Timescales of IP(3)-evoked Ca(2+) spikes emerge from Ca(2+) puffs only at the cellular level

Cross Talking Calcium, IP3 and Buffer Dynamics Alters ATP and NADH Level in Obese and Normal Hepatocyte Cell

The cross talk between calcium (Ca2+), IP3 and buffer dynamics regulate various mechanisms in hepatocyte cells. The study of independent systems of calcium, IP3, and buffer signaling provides limited information about cell dynamics. In the current study, coupled reaction-diffusion equations are used to design a cross-talk model for IP3, buffer, and calcium dynamics in a hepatocyte cell. The one-way feedback of calcium, buffer, and IP3 in ATP production, ATP degradation, and NADH production rate is incorporated into the model. Numerical simulation has been done using the Finite Element Method (FEM) along the spatial direction and the Crank-Nicolson (C-N) method along the temporal direction. The numerical results are analysed to determine the effects of alterations in processes of cross-talking dynamics of IP3, buffer, and calcium on ATP and NADH production and degradation rate of ATP in a hepatocyte cell under normal and obesity conditions. The comparative analysis of these findings unveils notable distinctions induced by obesity in calcium dynamics, ATP and NADH synthesis, and ATP degradation kinetics.

An efficient reduced-lattice model of IP3R for probing Ca2+ dynamics

Numerous cellular processes are regulated by Ca2+ signals, and the endoplasmic reticulum (ER) membrane's inositol triphosphate receptor (IP3R) is critical for modulating intracellular Ca2+ dynamics. The IP3Rs are seen to be clustered in a variety of cell types. The combination of IP3Rs clustering and IP3Rs-mediated Ca2+-induced Ca2+ release results in the hierarchical organization of the Ca2+ signals, which challenges the numerical simulation given the multiple spatial and temporal scales that must be covered. The previous methods rather ignore the spatial feature of IP3Rs or fail to coordinate the conflicts between the real biological relevance and the computational cost. In this work, a general and efficient reduced-lattice model is presented for the simulation of IP3Rs-mediated multiscale Ca2+ dynamics. The model highlights biological details that make up the majority of the calcium events, including IP3Rs clustering and calcium domains, and it reduces the complexity by approximating the minor details. The model's extensibility provides fresh insights into the function of IP3Rs in producing global Ca2+ events and supports the research under more physiological circumstances. Our work contributes to a novel toolkit for modeling multiscale Ca2+ dynamics and advances knowledge of Ca2+ signals.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

Highly localized intracellular Ca2+ signals promote optimal salivary gland fluid secretion

Salivary fluid secretion involves an intricate choreography of membrane transporters to result in the trans-epithelial movement of NaCl and water into the acinus lumen. Current models are largely based on experimental observations in enzymatically isolated cells where the Ca²⁺ signal invariably propagates globally and thus appears ideally suited to activate spatially separated Cl and K channels, present on the apical and basolateral plasma membrane, respectively. We monitored Ca²⁺ signals and salivary secretion in live mice expressing GCamp6F, following stimulation of the nerves innervating the submandibular gland. Consistent with in vitro studies, Ca²⁺ signals were initiated in the apical endoplasmic reticulum. In marked contrast to in vitro data, highly localized trains of Ca²⁺ transients that failed to fully propagate from the apical region were observed. Following stimuli optimum for secretion, large apical-basal gradients were elicited. A new mathematical model, incorporating these data was constructed to probe how salivary secretion can be optimally stimulated by apical Ca²⁺ signals.

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Calcium (Ca²⁺) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP₃)-induced Ca²⁺ release. These Ca²⁺ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca²⁺, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca²⁺ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca²⁺ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca²⁺ transients in different dynamical situations. We test two different models of the IP₃ receptor (IP₃R). The model with nonlinear concentration response of binding of activating Ca²⁺ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca²⁺-dependent inhibition of the IP₃R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca²⁺ and IP₃ production sensitizing IP₃R for Ca²⁺-induced Ca²⁺ release. We also model ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca²⁺ transients. Finally, we propose possible ways of recovering Ca²⁺ transients under ataxia.

A Tale of two receptors

Calcium (Ca²⁺) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca²⁺ oscillations. We present a new model of Ca²⁺ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca²⁺ oscillations. The model accounts for Ca²⁺ regulation of the IP₃ receptor (IP₃R), the positive feedback from Ca²⁺ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca²⁺ oscillations in response to P2Y receptor activation.

A Statistical View on Calcium Oscillations

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

Deterministic Limit of Intracellular Calcium Spikes

In nonexcitable cells, global Ca^{2+} spikes emerge from the collective dynamics of clusters of Ca^{2+} channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin, while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Disentangling astroglial physiology with a realistic cell model in silico

Electrically non-excitable astroglia take up neurotransmitters, buffer extracellular K+ and generate Ca2+ signals that release molecular regulators of neural circuitry. The underlying machinery remains enigmatic, mainly because the sponge-like astrocyte morphology has been difficult to access experimentally or explore theoretically. Here, we systematically incorporate multi-scale, tri-dimensional astroglial architecture into a realistic multi-compartmental cell model, which we constrain by empirical tests and integrate into the NEURON computational biophysical environment. This approach is implemented as a flexible astrocyte-model builder ASTRO. As a proof-of-concept, we explore an in silico astrocyte to evaluate basic cell physiology features inaccessible experimentally. Our simulations suggest that currents generated by glutamate transporters or K+ channels have negligible distant effects on membrane voltage and that individual astrocytes can successfully handle extracellular K+ hotspots. We show how intracellular Ca2+ buffers affect Ca2+ waves and why the classical Ca2+ sparks-and-puffs mechanism is theoretically compatible with common readouts of astroglial Ca2+ imaging.

On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling

The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP₃ induced Ca²⁺signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation.

Modeling Cell-to-Cell Communication Networks Using Response-Time Distributions

Cell-to-cell communication networks have critical roles in coordinating diverse organismal processes, such as tissue development or immune cell response. However, compared with intracellular signal transduction networks, the function and engineering principles of cell-to-cell communication networks are far less understood. Major complications include: cells are themselves regulated by complex intracellular signaling networks; individual cells are heterogeneous; and output of any one cell can recursively become an additional input signal to other cells. Here, we make use of a framework that treats intracellular signal transduction networks as "black boxes" with characterized input-to-output response relationships. We study simple cell-to-cell communication circuit motifs and find conditions that generate bimodal responses in time, as well as mechanisms for independently controlling synchronization and delay of cell-population responses. We apply our modeling approach to explain otherwise puzzling data on cytokine secretion onset times in T cells. Our approach can be used to predict communication network structure using experimentally accessible input-to-output measurements and without detailed knowledge of intermediate steps.

A Bayesian approach to modelling heterogeneous calcium responses in cell populations

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimulus time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environment.

Upon stimulation a large number of cell types respond with transient increases of the intracellular calcium concentration, which often take the form of repetitive spikes. It is therefore believed that calcium spikes play a central role in cellular signal transduction. A critical feature of these calcium spikes is that they occur randomly, which raises the question of how we can predict the timing of calcium spikes. We here show that by using Bayesian ideas and concepts from stochastic processes, we can quantitatively compute the calcium spike rate for a given stimulus. Our analysis also demonstrates that traditional methods for spike rate estimation perform less favourably compared to a Bayesian approach when small numbers of cells are investigated. To test our methodology under conditions that closely mimic those experienced in vivo we challenged cells with agonist concentrations that vary both in space and time. We find that cells that experience similar stimulus profiles are described by similar calcium spike rates. This suggests that calcium spike rates may constitute a quantitative description of whole-cell calcium spiking that reflects both the randomness and the spatiotemporal organisation of the calcium signalling machinery.

Mapping Interpuff Interval Distribution to the Properties of Inositol Trisphosphate Receptors

Tightly clustered inositol trisphosphate receptors (IP₃Rs) control localized Ca²⁺ liberation from the endoplasmic reticulum to generate repetitive Ca²⁺ puffs. Distributions of the interpuff interval (IPI), i.e., the waiting time between successive puffs, are found to be well characterized by a probability density function involving only two parameters, λ and ξ, which represent the basal rate of puff generation and the recovery rate from refractoriness, respectively. However, how the two parameters depend on the kinetic parameters of single IP₃Rs in a cluster is still unclear. In this article, using a stochastic puff model and a single-channel data-based IP₃R model, we establish the dependencies of λ and ξ on two important IP₃R model parameters, IP₃ concentration ([IP₃]) and the recovery rate from Ca²⁺ inhibition (r_low). By varying [IP₃] and r_low in physiologically plausible ranges, we find that the ξ-λ plane is comprised of only two disjoint regions, a biologically impermissible region and a region where each parameter set (ξ, λ) can be caused by using two different combinations of [IP₃] and r_low. The two combinations utilize very different mechanisms to maintain the same IPI distribution, and the mechanistic difference provides a way of identifying IP₃R kinetic parameters by observing properties of the IPI.

Suppressing effect of Ca^{2+} blips on puff amplitudes by inhibiting channels to prevent recovery

As local signals, calcium puffs arise from the concerted opening of a few nearby inositol 1,4,5-trisphospate receptor channels to release Ca^{2+} ions from the endoplasmic reticulum. Although Ca^{2+} puffs have been well studied, little is known about the modulation of cytosolic basal Ca^{2+} concentration ([Ca^{2+}]_{Basal}) on puff dynamics. In this paper we consider a puff model to study how the statistical properties of puffs are modulated by [Ca^{2+}]_{Basal}. The puff frequency and lifetime trivially increase with the increasing [Ca^{2+}]_{Basal}, but an unexpected result is that the puff amplitude and the maximum open-channel number of the puff show decreasing relationship with the increasing [Ca^{2+}]_{Basal}. The underlying dynamics is related not only to the increasing puff frequency which gives a shorter recovery time, but also to the increasing frequency of blips with only one channel open. We indicate that Ca^{2+} blips cause the channels to be inhibited and prevent their recovery during interpuff intervals, resulting in the suppressing effect on puff amplitudes. With increasing [Ca^{2+}]_{Basal}, more blips occur to cause more channels to be inhibited, leaving fewer channels available for puff events. This study shows that the blips may play relevant functions in global Ca^{2+} waves through modulating puff dynamics.

Damped oscillations in a multiple delayed feedback NF-κB signaling module

The NF-[Formula: see text]B signaling system regulates a number of cellular processes. Recent studies with simplified models found a damped function of the dual delayed feedback NF-κB signaling module. We use a computational model to investigate how multiple delayed feedback aids achieving damping oscillation in the system and how internal noise can influence the damping function. A curve-fitting method (CFM) is introduced to quantify the damped oscillation. Our results show that (1) the structure of multiple delayed feedback, containing double or triple significantly delayed feedback, determines achieving damped oscillation. (2) Internal noise could aid the system to achieve damped oscillation under almost all circumstances.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Factors determining the recruitment of inositol trisphosphate receptor channels during calcium puffs

Puffs are localized, transient elevations in cytosolic Ca(2+) that serve both as the building blocks of global cellular Ca(2+) signals and as local signals in their own right. They arise from clustered inositol 1,4,5-trisphosphate receptor/channels (IP3Rs), whose openings are coordinated by Ca(2+)-induced Ca(2+) release (CICR). We utilized total internal reflection fluorescence imaging of Ca(2+) signals in neuroblastoma cells with single-channel resolution to elucidate the mechanisms determining the triggering, amplitudes, kinetics, and spatial spread of puffs. We find that any given channel in a cluster has a mean probability of ∼66% of opening following opening of an initial "trigger" channel, and the probability of puff triggering thus increases steeply with increasing number of channels in a cluster (cluster size). Mean puff amplitudes scale with cluster size, but individual amplitudes vary widely, even at sites of similar cluster size, displaying similar proportions of events involving any given number of the channels in the cluster. Stochastic variation in numbers of Ca(2+)-inhibited IP3Rs likely contributes to the variability of amplitudes of repeated puffs at a site but the amplitudes of successive puffs were uncorrelated, even though we observed statistical correlations between interpuff intervals and puff amplitudes. Initial puffs evoked following photorelease of IP3-which would not be subject to earlier Ca(2+)-inhibition-also showed wide variability, indicating that mechanisms such as stochastic variation in IP3 binding and channel recruitment by CICR further determine puff amplitudes. The mean termination time of puffs lengthened with increasing puff amplitude size, consistent with independent closings of channels after a given mean open time, but we found no correlation of termination time with cluster size independent of puff amplitude. The spatial extent of puffs increased with their amplitude, and puffs of similar size were of similar width, independent of cluster size.

An algorithm for automated detection, localization and measurement of local calcium signals from camera-based imaging

Local Ca(2+) transients such as puffs and sparks form the building blocks of cellular Ca(2+) signaling in numerous cell types. They have traditionally been studied by linescan confocal microscopy, but advances in TIRF microscopy together with improved electron-multiplied CCD (EMCCD) cameras now enable rapid (>500 frames s(-1)) imaging of subcellular Ca(2+) signals with high spatial resolution in two dimensions. This approach yields vastly more information (ca. 1 Gb min(-1)) than linescan imaging, rendering visual identification and analysis of local events imaged both laborious and subject to user bias. Here we describe a routine to rapidly automate identification and analysis of local Ca(2+) events. This features an intuitive graphical user-interfaces and runs under Matlab and the open-source Python software. The underlying algorithm features spatial and temporal noise filtering to reliably detect even small events in the presence of noisy and fluctuating baselines; localizes sites of Ca(2+) release with sub-pixel resolution; facilitates user review and editing of data; and outputs time-sequences of fluorescence ratio signals for identified event sites along with Excel-compatible tables listing amplitudes and kinetics of events.

Reliable encoding of stimulus intensities within random sequences of intracellular Ca2+ spikes

Ca(2+) is a ubiquitous intracellular messenger that regulates diverse cellular activities. Extracellular stimuli often evoke sequences of intracellular Ca(2+) spikes, and spike frequency may encode stimulus intensity. However, the timing of spikes within a cell is random because each interspike interval has a large stochastic component. In human embryonic kidney (HEK) 293 cells and rat primary hepatocytes, we found that the average interspike interval also varied between individual cells. To evaluate how individual cells reliably encoded stimuli when Ca(2+) spikes exhibited such unpredictability, we combined Ca(2+) imaging of single cells with mathematical analyses of the Ca(2+) spikes evoked by receptors that stimulate formation of inositol 1,4,5-trisphosphate (IP3). This analysis revealed that signal-to-noise ratios were improved by slow recovery from feedback inhibition of Ca(2+) spiking operating at the whole-cell level and that they were robust against perturbations of the signaling pathway. Despite variability in the frequency of Ca(2+) spikes between cells, steps in stimulus intensity caused the stochastic period of the interspike interval to change by the same factor in all cells. These fold changes reliably encoded changes in stimulus intensity, and they resulted in an exponential dependence of average interspike interval on stimulation strength. We conclude that Ca(2+) spikes enable reliable signaling in a cell population despite randomness and cell-to-cell variability, because global feedback reduces noise, and changes in stimulus intensity are represented by fold changes in the stochastic period of the interspike interval.

Stochasticity in Ca2+ increase in spines enables robust and sensitive information coding

A dendritic spine is a very small structure (∼0.1 µm³) of a neuron that processes input timing information. Why are spines so small? Here, we provide functional reasons; the size of spines is optimal for information coding. Spines code input timing information by the probability of Ca²⁺ increases, which makes robust and sensitive information coding possible. We created a stochastic simulation model of input timing-dependent Ca²⁺ increases in a cerebellar Purkinje cell's spine. Spines used probability coding of Ca²⁺ increases rather than amplitude coding for input timing detection via stochastic facilitation by utilizing the small number of molecules in a spine volume, where information per volume appeared optimal. Probability coding of Ca²⁺ increases in a spine volume was more robust against input fluctuation and more sensitive to input numbers than amplitude coding of Ca²⁺ increases in a cell volume. Thus, stochasticity is a strategy by which neurons robustly and sensitively code information.

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.

Calcium-dependent regulation of glucose homeostasis in the liver

A major role of the liver is to integrate multiple signals to maintain normal blood glucose levels. The balance between glucose storage and mobilization is primarily regulated by the counteracting effects of insulin and glucagon. However, numerous signals converge in the liver to ensure energy demand matches the physiological status of the organism. Many circulating hormones regulate glycogenolysis, gluconeogenesis and mitochondrial metabolism by calcium-dependent signaling mechanisms that manifest as cytosolic Ca(2+) oscillations. Stimulus-strength is encoded in the Ca(2+) oscillation frequency, and also by the range of intercellular Ca(2+) wave propagation in the intact liver. In this article, we describe how Ca(2+) oscillations and waves can regulate glucose output and oxidative metabolism in the intact liver; how multiple stimuli are decoded though Ca(2+) signaling at the organ level, and the implications of Ca(2+) signal dysregulation in diseases such as metabolic syndrome and non-alcoholic fatty liver disease.

The emergence of subcellular pacemaker sites for calcium waves and oscillations

Calcium (Ca(2+)) waves generating oscillatory Ca(2+) signals are widely observed in biological cells. Experimental studies have shown that under certain conditions, initiation of Ca(2+) waves is random in space and time, while under other conditions, waves occur repetitively from preferred locations (pacemaker sites) from which they entrain the whole cell. In this study, we use computer simulations to investigate the self-organization of Ca(2+) sparks into pacemaker sites generating Ca(2+) oscillations. In both ventricular myocyte experiments and computer simulations of a heterogeneous Ca(2+) release unit (CRU) network model, we show that Ca(2+) waves occur randomly in space and time when the Ca(2+) level is low, but as the Ca(2+) level increases, waves occur repetitively from the same sites. Our analysis indicates that this transition to entrainment can be attributed to the fact that random Ca(2+) sparks self-organize into Ca(2+) oscillations differently at low and high Ca(2+) levels. At low Ca(2+), the whole cell Ca(2+) oscillation frequency of the coupled CRU system is much slower than that of an isolated single CRU. Compared to a single CRU, the distribution of interspike intervals (ISIs) of the coupled CRU network exhibits a greater variation, and its ISI distribution is asymmetric with respect to the peak, exhibiting a fat tail. At high Ca(2+), however, the coupled CRU network has a faster frequency and lesser ISI variation compared to an individual CRU. The ISI distribution of the coupled network no longer exhibits a fat tail and is well-approximated by a Gaussian distribution. This same Ca(2+) oscillation behaviour can also be achieved by varying the number of ryanodine receptors per CRU or the distance between CRUs. Using these results, we develop a theory for the entrainment of random oscillators which provides a unified explanation for the experimental observations underlying the emergence of pacemaker sites and Ca(2+) oscillations.

A stochastic model of calcium puffs based on single-channel data

Calcium puffs are local transient Ca(2+) releases from internal Ca(2+) stores such as the endoplasmic reticulum or the sarcoplasmic reticulum. Such release occurs through a cluster of inositol 1,4,5-trisphosphate receptors (IP3Rs). Based on the IP3R model (which is determined by fitting to stationary single-channel data) and nonstationary single-channel data, we construct a new IP3R model that includes time-dependent rates of mode switches. A point-source model of Ca(2+) puffs is then constructed based on the new IP3R model and is solved by a hybrid Gillespie method with adaptive timing. Model results show that a relatively slow recovery of an IP3R from Ca(2+) inhibition is necessary to reproduce most of the experimental outcomes, especially the nonexponential interpuff interval distributions. The number of receptors in a cluster could be severely underestimated when the recovery is sufficiently slow. Furthermore, we find that, as the number of IP3Rs increases, the average duration of puffs initially increases but then becomes saturated, whereas the average decay time keeps increasing linearly. This gives rise to the observed asymmetric puff shape.

Hierarchic stochastic modelling applied to intracellular Ca(2+) signals

Important biological processes like cell signalling and gene expression have noisy components and are very complex at the same time. Mathematical analysis of such systems has often been limited to the study of isolated subsystems, or approximations are used that are difficult to justify. Here we extend a recently published method (Thurley and Falcke, PNAS 2011) which is formulated in observable system configurations instead of molecular transitions. This reduces the number of system states by several orders of magnitude and avoids fitting of kinetic parameters. The method is applied to Ca(2+) signalling. Ca(2+) is a ubiquitous second messenger transmitting information by stochastic sequences of concentration spikes, which arise by coupling of subcellular Ca(2+) release events (puffs). We derive analytical expressions for a mechanistic Ca(2+) model, based on recent data from live cell imaging, and calculate Ca(2+) spike statistics in dependence on cellular parameters like stimulus strength or number of Ca(2+) channels. The new approach substantiates a generic Ca(2+) model, which is a very convenient way to simulate Ca(2+) spike sequences with correct spiking statistics.

The probability of triggering calcium puffs is linearly related to the number of inositol trisphosphate receptors in a cluster

Puffs are local Ca(2+) signals that arise by Ca(2+) liberation from the endoplasmic reticulum through concerted opening of tightly clustered inositol trisphosphate receptor/channels (IP(3)R). They serve both local signaling functions and trigger global Ca(2+) waves. The numbers of functional IP(3)R within clusters differ appreciably between different puff sites, and we investigated how the probability of puff occurrence varies with cluster size. We imaged puffs in SH-SY5Y cells using total internal fluorescence microscopy, and estimated cluster sizes from the magnitude of the largest puff observed at each site relative to the signal from a single channel. We find that the initial triggering rate of puffs following photorelease of IP(3), and the average frequency of subsequent repetitive puffs, vary about linearly with cluster size. These data accord well with stochastic simulations in which opening of any individual IP(3)R channel within a cluster triggers a puff via Ca(2+)-induced Ca(2+) release. An important consequence is that the signaling power of a puff site (average amount of Ca(2+) released per puff × puff frequency) varies about the square of cluster size, implying that large clusters contribute disproportionately to cellular signaling and, because of their higher puff frequency, preferentially act as pacemakers to initiate Ca(2+) waves.

Intracellular calcium signals display an avalanche-like behavior over multiple lengthscales

Many natural phenomena display “self-organized criticality” (SOC), (Bak et al., 1987). This refers to spatially extended systems for which patterns of activity characterized by different lengthscales can occur with a probability density that follows a power law with pattern size. Differently from power laws at phase transitions, systems displaying SOC do not need the tuning of an external parameter. Here we analyze intracellular calcium (Ca²⁺) signals, a key component of the signaling toolkit of almost any cell type. Ca²⁺ signals can either be spatially restricted (local) or propagate throughout the cell (global). Different models have suggested that the transition from local to global signals is similar to that of directed percolation. Directed percolation has been associated, in turn, to the appearance of SOC. In this paper we discuss these issues within the framework of simple models of Ca²⁺ signal propagation. We also analyze the size distribution of local signals (“puffs”) observed in immature Xenopus Laevis oocytes. The puff amplitude distribution obtained from observed local signals is not Gaussian with a noticeable fraction of large size events. The experimental distribution of puff areas in the spatio-temporal record of the image has a long tail that is approximately log-normal. The distribution can also be fitted with a power law relationship albeit with a smaller goodness of fit. The power law behavior is encountered within a simple model that includes some coupling among individual signals for a wide range of parameter values. An analysis of the model shows that a global elevation of the Ca²⁺ concentration plays a major role in determining whether the puff size distribution is long-tailed or not. This suggests that Ca²⁺-clearing from the cytosol is key to determine whether IP₃-mediated Ca²⁺ signals can display a SOC-like behavior or not.

Criticality in intracellular calcium signaling in cardiac myocytes

Calcium (Ca) is a ubiquitous second messenger that regulates many biological functions. The elementary events of local Ca signaling are Ca sparks, which occur randomly in time and space, and integrate to produce global signaling events such as intra- and intercellular Ca waves and whole-cell Ca oscillations. Despite extensive experimental characterization in many systems, the transition from local random to global synchronous events is still poorly understood. Here we show that criticality, a ubiquitous dynamical phenomenon in nature, is responsible for the transition from local to global Ca signaling. We demonstrate this first in a computational model of Ca signaling in a cardiac myocyte and then experimentally in mouse ventricular myocytes, complemented by a theoretical agent-based model to delineate the underlying dynamics. We show that the interaction between the Ca release units via Ca-induced Ca release causes self-organization of Ca spark clusters. When the coupling between Ca release units is weak, the cluster-size distribution is exponential. As the interactions become strong, the cluster-size distribution changes to a power-law distribution, which is characteristic of criticality in thermodynamic and complex nonlinear systems, and facilitates the formation and propagation of Ca waves and whole-cell Ca oscillations. Our findings illustrate how criticality is harnessed by a biological cell to regulate Ca signaling via self-organization of random subcellular events into cellular-scale oscillations, and provide a general theoretical framework for the transition from local Ca signaling to global Ca signaling in biological cells.

Termination of Ca²+ release for clustered IP₃R channels

In many cell types, release of calcium ions is controlled by inositol 1,4,5-trisphosphate () receptor channels. Elevations in concentration after intracellular release through receptors (R) can either propagate in the form of waves spreading through the entire cell or produce spatially localized puffs. The appearance of waves and puffs is thought to implicate random initial openings of one or a few channels and subsequent activation of neighboring channels because of an “autocatalytic” feedback. It is much less clear, however, what determines the further time course of release, particularly since the lifetime is very different for waves (several seconds) and puffs (around 100 ms). Here we study the lifetime of signals and their dependence on residual microdomains. Our general idea is that microdomains are dynamical and mediate the effect of other physiological processes. Specifically, we focus on the mechanism by which binding proteins (buffers) alter the lifetime of signals. We use stochastic simulations of channel gating coupled to a coarse-grained description for the concentration. To describe the concentration in a phenomenological way, we here introduce a differential equation, which reflects the buffer characteristics by a few effective parameters. This non-stationary model for microdomains gives deep insight into the dynamical differences between puffs and waves. It provides a novel explanation for the different lifetimes of puffs and waves and suggests that puffs are terminated by inhibition while unbinding is responsible for termination of waves. Thus our analysis hints at an additional role of and shows how cells can make use of the full complexity in R gating behavior to achieve different signals.

Calcium signals are important for a host of cellular processes such as neurotransmitter release, cell contraction and gene expression. While the principles of activation and spreading of calcium signals have been largely understood, it is much less clear how their spatio-temporal appearance is shaped. This issue is of high relevance since the spatio-temporal signature is thought to carry the information content. In our paper we study the dynamical mechanisms that determine the time course of calcium release from receptor channels. We use a stochastic channel description combined with a recently developed model for the distribution of released calcium in a microdomain. The simulations uncover a complex control mechanism, which allows for the tuning of release from short frequent puffs to extended and less frequent wave-like release. Unexpectedly, the model predicts that for wave-like release the dissociation of from the receptors leads to termination of the calcium signal. This effect relies on a well-known gating property of R channels, which earlier has been regarded as superfluous in studies for groups of channels. Our results also provide a missing link to understand cellular response to calcium-binding proteins and present a novel mechanism for information processing by R channels.

Calcium signaling: from single channels to pathways

Ca(2+) is not only one of the most versatile and ubiquitous second messengers but also a well-established representative example of cell signaling. The identification of most key elements involved in Ca(2+) signaling enables a mechanistic and quantitative understanding of this particular pathway. Cellular behavior relies in general on the orchestration of molecular behavior leading to reliable cellular responses that allow for regulation and adaptation. Ca(2+) signaling uses a hierarchical organization to transform single molecule behavior into cell wide signals. We have recently shown experimentally that this organization carries single channel signatures onto the whole cell level and renders Ca(2+) oscillations stochastic. Here, we briefly review the co-evolution of experimental and theoretical studies in Ca(2+) -signaling and show how dynamic bottom-up modeling can be used to address -biological questions and illuminate biological principles of cell signaling.

Fundamental properties of Ca2+ signals

BACKGROUND

Ca2+ is a ubiquitous and versatile second messenger that transmits information through changes of the cytosolic Ca2+ concentration. Recent investigations changed basic ideas on the dynamic character of Ca2+ signals and challenge traditional ideas on information transmission.

SCOPE OF REVIEW

We present recent findings on key characteristics of the cytosolic Ca2+ dynamics and theoretical concepts that explain the wide range of experimentally observed Ca2+ signals. Further, we relate properties of the dynamical regulation of the cytosolic Ca2+ concentration to ideas about information transmission by stochastic signals.

MAJOR CONCLUSIONS

We demonstrate the importance of the hierarchal arrangement of Ca2+ release sites on the emergence of cellular Ca2+ spikes. Stochastic Ca2+ signals are functionally robust and adaptive to changing environmental conditions. Fluctuations of interspike intervals (ISIs) and the moment relation derived from ISI distributions contain information on the channel cluster open probability and on pathway properties.

GENERAL SIGNIFICANCE

Robust and reliable signal transduction pathways that entail Ca2+ dynamics are essential for eukaryotic organisms. Moreover, we expect that the design of a stochastic mechanism which provides robustness and adaptivity will be found also in other biological systems. Ca2+ dynamics demonstrate that the fluctuations of cellular signals contain information on molecular behavior. This article is part of a Special Issue entitled Biochemical, biophysical and genetic approaches to intracellular calcium signaling.