and practical aspects of spark formation and detection. The quest for the spark mechanisms [the activation, coordination, and termination of Ca(2+) release units (CRUs)] has met unexpected challenges, however, and raised vexing questions about CRU operation in situ. Ample evidence shows that Ca(2+) sparks catalyze many high-threshold Ca(2+) processes involved in cardiac and skeletal muscle excitation-contraction coupling, vascular tone regulation, membrane excitability, and neuronal secretion. Investigation of Ca(2+) sparks in diseases has also begun to provide novel insights into hypertension, cardiac arrhythmias, heart failure, and muscular dystrophy. An emerging view is that spatially and temporally patterned activation of the digital subsystem confers on intracellular Ca(2+) signaling an exquisite architecture in space, time, and intensity, which underpins signaling efficiency, stability, specificity, and diversity. These recent advances in "sparkology" thus promise to unify the simplicity and complexity of Ca(2+) signaling in biology.

Modeling Ca2+ signaling differentiation during oocyte maturation

Ca2+ is a fundamental intracellular signal that mediates a variety of disparate physiological functions often in the same cell. Ca2+ signals span a wide range of spatial and temporal scales, which endow them with the specificity required to induce defined cellular functions. Furthermore, Ca2+ signaling is highly plastic as it is modulated dynamically during normal physiological development and under pathological conditions. However, the molecular mechanisms underlying Ca2+ signaling differentiation during cellular development remain poorly understood. Oocyte maturation in preparation for fertilization provides an exceptionally well-suited model to elucidate Ca2+ signaling regulation during cellular development. This is because a Ca2+ signal with specialized spatial and temporal dynamics is universally essential for egg activation at fertilization. Here we use mathematical modeling to define the critical determinants of Ca2+ signaling differentiation during oocyte maturation. We show that increasing IP3 receptor (IP3R) affinity replicates both elementary and global Ca2+ dynamics observed experimentally following oocyte maturation. Furthermore, our model reveals that because of the Ca2+ dependency of both SERCA and the IP3R, increased IP3R affinity shifts the system's equilibrium to a new steady state of high cytosolic Ca2+, which is essential for fertilization. Therefore our model provides unique insights into how relatively small alterations of the basic molecular mechanisms of Ca2+ signaling components can lead to dramatic alterations in the spatio-temporal properties of Ca2+ dynamics.

VISUALIZING Ca2+ SPARKS AND SUBSTRUCTURE OF Ca2+ WAVES BY TOTAL INTERNAL REFLECTION FLUORESCENCE MICROSCOPY (TIRFM)

Based on near-field optical theory, total internal reflection fluorescence microscope shows a novel character that its picture has great signal-to-noise ratio and high temporal resolution achieved by high quality CCD camera. This allows us to analyze the spatiotemporal details of local Ca 2+ dynamics within the nanoscale microdomain surrounding different Ca 2+ channels. We have recently constructed a versatile objective TIRFM equipped with a high numerical aperture (NA = 1.45) objective. Using fluo-4 as the Ca 2+ indicator, we visualized the near-membrane profiles of Ca 2+ waves and elementary Ca 2+ sparks generated by Ca 2+ release channels in rat ventricular myocytes. Different from those detected using conventional or confocal microscopy, Ca 2+ waves observed with TIRFM exhibited fine inhomogenous substructures. The propagation of Ca 2+ waves with anfractuous routes of spark recruitment is much more complicated than previously imagined. We believe that TIRFM will provide a unique tool for dissecting the microscopic mechanisms of intracellular Ca 2+ signaling.

Models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We review deterministic and stochastic mathematical models of the IPR, from the earliest ones of the 1970s and 1980s, to the most recent. The effects of IPR stochasticity on Ca2+ dynamics are briefly discussed.

Three-dimensional simulation of calcium waves and contraction in cardiomyocytes using the finite element method

To investigate the characteristics and underlying mechanisms of Ca(2+) wave propagation, we developed a three-dimensional (3-D) simulator of cardiac myocytes, in which the sarcolemma, myofibril, and Z-line structure with Ca(2+) release sites were modeled as separate structures using the finite element method. Similarly to previous studies, we assumed that Ca(2+) diffusion from one release site to another and Ca(2+)-induced Ca(2+) release were the basic mechanisms, but use of the finite element method enabled us to simulate not only the wave propagation in 3-D space but also the active shortening of the myocytes. Therefore, in addition to the dependence of the Ca(2+) wave propagation velocity on the sarcoplasmic reticulum Ca(2+) content and affinity of troponin C for Ca(2+), we were able to evaluate the influence of active shortening on the propagation velocity. Furthermore, if the initial Ca(2+) release took place in the proximity of the nucleus, spiral Ca(2+) waves evolved and spread in a complex manner, suggesting that this phenomenon has the potential for arrhythmogenicity. The present 3-D simulator, with its ability to study the interaction between Ca(2+) waves and contraction, will serve as a useful tool for studying the mechanism of this complex phenomenon.

Modeling the dependence of the period of intracellular Ca2+ waves on SERCA expression

Contrary to intuitive expectations, overexpression of sarco-endoplasmic reticulum (ER) Ca(2+) ATPases (SERCAs) in Xenopus oocytes leads to a decrease in the period and an increase in the amplitude of intracellular Ca(2+) waves. Here we examine these experimental findings by modeling Ca(2+) release using a modified Othmer-Tang-model. An increase in the period and a reduction in the amplitude of Ca(2+) wave activity are obtained when increases in SERCA density are simulated while keeping all other parameters of the model constant. However, Ca(2+) wave period can be reduced and the wave amplitude and velocity can be significantly increased when an increase in the luminal ER Ca(2+) concentration due to SERCA overexpression is incorporated into the model. Increased luminal Ca(2+) occurs because increased SERCA activity lowers cytosolic Ca(2+), which is partially replenished by Ca(2+) influx across the plasma membrane. These simulations are supported by experimental data demonstrating higher luminal Ca(2+) levels, decreased periods, increased amplitude, and increased velocity of Ca(2+) waves in response to increased SERCA density.

The voltage-sensitive release mechanism of excitation contraction coupling in rabbit cardiac muscle is explained by calcium-induced calcium release

The putative voltage-sensitive release mechanism (VSRM) was investigated in rabbit cardiac myocytes at 37 degrees C with high resistance microelectrodes to minimize intracellular dialysis. When the holding potential was adjusted from -40 to -60 mV, the putative VSRM was expected to operate alongside CICR. Under these conditions however, we did not observe a plateau at positive potentials of the cell shortening versus voltage relationship. The threshold for cell shortening changed by -10 mV, but this resulted from a similar change of the threshold for activation of inward current. Cell shortening under conditions where the putative VSRM was expected to operate was blocked in a dose dependent way by nifedipine and CdCl2 and blocked completely by NiCl2. "Tail contractions" persisted in the presence of nifedipine and CdCl2 but were blocked completely by NiCl2. Block of early outward current by 4-aminopyridine and 4-acetoamido-4'-isothiocyanato-stilbene-2,2'-disulfonic acid (SITS) demonstrated persisting inward current during test depolarizations despite the presence of nifedipine and CdCl2. Inward current did not persist in the presence of NiCl2. A tonic component of cell shortening that was prominent during depolarizations to positive potentials under conditions selective for the putative VSRM was sensitive to rapidly applied changes in superfusate [Na+] and to the outward Na+/Ca2+ exchange current blocking drug KB-R7943. This component of cell shortening was thought to be the result of Na+/Ca2+ exchange-mediated excitation contraction coupling. Cell shortening recorded under conditions selective for the putative VSRM was increased by the enhanced state of phosphorylation induced by isoprenaline (1 microM) and by enhancing sarcoplasmic reticulum Ca2+ content by manipulation of the conditioning steps. Under these conditions, cell shortening at positive test depolarizations was converted from tonic to phasic. We conclude that the putative VSRM is explained by CICR with the Ca2+ "trigger" supplied by unblocked L-type Ca2+ channels and Na+/Ca2+ exchange.

Simulation of waves in calcium models with 3D spherical geometry

Waves of calcium ions are present in fertilized eggs of many species. Models for pulse and tidal wave propagation have usually been studied in one or two spatial coordinates only. We examine in three spatial coordinates some established models, based on Ca(2+)-induced Ca(2+)-release from both (assumed) continuously or heterogeneously distributed stores of endoplasmic reticulum (ER) through channels activated by inositol triphosphate (IP(3)). With continuous IP(3) distribution decreasing radially towards the interior, we obtain concave pulse shapes for waves penetrating the interior. Concave waves are also recorded in systems with ER confined to distributions of small spheres (microdomains) inside the cell, which we simulate for front waves (tides) in bistable systems.

Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling

This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into 'minimal' models containing two variables and 'extended' models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3--calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.

Formation of planar and spiral Ca2+ waves in isolated cardiac myocytes

A novel Nipkow-type confocal microscope was applied to image spontaneously propagating Ca2+ waves in isolated rat ventricular myocytes by means of fluo-3. The sarcolemma was imaged with di-8-ANEPPS and the nucleus with SYTO 11. Full frame images in different vertical sections were obtained at video frame rate by means of an intensified CCD camera. Three types of Ca2+ waves were identified: spherical waves, planar waves, and spiral waves. Both spherical waves and spiral waves could initiate a planar wave, and planar waves were not influenced by the presence of a nucleus. Spiral waves, however, were consistently found adjacent to a nucleus and displayed a slower propagation rate and slower rate of increase in Ca2+ concentration in the wave front than did spherical and planar waves. The planar waves were apparent throughout the vertical axis of the cell, whereas spiral waves appeared to have a vertical height of approximately 3 microm, less than the maximum thickness of the nucleus (5.0 +/- 0.3 microm). These results provide experimental confirmation of previous modeling studies which predicted an influence of the nucleus on spiral-type Ca2+ waves. When a spontaneous Ca2+ wave is small relative to the size of the nucleus, it appears that the Ca2+ buffering by the nucleus is sufficient to slow the rate of spontaneous propagation of the Ca2+ wave in close proximity to the nucleus. These findings thus support the idea that the nucleus can influence complex behavior of Ca2+ waves in isolated cardiac myocytes.

Localized intracellular calcium signaling in muscle: calcium sparks and calcium quarks

Subcellularly localized Ca2+ signals in cardiac and skeletal muscle have recently been identified as elementary Ca2+ signaling events. The signals, termed Ca2+ sparks and Ca2+ quarks, represent openings of Ca2+ release channels located in the membrane of the sarcoplasmic reticulum (SR). In cardiac muscle, the revolutionary discovery of Ca2+ sparks has allowed the development of a fundamentally different concept for the amplification of Ca2+ signals by Ca(2+)-induced Ca2+ release. In such a system, a graded amplification of the triggering Ca2+ signal entering the myocyte via L-type Ca2+ channels is accomplished by a recruitment process whereby individual SR Ca2+ release units are locally controlled by L-type Ca2+ channels. In skeletal muscle, the initial SR Ca2+ release is governed by voltage-sensors but subsequently activates additional Ca2+ sparks by Ca(2+)-induced Ca2+ release from the SR. Results from studies on elementary Ca2+ release events will improve our knowledge of muscle Ca2+ signaling at all levels of complexity, from the molecule to normal cellular function, and from the regulation of cardiac and skeletal muscle force to the pathophysiology of excitation-contraction coupling.

Theoretical insights into the mechanism of spiral Ca2+ wave initiation in Xenopus oocytes

Spiral waves of intracellular Ca2+ have often been observed in Xenopus oocytes. Such waves can be accounted for by most realistic models for Ca2+ oscillations taking diffusion of cytosolic Ca2+ into account, but their initiation requires rather demanding and unphysiological initial conditions. Here, it is shown by means of numerical simulations that these spiral Ca2+ waves naturally arise if the cytoplasm is assumed to be heterogeneous both at the level of the synthesis and metabolism of D-myo-inositol 1,4,5-trisphosphate [Ins(1,4,5)P3] and at the level of the distribution of the Ins(1,4, 5)P3 receptors. In such conditions, a spiral can be initiated in the simulations after an increase in Ins(1,4,5)P3 concentration, with the direction of rotation being determined by the position of the region of high receptor density with respect to the locus of Ins(1,4, 5)P3 production.

Spark-to-wave transition: saltatory transmission of calcium waves in cardiac myocytes

Using a modular approach, in which kinetic models of various mechanisms of calcium handling in cells are fine-tuned to in vivo and in vitro measurements before combining them into whole-cell models, three distinct modes of transmission of calcium waves in mature and immature frog eggs have been defined. Two modes of transmission are found in immature eggs, where the inositol 1,4,5-trisphosphate receptor (IP3R) controls release of calcium from the endoplasmic reticulum (ER). The first mode corresponds to an excitable physiological state of the cytoplasm and results in solitary waves that can appear as circular or spiral waves in two dimensions with the wave speed proportional to the square root of the diffusion constant of calcium. A second mode occurs when the state of the cytoplasm is oscillatory and because of the small size of the buffered diffusion constant for calcium, the wave speed can appear to be weakly dependent on diffusion. In the mature frog egg, where the sperm-induced Ca2+ fertilization wave is a propagating front, the cytoplasm appears to be bistable and in this mode the wave speed is also proportional to the square root of the diffusion constant. Here we investigate a fourth mode of propagation for cardiac myocytes, in which calcium release from the sarcoplasmic reticulum (SR) is dominated by clusters of ryanodine receptors spaced at regular intervals. In myocytes a stochastically excitable myoplasm leads to the spontaneous production of calcium 'sparks' that under certain conditions can merge into saltatory waves with a speed proportional to the diffusion constant.

Evolving perspectives during 12 years of electrical turbulence

This Focus issue describes a problem in electrical dynamics which has fascinated generations of physiologists. There are today so many views of fibrillation that only the rarest generalization can embrace all of them. Fifty-two prominent investigators collaborate here to present aspects of the problem in these eighteen articles (including this introduction) tailored for readers whose principal expertise lies elsewhere. In "The High One's Lay" (Norse Runes, ca. 800) Odin remarks, "Much too early I came to many places: the beer was not yet ready, or was already drunk em leader " but to this one we come at very nearly the right time in 1998. This introduction attempts to guide newcomers by noting the changed or multiple meanings of novel technical terms while sorting the key facts and ideas into an order that facilitates comparison and contrast with those of a dozen years ago. This Focus issue is authored by some of the foremost innovators of both theory and experiment in this area. By assimilating their presentations the readers of Chaos can become well poised to appreciate and evaluate the definitive evidence expected in the next few years. (c) 1998 American Institute of Physics.

Patch-clamp detection of macromolecular translocation along nuclear pores

The present paper reviews the application of patch-clamp principles to the detection and measurement of macromolecular translocation along the nuclear pores. We demonstrate that the tight-seal 'gigaseal' between the pipette tip and the nuclear membrane is possible in the presence of fully operational nuclear pores. We show that the ability to form a gigaseal in nucleus-attached configurations does not mean that only the activity of channels from the outer membrane of the nuclear envelope can be detected. Instead, we show that, in the presence of fully operational nuclear pores, it is likely that the large-conductance ion channel activity recorded derives from the nuclear pores. We conclude the technical section with the suggestion that the best way to demonstrate that the nuclear pores are responsible for ion channel activity is by showing with fluorescence microscopy the nuclear translocation of ions and small molecules and the exclusion of the same from the cisterna enclosed by the two membranes of the envelope. Since transcription factors and mRNAs, two major groups of nuclear macromolecules, use nuclear pores to enter and exit the nucleus and play essential roles in the control of gene activity and expression, this review should be useful to cell and molecular biologists interested in understanding how patch-clamp can be used to quantitate the translocation of such macromolecules into and out of the nucleus.

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.