A bidomain threshold model of propagating calcium waves

IP3 receptors: An "elementary" journey from structure to signals

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are large tetrameric channels which sit mostly in the membrane of the endoplasmic reticulum (ER) and mediate Ca2+ release from intracellular stores in response to extracellular stimuli in almost all cells. Dual regulation of IP3Rs by IP3 and Ca2+ itself, upstream "licensing", and the arrangement of IP3Rs into small clusters in the ER membrane, allow IP3Rs to generate spatially and temporally diverse Ca2+ signals. The characteristic biphasic regulation of IP3Rs by cytosolic Ca2+ concentration underpins regenerative Ca2+ signals by Ca2+-induced Ca2+-release, while also preventing uncontrolled explosive Ca2+ release. In this way, cells can harness a simple ion such as Ca2+ as a near-universal intracellular messenger to regulate diverse cellular functions, including those with conflicting outcomes such as cell survival and cell death. High-resolution structures of the IP3R bound to IP3 and Ca2+ in different combinations have together started to unravel the workings of this giant channel. Here we discuss, in the context of recently published structures, how the tight regulation of IP3Rs and their cellular geography lead to generation of "elementary" local Ca2+ signals known as Ca2+ "puffs", which form the fundamental bottleneck through which all IP3-mediated cytosolic Ca2+ signals must first pass.

Fundamentals of Cellular Calcium Signaling: A Primer

Ionized calcium (Ca²⁺) is the most versatile cellular messenger. All cells use Ca²⁺ signals to regulate their activities in response to extrinsic and intrinsic stimuli. Alterations in cellular Ca²⁺ signaling and/or Ca²⁺ homeostasis can subvert physiological processes into driving pathological outcomes. Imaging of living cells over the past decades has demonstrated that Ca²⁺ signals encode information in their frequency, kinetics, amplitude, and spatial extent. These parameters alter depending on the type and intensity of stimulation, and cellular context. Moreover, it is evident that different cell types produce widely varying Ca²⁺ signals, with properties that suit their physiological functions. This primer discusses basic principles and mechanisms underlying cellular Ca²⁺ signaling and Ca²⁺ homeostasis. Consequently, we have cited some historical articles in addition to more recent findings. A brief summary of the core features of cellular Ca²⁺ signaling is provided, with particular focus on Ca²⁺ stores and Ca²⁺ transport across cellular membranes, as well as mechanisms by which Ca²⁺ signals activate downstream effector systems.

What Is Required for Neuronal Calcium Waves? A Numerical Parameter Study

Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be significantly increased by means of calcium-induced calcium release from internal calcium stores, notably the endoplasmic reticulum. The organelle, which can be thought of as a cell-within-a-cell, is able to sequester large amounts of cytosolic calcium ions via SERCA pumps and selectively release them into the cytosol through ryanodine receptor channels leading to the formation of calcium waves. In this study, we set out to investigate the basic properties of such dendritic calcium waves and how they depend on the three parameters dendrite radius, ER radius and ryanodine receptor density in the endoplasmic membrane. We demonstrate that there are stable and abortive regimes for calcium waves, depending on the above morphological and physiological parameters. In stable regimes, calcium waves can travel across long dendritic distances, similar to electrical action potentials. We further observe that abortive regimes exist, which could be relevant for spike-timing dependent plasticity, as travel distances and wave velocities vary with changing intracellular architecture. For some of these regimes, analytic functions could be derived that fit the simulation data. In parameter spaces, that are non-trivially influenced by the three-dimensional calcium concentration profile, we were not able to derive such a functional description, demonstrating the mathematical requirement to model and simulate biochemical signaling in three-dimensional space.

Unifying principles of calcium wave propagation - Insights from a three-dimensional model for atrial myocytes

Atrial myocytes in a number of species lack transverse tubules. As a consequence the intracellular calcium signals occurring during each heartbeat exhibit complex spatio-temporal dynamics. These calcium patterns arise from saltatory calcium waves that propagate via successive rounds of diffusion and calcium-induced calcium release. The many parameters that impinge on calcium-induced calcium release and calcium signal propagation make it difficult to know a priori whether calcium waves will successfully travel, or be extinguished. In this study, we describe in detail a mathematical model of calcium signalling that allows the effect of such parameters to be independently assessed. A key aspect of the model is to follow the triggering and evolution of calcium signals within a realistic three-dimensional cellular volume of an atrial myocyte, but with low computational costs. This is achieved by solving the linear transport equation for calcium analytically between calcium release events and by expressing the onset of calcium liberation as a threshold process. The model makes non-intuitive predictions about calcium signal propagation. For example, our modelling illustrates that the boundary of a cell produces a wave-guiding effect that enables calcium ions to propagate further and for longer, and can subtly alter the pattern of calcium wave movement. The high spatial resolution of the modelling framework allows the study of any arrangement of calcium release sites. We demonstrate that even small variations in randomly positioned release sites cause highly heterogeneous cellular responses. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

Basis for a neuronal version of Grover's quantum algorithm

Grover's quantum (search) algorithm exploits principles of quantum information theory and computation to surpass the strong Church–Turing limit governing classical computers. The algorithm initializes a search field into superposed N (eigen)states to later execute nonclassical “subroutines” involving unitary phase shifts of measured states and to produce root-rate or quadratic gain in the algorithmic time (O(N^1/2)) needed to find some “target” solution m. Akin to this fast technological search algorithm, single eukaryotic cells, such as differentiated neurons, perform natural quadratic speed-up in the search for appropriate store-operated Ca²⁺ response regulation of, among other processes, protein and lipid biosynthesis, cell energetics, stress responses, cell fate and death, synaptic plasticity, and immunoprotection. Such speed-up in cellular decision making results from spatiotemporal dynamics of networked intracellular Ca²⁺-induced Ca²⁺ release and the search (or signaling) velocity of Ca²⁺ wave propagation. As chemical processes, such as the duration of Ca²⁺ mobilization, become rate-limiting over interstore distances, Ca²⁺ waves quadratically decrease interstore-travel time from slow saltatory to fast continuous gradients proportional to the square-root of the classical Ca²⁺ diffusion coefficient, D^1/2, matching the computing efficiency of Grover's quantum algorithm. In this Hypothesis and Theory article, I elaborate on these traits using a fire-diffuse-fire model of store-operated cytosolic Ca²⁺ signaling valid for glutamatergic neurons. Salient model features corresponding to Grover's quantum algorithm are parameterized to meet requirements for the Oracle Hadamard transform and Grover's iteration. A neuronal version of Grover's quantum algorithm figures to benefit signal coincidence detection and integration, bidirectional synaptic plasticity, and other vital cell functions by rapidly selecting, ordering, and/or counting optional response regulation choices.

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Spatial Quantification of Cytosolic Ca²⁺ Accumulation in Nonexcitable Cells: An Analytical Study

Calcium ions act as messengers in a broad range of processes such as learning, apoptosis, and muscular movement. The transient profile and the temporal accumulation of calcium signals have been suggested as the two main characteristics in which calcium cues encode messages to be forwarded to downstream pathways. We address the analytical quantification of calcium temporal-accumulation in a long, thin section of a nonexcitable cell by solving a boundary value problem. In these expressions we note that the cytosolic Ca(2+) accumulation is independent of every intracellular calcium flux and depends on the Ca(2+) exchange across the membrane, cytosolic calcium diffusion, geometry of the cell, extracellular calcium perturbation, and initial concentrations. In particular, we analyse the time-integrated response of cytosolic calcium due to i) a localised initial concentration of cytosolic calcium and ii) transient extracellular perturbation of calcium. In these scenarios, we conclude that i) the range of calcium progression is confined to the vicinity of the initial concentration, thereby creating calcium microdomains; and ii) we observe a low-pass filtering effect in the response driven by extracellular Ca(2+) perturbations. Additionally, we note that our methodology can be used to analyse a broader range of stimuli and scenarios.

Synchrony of cardiomyocyte Ca(2+) release is controlled by T-tubule organization, SR Ca(2+) content, and ryanodine receptor Ca(2+) sensitivity

Recent work has demonstrated that cardiomyocyte Ca(2+)release is desynchronized in several pathological conditions. Loss of Ca(2+) release synchrony has been attributed to t-tubule disruption, but it is unknown if other factors also contribute. We investigated this issue in normal and failing myocytes by integrating experimental data with a mathematical model describing spatiotemporal dynamics of Ca(2+) in the cytosol and sarcoplasmic reticulum (SR). Heart failure development in postinfarction mice was associated with progressive t-tubule disorganization, as quantified by fast-Fourier transforms. Data from fast-Fourier transforms were then incorporated in the model as a dyadic organization index, reflecting the proportion of ryanodine receptors located in dyads. With decreasing dyadic-organization index, the model predicted greater dyssynchrony of Ca(2+) release, which exceeded that observed in experimental line-scan images. Model and experiment were reconciled by reducing the threshold for Ca(2+) release in the model, suggesting that increased RyR sensitivity partially offsets the desynchronizing effects of t-tubule disruption in heart failure. Reducing the magnitude of SR Ca(2+) content and release, whether experimentally by thapsigargin treatment, or in the model, desynchronized the Ca(2+) transient. However, in cardiomyocytes isolated from SERCA2 knockout mice, RyR sensitization offset such effects. A similar interplay between RyR sensitivity and SR content was observed during treatment of myocytes with low-dose caffeine. Initial synchronization of Ca(2+) release during caffeine was reversed as SR content declined due to enhanced RyR leak. Thus, synchrony of cardiomyocyte Ca(2+) release is not only determined by t-tubule organization but also by the interplay between RyR sensitivity and SR Ca(2+) content.

Ca²⁺ waves in the heart

Ca(2+) waves were probably first observed in the early 1940s. Since then Ca(2+) waves have captured the attention of an eclectic mixture of mathematicians, neuroscientists, muscle physiologists, developmental biologists, and clinical cardiologists. This review discusses the current state of mathematical models of Ca(2+) waves, the normal physiological functions Ca(2+) waves might serve in cardiac cells, as well as how the spatial arrangement of Ca(2+) release channels shape Ca(2+) waves, and we introduce the idea of Ca(2+) phase waves that might provide a useful framework for understanding triggered arrhythmias.

Persistence of pro-arrhythmic spatio-temporal calcium patterns in atrial myocytes: a computational study of ping waves

Clusters of ryanodine receptors within atrial myocytes are confined to spatially separated layers. In many species, these layers are not juxtaposed by invaginations of the plasma membrane (transverse tubules; 'T-tubules'), so that calcium-induced-calcium signals rely on centripetal propagation rather than voltage-synchronized channel openings to invade the interior of the cell and trigger contraction. The combination of this specific cellular geometry and dynamics of calcium release can lead to novel autonomous spatio-temporal calcium waves, and in particular ping waves. These are waves of calcium release activity that spread as counter-rotating sectors of elevated calcium within a single layer of ryanodine receptors, and can seed further longitudinal calcium waves. Here we show, using a computational model, that these calcium waves can dominate the response of a cell to electrical pacing and hence are pro-arrhythmic. This highlights the importance of modeling internal cellular structures when investigating mechanisms of cardiac dysfunction such as atrial arrhythmia.

Dynamic local changes in sarcoplasmic reticulum calcium: physiological and pathophysiological roles

Evidence obtained in recent years indicates that, in cardiac myocytes, release of Ca(2+) from the sarcoplasmic reticulum (SR) is regulated by changes in the concentration of Ca(2+) within the SR. In this review, we summarize recent advances in our understanding of this regulatory role, with a particular emphasis on dynamic and local changes in SR [Ca(2+)]. We focus on five important questions that are to some extent unresolved and controversial. These questions concern: (1) the importance of SR [Ca(2+)] depletion in the termination of Ca(2+) release; (2) the quantitative extent of depletion during local release events such as Ca(2+) sparks; (3) the influence of SR [Ca(2+)] refilling on release refractoriness and the propensity for pathological Ca(2+) release; (4) dynamic changes in SR [Ca(2+)] during propagating Ca(2+) waves; and (5) the speed of Ca(2+) diffusion within the SR. With each issue, we discuss data supporting alternative viewpoints, and we identify fundamental questions that are being actively investigated. We conclude with a discussion of experimental and computational advances that will help to resolve controversies. This article is part of a special issue entitled "Local Signaling in Myocytes."

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.

Calcium oscillations

Calcium signaling results from a complex interplay between activation and inactivation of intracellular and extracellular calcium permeable channels. This complexity is obvious from the pattern of calcium signals observed with modest, physiological concentrations of calcium-mobilizing agonists, which typically present as sequential regenerative discharges of stored calcium, a process referred to as calcium oscillations. In this review, we discuss recent advances in understanding the underlying mechanism of calcium oscillations through the power of mathematical modeling. We also summarize recent findings on the role of calcium entry through store-operated channels in sustaining calcium oscillations and in the mechanism by which calcium oscillations couple to downstream effectors.

Dynamic calcium movement inside cardiac sarcoplasmic reticulum during release

RATIONALE

Intra-sarcoplasmic reticulum (SR) free [Ca] ([Ca](SR)) provides the driving force for SR Ca release and is a key regulator of SR Ca release channel gating during normal SR Ca release or arrhythmogenic spontaneous Ca release events. However, little is known about [Ca](SR) spatiotemporal dynamics.

OBJECTIVE

To directly measure local [Ca](SR) with subsarcomeric spatiotemporal resolution during both normal global SR Ca release and spontaneous Ca sparks and to evaluate the quantitative implications of spatial [Ca](SR) gradients.

METHODS AND RESULTS

Intact and permeabilized rabbit ventricular myocytes were subjected to direct simultaneous measurement of cytosolic [Ca] and [Ca](SR) and FRAP (fluorescence recovery after photobleach). We found no detectable [Ca](SR) gradients between SR release sites (junctional SR) and Ca uptake sites (free SR) during normal global Ca release, clear spatiotemporal [Ca](SR) gradients during isolated Ca blinks, faster intra-SR diffusion in the longitudinal versus transverse direction, 3- to 4-fold slower diffusion of fluorophores in the SR than in cytosol, and that intra-SR Ca diffusion varies locally, dependent on local SR connectivity. A computational model clarified why spatiotemporal gradients are more detectable in isolated local releases versus global releases and provides a quantitative framework for understanding intra-SR Ca diffusion.

CONCLUSIONS

Intra-SR Ca diffusion is rapid, limiting spatial [Ca](SR) gradients during excitation-contraction coupling. Spatiotemporal [Ca](SR) gradients are apparent during Ca sparks, and these observations constrain models of dynamic Ca movement inside the SR. This has important implications for myocyte SR Ca handling, synchrony, and potentially arrhythmogenic spontaneous contraction.

Intercellular calcium waves in the fire-diffuse-fire framework: Green's function for gap-junctional coupling

Calcium is a crucial component in a plethora of cellular processes involved in cell birth, life, and death. Intercellular calcium waves that can spread through multiple cells provide one form of cellular communication mechanism between various parts of cell tissues. Here we introduce a simple, yet biophysically realistic model for the propagation of intercellular calcium waves based on the fire-diffuse-fire type model for calcium dynamics. Calcium release sites are considered to be discretely distributed along individual linear cells that are connected by gap junctions and a solution of this model can be found in terms of the Green's function for this system. We develop the "sum-over-trips" formalism that takes into account the boundary conditions at gap junctions providing a generalization of the original sum-over-trips approach for constructing the response function for branched neural dendrites. We obtain the exact solution of the Green's function in the Laplace (frequency) domain for an infinite array of cells and show that this Green's function can be well approximated by its truncated version. This allows us to obtain an analytical traveling wave solution for an intercellular calcium wave and analyze the speed of solitary wave propagation as a function of physiologically important system parameters. Periodic and irregular traveling waves can be also sustained by the proposed model.

Predicting local SR Ca(2+) dynamics during Ca(2+) wave propagation in ventricular myocytes

Of the many ongoing controversies regarding the workings of the sarcoplasmic reticulum (SR) in cardiac myocytes, two unresolved and interconnected topics are 1), mechanisms of calcium (Ca(2+)) wave propagation, and 2), speed of Ca(2+) diffusion within the SR. Ca(2+) waves are initiated when a spontaneous local SR Ca(2+) release event triggers additional release from neighboring clusters of SR release channels (ryanodine receptors (RyRs)). A lack of consensus regarding the effective Ca(2+) diffusion constant in the SR (D(Ca,SR)) severely complicates our understanding of whether dynamic local changes in SR [Ca(2+)] can influence wave propagation. To address this problem, we have implemented a computational model of cytosolic and SR [Ca(2+)] during Ca(2+) waves. Simulations have investigated how dynamic local changes in SR [Ca(2+)] are influenced by 1), D(Ca,SR); 2), the distance between RyR clusters; 3), partial inhibition or stimulation of SR Ca(2+) pumps; 4), SR Ca(2+) pump dependence on cytosolic [Ca(2+)]; and 5), the rate of transfer between network and junctional SR. Of these factors, D(Ca,SR) is the primary determinant of how release from one RyR cluster alters SR [Ca(2+)] in nearby regions. Specifically, our results show that local increases in SR [Ca(2+)] ahead of the wave can potentially facilitate Ca(2+) wave propagation, but only if SR diffusion is relatively slow. These simulations help to delineate what changes in [Ca(2+)] are possible during SR Ca(2+)release, and they broaden our understanding of the regulatory role played by dynamic changes in [Ca(2+)](SR).

Calsequestrin mediates changes in spontaneous calcium release profiles

Calsequestrin (CSQ) is the primary calcium buffer within the sarcoplasmic reticulum (SR) of cardiac cells. It has also been identified as a regulator of Ryanodine receptor (RyR) calcium release channels by serving as a SR luminal sensor. When calsequestrin is free and unbound to calcium, it can bind to RyR and desensitize the channel from cytoplasmic calcium activation. In this paper, we study the role of CSQ as a buffer and RyR luminal sensor using a mechanistic model of RyR-CSQ interaction. By using various asymptotic approximations and mean first exit time calculation, we derive a minimal model of a calcium release unit which includes CSQ dependence. Using this model, we then analyze the effect of changing CSQ expression on the calcium release profile and the rate of spontaneous calcium release. We show that because of its buffering capability, increasing CSQ increases the spark duration and size. However, because of luminal sensing effects, increasing CSQ depresses the basal spark rate and increases the critical SR level for calcium release termination. Finally, we show that with increased bulk cytoplasmic calcium concentration, the CRU model exhibits deterministic oscillations.

Cardiac cell modelling: observations from the heart of the cardiac physiome project

In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.

Calcium oscillations

Changes in cellular Ca2+ concentration control a wide range of physiological processes, from the subsecond release of synaptic neurotransmitters, to the regulation of gene expression over months or years. Ca2+ can also trigger cell death through both apoptosis and necrosis, and so the regulation of cellular Ca2+ concentration must be tightly controlled through the concerted action of pumps, channels and buffers that transport Ca2+ into and out of the cell cytoplasm. A hallmark of cellular Ca2+ signalling is its spatiotemporal complexity: stimulation of cells by a hormone or neurotransmitter leads to oscillations in cytoplasmic Ca2+ concentration that can vary markedly in time course, amplitude, frequency, and spatial range. In this chapter we review some of the biological roles of Ca2+, the experimental characterisation of complex dynamic changes in Ca2+ concentration, and attempts to explain this complexity using computational models. We consider the 'toolkit' of cellular proteins which influence Ca2+ concentrarion, describe mechanistic models of key elements of the toolkit, and fit these into the framework of whole cell models of Ca2+ oscillations and waves. Finally, we will touch on recent efforts to use stochastic modelling to elucidate elementary Ca2+ signal events, and how these may evolve into global signals.