Models of IP3 and Ca2+ oscillations: frequency encoding and identification of underlying feedbacks

A molecular systems perspective on calcium oscillations beyond ion fluxes

Calcium (Ca2+) oscillations, marked by periodic fluctuations in cytosolic Ca2+ levels, are a universal feature of both excitable and non-excitable cells, regulating key functions like immune responses, neuronal activity and oocyte activation. Despite significant progress over the past few decades in identifying the molecular toolkits involved in Ca2+ mobilization, fundamental questions remain unresolved: How do Ca2+oscillations arise? In dynamical systems, oscillations arise as closed-loop trajectories in phase space, known as limit cycles. In this framework, [Ca2+] is the variable that oscillates along the limit cycle. Is [Ca2+] also the control parameter that defines the system's stability? Understanding how oscillations arise and how instability is controlled are essential for determining what these oscillations encode. This review revisits classic categorizations of Ca2+ oscillation models, focusing on the minimal mathematical models, their assumptions and gaps linking models with experimental data. We examine historical arguments in light of recent discoveries of plasma membrane lipid oscillations in non-excitable cells. While growing evidence support the pivotal role of lipid signaling in regulating Ca2+ dynamics, they mostly focused on the upstream role of signaling in Ca2+ mobilization, rather than viewing membrane-dependent signal transduction as the core control loop that is responsible for oscillatory Ca2+ dynamics. Here we summarize recent molecular studies of phosphoinositide signaling in modulating Ca2+ dynamics, by considering a broader chemical perspective as essential for understanding Ca2+ oscillations beyond ion fluxes.

Computational Model of Complex Calcium Dynamics: Store Operated Ca2+ Channels and Mitochondrial Associated Membranes in Pancreatic Acinar Cells

This proposed model explores the intricate Ca2+ dynamics within the pancreatic acinar cells (PACs) by emphasizing the role of store-operated Ca2+ entry (SOCE) and the mitochondrial-associated membranes (MAMs) in the secretory region (apical) of the PACs. Traditionally, Ca2+ releases from the endoplasmic reticulum (ER) via calcium-induced calcium release (CICR). It has been shown to be important in regulating functions such as secretion of digestive enzymes in PACs. However, this model posits that upon the depletion of Ca2+ in the ER, the signaling protein stromal interaction molecule (STIM1) is activated. Activated STIM1, then facilitates the opening of Orai channels, allowing Ca2+ influx through the store-operated calcium channels (SOCCs). The model highlights the complexity of the Ca2+ dynamics, and the importance of SOCE and MAMs in the PACs Ca2+ homeostasis. The numerical and bifurcation analysis illustrate how changes in agonist concentrations can lead to the diverse Ca2+ oscillation patterns, such as thin to broader oscillations, sinusoidal patterns, and baseline fluctuations, driven by the feedback mechanisms involving Ca2+ and inositol 1,4,5 trisphosphate (IP3). This understanding could have broader implications for cellular physiology and the development of therapies targeting Ca2+ signaling pathways.

The characteristics of the calcium signals that activate mammalian eggs at fertilization

Gamete membrane fusion in mammals brings the paternal genome into the cytoplasm of the egg. It also enables signals to pass from the sperm into the egg to trigger the completion of meiosis and the start of embryo development. The essential signal to activate development in all mammals studied, consists of a series of transient increases in the cytosolic Ca2+ concentration driven by cycles of InsP3 production. This review focusses on the characteristics of these sperm-induced Ca2+ signals. I consider how some specific features of sperm-derived phospholipase C-zeta (PLCζ), along with the known properties of the type 1 InsP3 receptor, provide a basis for understanding the mechanisms of the dynamic changes in Ca2+ observed in fertilizing eggs. I describe how the PLCζ targeting of cytoplasmic vesicles in the egg cytoplasm, that contain PI(4,5)P2, is necessary to explain the rapid waves associated with the rising phase of each Ca2+ transient. I also discuss the importance of the repetitive Ca2+ rises for egg activation and the way mitochondrial ATP production may modulate Ca2+ release in eggs. Finally, I consider the role that a sperm-induced ATP increase may play in the egg activation process.

The signal that stimulates mammalian embryo development

Embryo development is stimulated by calcium (Ca2+) signals that are generated in the egg cytoplasm by the fertilizing sperm. Eggs are formed via oogenesis. They go through a cell division known as meiosis, during which their diploid chromosome number is halved and new genetic combinations are created by crossing over. During formation the eggs also acquire cellular components that are necessary to produce the Ca2+ signal and also, to support development of the newly formed embryo. Ionized calcium is a universal second messenger used by cells in a plethora of biological processes and the eggs develop a "toolkit", a set of molecules needed for signaling. Meiosis stops twice and these arrests are controlled by a complex interaction of regulatory proteins. The first meiotic arrest lasts until after puberty, when a luteinizing hormone surge stimulates meiotic resumption. The cell cycle proceeds to stop again in the middle of the second meiotic division, right before ovulation. The union of the female and male gametes takes place in the oviduct. Following gamete fusion, the sperm triggers the release of Ca2+ from the egg's intracellular stores which in mammals is followed by repetitive Ca2+ spikes known as Ca2+ oscillations in the cytosol that last for several hours. Downstream sensor proteins help decoding the signal and stimulate other molecules whose actions are required for proper development including those that help to prevent the fusion of additional sperm cells to the egg and those that assist in the release from the second meiotic arrest, completion of meiosis and entering the first mitotic cell division. Here I review the major steps of egg formation, discuss the signaling toolkit that is essential to generate the Ca2+ signal and describe the steps of the signal transduction mechanism that activates the egg's developmental program and turns it into an embryo.

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.

Model of Calcium Dynamics Regulating I P 3 , ATP and Insulin Production in a Pancreatic β -Cell

The calcium signals regulate the production and secretion of many signaling molecules like inositol trisphosphate (I P 3 ) and adenosine triphosphate (ATP) in various cells including pancreaticβ -cells. The calcium signaling mechanisms regulatingI P 3 , ATP and insulin responsible for various functions ofβ -cells are still not well understood. Any disturbance in these mechanisms can alter the functions ofβ -cells leading to diabetes and metabolic disorders. Therefore, a mathematical model is proposed by incorporating the reaction-diffusion equation for calcium dynamics and a system of first-order differential equations forI P 3 , ATP-production and insulin secretion with initial and boundary conditions. The model incorporates the temporal dependence ofI P 3 -production and degradation, ATP production and insulin secretion on calcium dynamics in aβ -cell. The piecewise linear finite element method has been used for the spatial dimension and the Crank-Nicolson scheme for the temporal dimension to obtain numerical results. The effect of changes in source influxes and buffers on calcium dynamics and production ofI P 3 , ATP and insulin levels in aβ -cell has been analyzed. It is concluded that the dysfunction of source influx and buffers can cause significant variations in calcium levels and dysregulation ofI P 3 , ATP and insulin production, which can lead to various metabolic disorders, diabetes, obesity, etc. The proposed model provides crucial information about the changes in mechanisms of calcium dynamics causing proportionate disturbances inI P 3 , ATP and insulin levels in pancreatic cells, which can be helpful for devising protocols for diagnosis and treatment of various metabolic diseases.

Sperm-Induced Ca2+ Release in Mammalian Eggs: The Roles of PLCζ, InsP3, and ATP

Mammalian egg activation at fertilization is triggered by a long-lasting series of increases in cytosolic Ca2+ concentration. These Ca2+ oscillations are due to the production of InsP3 within the egg and the subsequent release of Ca2+ from the endoplasmic reticulum into the cytosol. The generation of InsP3 is initiated by the diffusion of sperm-specific phospholipase Czeta1 (PLCζ) into the egg after gamete fusion. PLCζ enables a positive feedback loop of InsP3 production and Ca2+ release which then stimulates further InsP3 production. Most cytosolic Ca2+ increases in eggs at fertilization involve a fast Ca2+ wave; however, due to the limited diffusion of InsP3, this means that InsP3 must be generated from an intracellular source rather than at the plasma membrane. All mammalian eggs studied generated Ca2+ oscillations in response to PLCζ, but the sensitivity of eggs to PLCζ and to some other stimuli varies between species. This is illustrated by the finding that incubation in Sr2+ medium stimulates Ca2+ oscillations in mouse and rat eggs but not eggs from other mammalian species. This difference appears to be due to the sensitivity of the type 1 InsP3 receptor (IP3R1). I suggest that ATP production from mitochondria modulates the sensitivity of the IP3R1 in a manner that could account for the differential sensitivity of eggs to stimuli that generate Ca2+ oscillations.

Numerical simulation of calcium dynamics dependent ATP degradation, IP3 and NADH production due to obesity in a hepatocyte cell

Calcium (Ca[Formula: see text]) signals have a crucial role in regulating various processes of almost every cell to maintain its structure and function. Calcium dynamics has been studied in various cells including hepatocytes by many researchers, but the mechanisms of calcium signals involved in regulation and dysregulation of various processes like ATP degradation rate, IP[Formula: see text] and NADH production rate respectively in normal and obese cells are still poorly understood. In this paper, a reaction diffusion equation of calcium is employed to propose a model of calcium dynamics by coupling ATP degradation rate, IP[Formula: see text] and NADH production rate in hepatocyte cells under normal and obese conditions. The processes like source influx, buffer, endoplasmic reticulum (ER), mitochondrial calcium uniporters (MCU) and Na[Formula: see text]/Ca[Formula: see text] exchanger (NCX) have been incorporated in the model. Linear finite element method is used along spatial dimension, and Crank-Nicolson method is used along temporal dimension for numerical simulation. The results have been obtained for the normal hepatocyte cells and for cells due to obesity. The comparative study of these results reveal significant difference caused due to obesity in Ca[Formula: see text] dynamics as well as in ATP degradation rate, IP[Formula: see text] and NADH production rate.

Modulating Biological Rhythms: A Noncomputational Strategy Harnessing Nonlinearity and Decoupling Frequency and Amplitude

Understanding and achieving concurrent modulation of amplitude and frequency, particularly adjusting one quantity and simultaneously sustaining the other at an invariant level, are of paramount importance for complex biophysical systems, including the signal pathway where different frequency indicates different upstream signal yielding a certain downstream physiological function while different amplitude further determines different efficacy of a downstream output. However, such modulators with clearly described and universally valid mechanisms are still lacking. Here, we rigorously propose an easy-to-use control strategy containing only one or two steps, leveraging the nonlinearity in the modulated systems to decouple frequency and amplitude in a noncomputational manner. The strategy's efficacy is demonstrated using representative biochemical systems and, thus, it could be potentially applicable to modulating rhythms in experiments of biochemistry and synthetic biology.

Modeling IP3-induced Ca2+ signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

Orai1 is an Entotic Ca2+ Channel for Non-Apoptotic Cell Death, Entosis in Cancer Development

Entosis is a non-apoptotic cell death process that forms characteristic cell-in-cell structures in cancers, killing invading cells. Intracellular Ca²⁺ dynamics are essential for cellular processes, including actomyosin contractility, migration, and autophagy. However, the significance of Ca²⁺ and Ca²⁺ channels participating in entosis is unclear. Here, it is shown that intracellular Ca²⁺ signaling regulates entosis via SEPTIN-Orai1-Ca²⁺ /CaM-MLCK-actomyosin axis. Intracellular Ca²⁺ oscillations in entotic cells show spatiotemporal variations during engulfment, mediated by Orai1 Ca²⁺ channels in plasma membranes. SEPTIN controlled polarized distribution of Orai1 for local MLCK activation, resulting in MLC phosphorylation and actomyosin contraction, leads to internalization of invasive cells. Ca²⁺ chelators and SEPTIN, Orai1, and MLCK inhibitors suppress entosis. This study identifies potential targets for treating entosis-associated tumors, showing that Orai1 is an entotic Ca²⁺ channel that provides essential Ca²⁺ signaling and sheds light on the molecular mechanism underlying entosis that involves SEPTIN filaments, Orai1, and MLCK.

Computational investigation of IP3 diffusion

Inositol 1,4,5-trisphosphate (IP₃) plays a key role in calcium signaling. After stimulation, it diffuses from the plasma membrane where it is produced to the endoplasmic reticulum where its receptors are localized. Based on in vitro measurements, IP₃ was long thought to be a global messenger characterized by a diffusion coefficient of ~ 280 μm²s^-1. However, in vivo observations revealed that this value does not match with the timing of localized Ca²⁺ increases induced by the confined release of a non-metabolizable IP₃ analog. A theoretical analysis of these data concluded that in intact cells diffusion of IP₃ is strongly hindered, leading to a 30-fold reduction of the diffusion coefficient. Here, we performed a new computational analysis of the same observations using a stochastic model of Ca²⁺ puffs. Our simulations concluded that the value of the effective IP₃ diffusion coefficient is close to 100 μm²s^-1. Such moderate reduction with respect to in vitro estimations quantitatively agrees with a buffering effect by non-fully bound inactive IP₃ receptors. The model also reveals that IP₃ spreading is not much affected by the endoplasmic reticulum, which represents an obstacle to the free displacement of molecules, but can be significantly increased in cells displaying elongated, 1-dimensional like geometries.

Muscarinic acetylcholine receptor-dependent and NMDA receptor-dependent LTP and LTD share the common AMPAR trafficking pathway

The forebrain cholinergic system promotes higher brain function in part by signaling through the M₁ muscarinic acetylcholine receptor (mAChR). Long-term potentiation (LTP) and long-term depression (LTD) of excitatory synaptic transmission in the hippocampus are also induced by mAChR. An AMPA receptor (AMPAR) trafficking model for hippocampal neurons has been proposed to simulate N-methyl-D-aspartate receptor (NMDAR)-dependent synaptic plasticity in the early phase. In this study, we demonstrated the validity of the hypothesis that the mAChR-dependent LTP/LTD shares a common AMPAR trafficking pathway associated with NMDAR-dependent LTP/LTD. However, unlike NMDAR, Ca²⁺ influx into the spine cytosol occurs owing to the Ca²⁺ stored inside the ER and is induced via the activation of inositol 1,4,5-trisphosphate (IP3) receptors during M1 mAChR activation. Moreover, the AMPAR trafficking model implies that alterations in LTP and LTD observed in Alzheimer's disease could be attributed to age-dependent reductions in AMPAR expression levels.

Mathematical modeling of intracellular calcium in presence of receptor: a homeostatic model for endothelial cell

Calcium is a ubiquitous molecule and second messenger that regulates many cellular functions ranging from exocytosis to cell proliferation at different time scales. In the vasculature, a constant adenosine triphosphate (ATP) concentration is maintained because of ATP released by red blood cells (RBCs). These ATP molecules continuously react with purinergic receptors on the surface of endothelial cells (ECs). Consequently, a cascade of chemical reactions are triggered that result in a transient cytoplasmic calcium (Ca[Formula: see text]), followed by return to its basal concentration. The mathematical models proposed in the literature are able to reproduce the transient peak. However, the trailing concentration is always higher than the basal cytoplasmic Ca[Formula: see text] concentrations, and the Ca[Formula: see text] concentration in endoplasmic reticulum (ER) remains lower than its initial concentration. This means that the intracellular homeostasis is not recovered. We propose, herein, a minimal model of calcium kinetics. We find that the desensitization of EC surface receptors due to phosphorylation and recycling plays a vital role in maintaining calcium homeostasis in the presence of a constant stimulus (ATP). The model is able to capture several experimental observations such as refilling of Ca[Formula: see text] in the ER, variation of cytoplasmic Ca[Formula: see text] transient peak in ECs, the resting cytoplasmic Ca[Formula: see text] concentration, the effect of removing ATP from the plasma on Ca[Formula: see text] homeostasis, and the saturation of cytoplasmic Ca[Formula: see text] transient peak with increase in ATP concentration. Direct confrontation with several experimental results is conducted. This work paves the way for systematic studies on coupling between blood flow and chemical signaling, and should contribute to a better understanding of the relation between (patho)physiological conditions and Ca[Formula: see text] kinetics.

Upregulated Ca2+ Release from the Endoplasmic Reticulum Leads to Impaired Presynaptic Function in Familial Alzheimer's Disease

Neurotransmitter release from presynaptic terminals is primarily regulated by rapid Ca2+ influx through membrane-resident voltage-gated Ca2+ channels (VGCCs). Moreover, accumulating evidence indicates that the endoplasmic reticulum (ER) is extensively present in axonal terminals of neurons and plays a modulatory role in synaptic transmission by regulating Ca2+ levels. Familial Alzheimer's disease (FAD) is marked by enhanced Ca2+ release from the ER and downregulation of Ca2+ buffering proteins. However, the precise consequence of impaired Ca2+ signaling within the vicinity of VGCCs (active zone (AZ)) on exocytosis is poorly understood. Here, we perform in silico experiments of intracellular Ca2+ signaling and exocytosis in a detailed biophysical model of hippocampal synapses to investigate the effect of aberrant Ca2+ signaling on neurotransmitter release in FAD. Our model predicts that enhanced Ca2+ release from the ER increases the probability of neurotransmitter release in FAD. Moreover, over very short timescales (30-60 ms), the model exhibits activity-dependent and enhanced short-term plasticity in FAD, indicating neuronal hyperactivity-a hallmark of the disease. Similar to previous observations in AD animal models, our model reveals that during prolonged stimulation (~450 ms), pathological Ca2+ signaling increases depression and desynchronization with stimulus, causing affected synapses to operate unreliably. Overall, our work provides direct evidence in support of a crucial role played by altered Ca2+ homeostasis mediated by intracellular stores in FAD.

Mathematical Analysis of Membrane Transporters Dynamics: A Calcium Fluxes Case Study

A tight control of intracellular [Ca[Formula: see text]] is essential for the survival and normal function of cells. In this study we investigate key mechanistic steps by which calcium is regulated and calcium oscillations could occur using in silico modeling of membrane transporters. To do so we give a deterministic description of intracellular Ca[Formula: see text] dynamics using nonlinear dynamics in order to understand Ca[Formula: see text] signaling. We first present the ordinary differential equations (ODEs) system for cell calcium kinetics and make a preliminary work on Sobol indices. We then describe and analyze complex transporters action. Besides, we analyze the whole system. We finally perform numerical simulations and compare our results to real data.

From spikes to intercellular waves: Tuning intercellular calcium signaling dynamics modulates organ size control

Information flow within and between cells depends significantly on calcium (Ca²⁺) signaling dynamics. However, the biophysical mechanisms that govern emergent patterns of Ca²⁺ signaling dynamics at the organ level remain elusive. Recent experimental studies in developing Drosophila wing imaginal discs demonstrate the emergence of four distinct patterns of Ca²⁺ activity: Ca²⁺ spikes, intercellular Ca²⁺ transients, tissue-level Ca²⁺ waves, and a global “fluttering” state. Here, we used a combination of computational modeling and experimental approaches to identify two different populations of cells within tissues that are connected by gap junction proteins. We term these two subpopulations “initiator cells,” defined by elevated levels of Phospholipase C (PLC) activity, and “standby cells,” which exhibit baseline activity. We found that the type and strength of hormonal stimulation and extent of gap junctional communication jointly determine the predominate class of Ca²⁺ signaling activity. Further, single-cell Ca²⁺ spikes are stimulated by insulin, while intercellular Ca²⁺ waves depend on Gαq activity. Our computational model successfully reproduces how the dynamics of Ca²⁺ transients varies during organ growth. Phenotypic analysis of perturbations to Gαq and insulin signaling support an integrated model of cytoplasmic Ca²⁺ as a dynamic reporter of overall tissue growth. Further, we show that perturbations to Ca²⁺ signaling tune the final size of organs. This work provides a platform to further study how organ size regulation emerges from the crosstalk between biochemical growth signals and heterogeneous cell signaling states.

Calcium (Ca²⁺) is a universal second messenger that regulates a myriad of cellular processes such as cell division, cell proliferation and apoptosis. Multiple patterns of Ca²⁺ signaling including single-cell spikes, multicellular Ca²⁺ transients, large-scale Ca²⁺ waves, and global “fluttering” have been observed in epithelial systems during organ development. Key molecular players and biophysical mechanisms involved in formation of these patterns during organ development are not well understood. In this work, we developed a generalized multicellular model of Ca²⁺ that captures all the key categories of Ca²⁺ activity as a function of key hormonal signals. Integration of model predictions and experiments reveals two subclasses of cell populations and demonstrates that Ca²⁺ signaling activity at the organ scale is defined by a general decrease in gap junction communication as an organ grows. Our experiments also reveal that a “goldilocks zone” of optimal Ca²⁺ activity is required to achieve optimal growth at the organ level.

Receptor-specific Ca2+ oscillation patterns mediated by differential regulation of P2Y purinergic receptors in rat hepatocytes

Extracellular agonists linked to inositol-1,4,5-trisphosphate (IP₃) formation elicit cytosolic Ca²⁺ oscillations in many cell types, but despite a common signaling pathway, distinct agonist-specific Ca²⁺ spike patterns are observed. Using qPCR, we show that rat hepatocytes express multiple purinergic P2Y and P2X receptors (R). ADP acting through P2Y1R elicits narrow Ca²⁺ oscillations, whereas UTP acting through P2Y2R elicits broad Ca²⁺ oscillations, with composite patterns observed for ATP. P2XRs do not play a role at physiological agonist levels. The discrete Ca²⁺ signatures reflect differential effects of protein kinase C (PKC), which selectively modifies the falling phase of the Ca²⁺ spikes. Negative feedback by PKC limits the duration of P2Y1R-induced Ca²⁺ spikes in a manner that requires extracellular Ca²⁺. By contrast, P2Y2R is resistant to PKC negative feedback. Thus, the PKC leg of the bifurcated IP₃ signaling pathway shapes unique Ca²⁺ oscillation patterns that allows for distinct cellular responses to different agonists.

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Distinct stereotypic Ca²⁺ oscillations are elicited by P2Y1 and P2Y2 receptors

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P2X receptors do not contribute to the generation of Ca²⁺ oscillations

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Agonist-specific Ca²⁺ spike shapes reflect discrete modes of PKC negative feedback

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Bifurcation of IP₃/PKC signaling yields unique Ca²⁺ oscillation signatures

A frequency-amplitude coordinator and its optimal energy consumption for biological oscillators

Biorhythm including neuron firing and protein-mRNA interaction are fundamental activities with diffusive effect. Their well-balanced spatiotemporal dynamics are beneficial for healthy sustainability. Therefore, calibrating both anomalous frequency and amplitude of biorhythm prevents physiological dysfunctions or diseases. However, many works were devoted to modulate frequency exclusively whereas amplitude is usually ignored, although both quantities are equally significant for coordinating biological functions and outputs. Especially, a feasible method coordinating the two quantities concurrently and precisely is still lacking. Here, for the first time, we propose a universal approach to design a frequency-amplitude coordinator rigorously via dynamical systems tools. We consider both spatial and temporal information. With a single well-designed coordinator, they can be calibrated to desired levels simultaneously and precisely. The practical usefulness and efficacy of our method are demonstrated in representative neuronal and gene regulatory models. We further reveal its fundamental mechanism and optimal energy consumption providing inspiration for biorhythm regulation in future.

Quantifying the dose-dependent impact of intracellular amyloid beta in a mathematical model of calcium regulation in xenopus oocyte

Alzheimer’s disease (AD) is a devastating illness affecting over 40 million people worldwide. Intraneuronal rise of amyloid beta in its oligomeric forms (iAβOs), has been linked to the pathogenesis of AD by disrupting cytosolic Ca²⁺ homeostasis. However, the specific mechanisms of action are still under debate and intense effort is ongoing to improve our understanding of the crucial steps involved in the mechanisms of AβOs toxicity. We report the development of a mathematical model describing a proposed mechanism by which stimulation of Phospholipase C (PLC) by iAβO, triggers production of IP₃ with consequent abnormal release of Ca²⁺ from the endoplasmic reticulum (ER) through activation of IP₃ receptor (IP₃R) Ca²⁺ channels. After validating the model using experimental data, we quantify the effects of intracellular rise in iAβOs on model solutions. Our model validates a dose-dependent influence of iAβOs on IP₃-mediated Ca²⁺ signaling. We investigate Ca²⁺ signaling patterns for small and large iAβOs doses and study the role of various parameters on Ca²⁺ signals. Uncertainty quantification and partial rank correlation coefficients are used to better understand how the model behaves under various parameter regimes. Our model predicts that iAβO alter IP₃R sensitivity to IP₃ for large doses. Our analysis also shows that the upstream production of IP₃ can influence Aβ-driven solution patterns in a dose-dependent manner. Model results illustrate and confirm the detrimental impact of iAβOs on IP₃ signaling.

IP3-Dependent Ca2+ Oscillations Switch into a Dual Oscillator Mechanism in the Presence of PLC-Linked Hormones

Ca2+ oscillations that depend on inositol-1,4,5-trisphosphate (IP3) have been ascribed to biphasic Ca2+ regulation of the IP3 receptor (IP3R) or feedback mechanisms controlling IP3 levels in different cell types. IP3 uncaging in hepatocytes elicits Ca2+ transients that are often localized at the subcellular level and increase in magnitude with stimulus strength. However, this does not reproduce the broad baseline-separated global Ca2+ oscillations elicited by vasopressin. Addition of hormone to cells activated by IP3 uncaging initiates a qualitative transition from high-frequency spatially disorganized Ca2+ transients, to low-frequency, oscillatory Ca2+ waves that propagate throughout the cell. A mathematical model with dual coupled oscillators that integrates Ca2+-induced Ca2+ release at the IP3R and mutual feedback mechanisms of cross-coupling between Ca2+ and IP3 reproduces this behavior. Thus, multiple Ca2+ oscillation modes can coexist in the same cell, and hormonal stimulation can switch from the simpler to the more complex to yield robust signaling.

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.

Roles for the Endoplasmic Reticulum in Regulation of Neuronal Calcium Homeostasis

By influencing Ca²⁺ homeostasis in spatially and architecturally distinct neuronal compartments, the endoplasmic reticulum (ER) illustrates the notion that form and function are intimately related. The contribution of ER to neuronal Ca²⁺ homeostasis is attributed to the organelle being the largest reservoir of intracellular Ca²⁺ and having a high density of Ca²⁺ channels and transporters. As such, ER Ca²⁺ has incontrovertible roles in the regulation of axodendritic growth and morphology, synaptic vesicle release, and neural activity dependent gene expression, synaptic plasticity, and mitochondrial bioenergetics. Not surprisingly, many neurological diseases arise from ER Ca²⁺ dyshomeostasis, either directly due to alterations in ER resident proteins, or indirectly via processes that are coupled to the regulators of ER Ca²⁺ dynamics. In this review, we describe the mechanisms involved in the establishment of ER Ca²⁺ homeostasis in neurons. We elaborate upon how changes in the spatiotemporal dynamics of Ca²⁺ exchange between the ER and other organelles sculpt neuronal function and provide examples that demonstrate the involvement of ER Ca²⁺ dyshomeostasis in a range of neurological and neurodegenerative diseases.

Soluble adenylyl cyclase (sAC) regulates calcium signaling in the vascular endothelium

The vascular endothelium acts as a selective barrier between the bloodstream and extravascular tissues. Intracellular [Ca²⁺]_i signaling is essential for vasoactive agonist-induced stimulation of endothelial cells (ECs), typically including Ca²⁺ release from the endoplasmic reticulum (ER). Although it is known that interactions of Ca²⁺ and cAMP as ubiquitous messengers are involved in this process, the individual contribution of cAMP-generating adenylyl cyclases (ACs), including the only soluble AC (sAC; ADCY10), remains less clear. Using life-cell microscopy and plate reader-based [Ca²⁺]_i measurements, we found that human immortalized ECs, primary aortic and cardiac microvascular ECs, and primary vascular smooth muscle cells treated with sAC-specific inhibitor KH7 or anti-sAC-small interfering RNA did not show endogenous or exogenous ATP-induced [Ca²⁺]_i elevation. Of note, a transmembrane AC (tmAC) inhibitor did not prevent ATP-induced [Ca²⁺]_i elevation in ECs. Moreover, l-phenylephrine-dependent constriction of ex vivo mouse aortic ring segments was also reduced by KH7. Analysis of the inositol-1,4,5-trisphosphate (IP₃) pathway revealed reduced IP₃ receptor phosphorylation after KH7 application, which also prevented [Ca²⁺]_i elevation induced by IP₃ receptor agonist adenophostin A. Our results suggest that sAC rather than tmAC controls the agonist-induced ER-dependent Ca²⁺ response in ECs and may represent a treatment target in arterial hypertension and heart failure.-Mewes, M., Lenders, M., Stappers, F., Scharnetzki, D., Nedele, J., Fels, J., Wedlich-Söldner, R., Brand, S.-M., Schmitz, B., Brand, E. Soluble adenylyl cyclase (sAC) regulates calcium signaling in the vascular endothelium.

A Model of Ca 2 + Dynamics in an Accurate Reconstruction of Parotid Acinar Cells

We have constructed a spatiotemporal model ofCa 2 + dynamics in parotid acinar cells, based on new data about the distribution of inositol trisphophate receptors (IPR). The model is solved numerically on a mesh reconstructed from images of a cluster of parotid acinar cells. In contrast to our earlier model (Sneyd et al. in J Theor Biol 419:383-393. https://doi.org/10.1016/j.jtbi.2016.04.030 , 2017b), which cannot generate realisticCa 2 + oscillations with the new data on IPR distribution, our new model reproduces theCa 2 + dynamics observed in parotid acinar cells. This model is then coupled with a fluid secretion model described in detail in a companion paper: A mathematical model of fluid transport in an accurate reconstruction of a parotid acinar cell (Vera-Sigüenza et al. in Bull Math Biol. https://doi.org/10.1007/s11538-018-0534-z , 2018b). Based on the new measurements of IPR distribution, we show that Class I models (whereCa 2 + oscillations can occur at constant [IP 3 ]) can produceCa 2 + oscillations in parotid acinar cells, whereas Class II models (where [IP 3 ] needs to oscillate in order to produceCa 2 + oscillations) are unlikely to do so. In addition, we demonstrate that coupling fluid flow secretion with theCa 2 + signalling model changes the dynamics of theCa 2 + oscillations significantly, which indicates thatCa 2 + dynamics and fluid flow cannot be accurately modelled independently. Further, we determine that an active propagation mechanism based on calcium-induced calcium release channels is needed to propagate theCa 2 + wave from the apical region to the basal region of the acinar cell.

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.

Coding and decoding of oscillatory Ca2+ signals

About 30 years after their first observation, Ca²⁺ oscillations are now recognised as a universal mechanism of signal transduction. These oscillations are driven by periodic cycles of release and uptake of Ca²⁺ between the cytoplasm and the endoplasmic reticulum. Their frequency often increases with the level of stimulation, which can be decoded by some molecules. However, it is becoming increasingly evident that the widespread core oscillatory mechanism is modulated in many ways, depending on the cell type and on the physiological conditions. Interplay with inositol 1,4,5-trisphosphate metabolism and with other Ca²⁺ stores as the extracellular medium or mitochondria can much affect the properties of these oscillations. In many cases, these finely tuned characteristics of Ca²⁺ oscillations impact the physiological response that is triggered by the signal. Moreover, oscillations are intrinsically irregular. This randomness can also be exploited by the cell. In this review, we discuss evidences of these additional manifestations of the versatility of Ca²⁺ signalling.

Inositol phosphate kinases: Expanding the biological significance of the universal core of the protein kinase fold

The protein kinase family is characterized by substantial conservation of architectural elements that are required for both ATP binding and phosphotransferase activity. Many of these structural features have also been identified in homologous enzymes that phosphorylate a variety of alternative, non-protein substrates. A comparative structural analysis of these different kinase sub-classes is a portal to a greater understanding of reaction mechanisms, enzyme regulation, inhibitor-development strategies, and superfamily-level evolutionary relationships. To serve such advances, we review structural elements of the protein kinase fold that are conserved in the subfamily of inositol phosphate kinases (InsPKs) that share a PxxxDxKxG catalytic signature: inositol 1,4,5-trisphosphate kinase (IP3K), inositol hexakisphosphate kinase (IP6K), and inositol polyphosphate multikinase (IPMK). We describe conservation of the fundamental two-lobe kinase architecture: an N-lobe constructed upon an anti-parallel β-strand scaffold, which is coupled to a largely helical C-lobe by a single, adenine-binding hinge. This equivalency also includes a G-loop that embraces the β/γ-phosphates of ATP, a transition-state stabilizing residue (Lys/His), and a Mg-positioning aspartate residue within a catalytic triad. Furthermore, we expand this list of conserved structural features to include some not previously identified in InsPKs: a 'gatekeeper' residue in the N-lobe, and an 'αF'-like helix in the C-lobe that anchors two structurally-stabilizing, hydrophobic spines, formed from non-consecutive residues that span the two lobes. We describe how this wide-ranging structural homology can be exploited to develop lead inhibitors of IP6K and IPMK, by using strategies similar to those that have generated ATP-competing inhibitors of protein-kinases. We provide several examples to illustrate how such an approach could benefit human health.

Calcium Signaling, PKC Gamma, IP3R1 and CAR8 Link Spinocerebellar Ataxias and Purkinje Cell Dendritic Development

Background

Spinocerebellar ataxias (SCAs) are a group of cerebellar diseases characterized by progressive ataxia and cerebellar atrophy. Several forms of SCAs are caused by missense mutations or deletions in genes related to calcium signaling in Purkinje cells. Among them, spinocerebellar ataxia type 14 (SCA14) is caused by missense mutations in PRKCG gene which encodes protein kinase C gamma (PKCγ). It is remarkable that in several cases in which SCA is caused by point mutations in an individual gene, the affected genes are involved in the PKCγ signaling pathway and calcium signaling which is not only crucial for proper Purkinje cell function but is also involved in the control of Purkinje cell dendritic development. In this review, we will focus on the PKCγ signaling related genes and calcium signaling related genes then discuss their role for both Purkinje cell dendritic development and cerebellar ataxia.

Methods

Research related to SCAs and Purkinje cell dendritic development is reviewed.

Results

PKCγ dysregulation causes abnormal Purkinje cell dendritic development and SCA14. Carbonic anhydrase related protein 8 (Car8) encoding CAR8 and Itpr1 encoding IP3R1were identified as upregulated genes in one of SCA14 mouse model. IP3R1, CAR8 and PKCγ proteins are strongly and specifically expressed in Purkinje cells. The common function among them is that they are involved in the regulation of calcium homeostasis in Purkinje cells and their dysfunction causes ataxia in mouse and human. Furthermore, disruption of intracellular calcium homeostasis caused by mutations in some calcium channels in Purkinje cells links to abnormal Purkinje cell dendritic development and the pathogenesis of several SCAs.

Conclusion

Once PKCγ signaling related genes and calcium signaling related genes are disturbed, the normal dendritic development of Purkinje cells is impaired as well as the integration of signals from other neurons, resulting in abnormal development, cerebellar dysfunction and eventually Purkinje cell loss.

A mathematical model for the effects of amyloid beta on intracellular calcium

The accumulation of Alzheimer's disease (AD) associated Amyloid beta (Aβ) oligomers can trigger aberrant intracellular calcium (Ca2+) levels by disrupting the intrinsic Ca2+ regulatory mechanism within cells. These disruptions can cause changes in homeostasis levels that can have detrimental effects on cell function and survival. Although studies have shown that Aβ can interfere with various Ca2+ fluxes, the complexity of these interactions remains elusive. We have constructed a mathematical model that simulates Ca2+ patterns under the influence of Aβ. Our simulations shows that Aβ can increase regions of mixed-mode oscillations leading to aberrant signals under various conditions. We investigate how Aβ affects individual flux contributions through inositol triphosphate (IP3) receptors, ryanodine receptors, and membrane pores. We demonstrate that controlling for the ryanodine receptor's maximal kinetic reaction rate may provide a biophysical way of managing aberrant Ca2+ signals. The influence of a dynamic model for IP3 production is also investigated under various conditions as well as the impact of changes in membrane potential. Our model is one of the first to investigate the effects of Aβ on a variety of cellular mechanisms providing a base modeling scheme from which further studies can draw on to better understand Ca2+ regulation in an AD environment.

[Advances in methods for analyzing IP3 signaling and understanding of coupled Ca2+ and IP3 oscillations]

Inositol 1,4,5-trisphosphate (IP₃) is an important intracellular messenger produced by phospholipase C via the activation of G-protein-coupled receptor- or receptor-tyrosine-kinase-mediated pathways, and is involved in numerous responses to hormones, neurotransmitters, and growth factors through the releases of Ca²⁺ from intracellular stores via IP₃ receptors. IP₃-mediated Ca²⁺ signals often exhibit complex spatial and temporal organizations, such as Ca²⁺ oscillations. Recently, new methods have become available to measure IP₃ concentration ([IP₃]) using AlphaScreen technology, fluorescence polarization, and competitive ligand binding assay (CFLA). These methods are useful for the high throughput screening in drug discovery. Calcium ions generate versatile intracellular signals such as Ca²⁺ oscillations and waves. Fluorescent sensors molecules to monitor changes in [IP₃] in single living cells are crucial to study the mechanism for the spatially and temporally regulated Ca²⁺ signals. In particular, FRET-based IP₃ sensors are useful for the quantitative monitoring intracellular [IP₃], and allowed to uncovered the oscillatory IP₃ dynamics in association with Ca²⁺ oscillations. A mathematical model of coupled Ca²⁺ and IP₃ oscillations predicts that Ca²⁺ oscillations are the result of modulation of the IP₃ receptor by intracellular Ca²⁺, and that the period is modulated by the accompanying IP₃ oscillations. These model predictions have also been confirmed experimentally. At present, however, usefulness of FRET-based IP₃ sensors are limited by their relatively small change in fluorescence. Development of novel IP₃ sensors with improve dynamic range would be important for understanding the regulatory mechanism of Ca²⁺ signaling and for in vivo IP₃ imaging.

Effect of Electroacupuncture on 99mTc-Sodium Pertechnetate Uptake and Extracellular Fluid Free Molecules in the Stomach in Acupoint ST36 and ST39

Electroacupuncture (EA) is a therapeutic modality in which the electrical stimulation is integrated with concepts of acupuncture to treat diseases. This study was designed to evaluate the connection between the electro-acupuncture induced increase in Na^99mTcO4 uptake in the stomach wall, and the ionic molecule levels in the extracellular fluid in the acupoints. Wistar rats were treated by 2 or 100 Hz EA at Zusanli (ST 36) and Xiajuxu (ST 39) bilaterally for 60 minutes. The accumulation of Na^99mTcO4 in the gastric wall and the free ions, including Ca²⁺, K⁺, Na⁺, and Cl⁻, in the acupoints were measured every 60 minutes. The radioactivity uptake in the stomach was significantly increased during EA, reaching peak at 180 minutes after the EA. The concentration of extracellular ions was also significantly increased during EA. The Ca²⁺ level continued to rise until 60 minutes after EA, then started to decrease at 120 minutes post-EA. The results suggest this up-regulatory effect of EA on gastric activity might be triggered by the increase of the extracellular ion levels, this effect lasts longer than stimulating the release of transmembrane Ca²⁺ flow alone. This might aid in providing a better understanding of the long-lasting effect claimed in acupuncture treatment.

PLCζ Induced Ca2+ Oscillations in Mouse Eggs Involve a Positive Feedback Cycle of Ca2+ Induced InsP3 Formation From Cytoplasmic PIP2

Egg activation at fertilization in mammalian eggs is caused by a series of transient increases in the cytosolic free Ca²⁺ concentration, referred to as Ca²⁺ oscillations. It is widely accepted that these Ca²⁺ oscillations are initiated by a sperm derived phospholipase C isoform, PLCζ that hydrolyses its substrate PIP₂ to produce the Ca²⁺ releasing messenger InsP₃. However, it is not clear whether PLCζ induced InsP₃ formation is periodic or monotonic, and whether the PIP₂ source for generating InsP₃ from PLCζ is in the plasma membrane or the cytoplasm. In this study we have uncaged InsP₃ at different points of the Ca²⁺ oscillation cycle to show that PLCζ causes Ca²⁺ oscillations by a mechanism which requires Ca²⁺ induced InsP₃ formation. In contrast, incubation in Sr²⁺ media, which also induces Ca²⁺ oscillations in mouse eggs, sensitizes InsP₃-induced Ca²⁺ release. We also show that the cytosolic level Ca²⁺ is a key factor in setting the frequency of Ca²⁺ oscillations since low concentrations of the Ca²⁺ pump inhibitor, thapsigargin, accelerates the frequency of PLCζ induced Ca²⁺ oscillations in eggs, even in Ca²⁺ free media. Given that Ca²⁺ induced InsP₃ formation causes a rapid wave during each Ca²⁺ rise, we use a mathematical model to show that InsP₃ generation, and hence PLCζ's substate PIP₂, has to be finely distributed throughout the egg cytoplasm. Evidence for PIP₂ distribution in vesicles throughout the egg cytoplasm is provided with a rhodamine-peptide probe, PBP10. The apparent level of PIP₂ in such vesicles could be reduced by incubating eggs in the drug propranolol which also reversibly inhibited PLCζ induced, but not Sr²⁺ induced, Ca²⁺ oscillations. These data suggest that the cytosolic Ca²⁺ level, rather than Ca²⁺ store content, is a key variable in setting the pace of PLCζ induced Ca²⁺ oscillations in eggs, and they imply that InsP₃ oscillates in synchrony with Ca²⁺ oscillations. Furthermore, they support the hypothesis that PLCζ and sperm induced Ca²⁺ oscillations in eggs requires the hydrolysis of PIP₂ from finely spaced cytoplasmic vesicles.

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.

The role of neuron-glia interactions in the emergence of ultra-slow oscillations

Ultra-slow cortical oscillatory activity of 1-100 mHz has been recorded in human by electroencephalography and in dissociated cultures of cortical rat neurons, but the underlying mechanisms remain to be elucidated. This study presents a computational model of ultra-slow oscillatory activity based on the interaction between neurons and astrocytes. We predict that the frequency of these oscillations closely depends on activation of astrocytes in the network, which is reflected by oscillations of their intracellular calcium concentrations with periods between tens of seconds and minutes. An increase of intracellular calcium in astrocytes triggers the release of adenosine triphosphate from these cells which may alter transmission at nearby synapses by increasing or decreasing neurotransmitter release. These results provide theoretical support for the emerging awareness of astrocytes as active players in the regulation of neural activity and identify neuron-astrocyte interactions as a potential primary mechanism for the emergence of ultra-slow cortical oscillations.

Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified through Experimental Measurements and Mathematical Modeling

Astrocytes are a major cell type in the mammalian brain. They are not electrically excitable, but generate prominent Ca²⁺ signals related to a wide variety of critical functions. The mechanisms driving these Ca²⁺ events remain incompletely understood. In this study, we integrate Ca²⁺ imaging, quantitative data analysis, and mechanistic computational modeling to study the spatial and temporal heterogeneity of cortical astrocyte Ca²⁺ transients evoked by focal application of ATP in mouse brain slices. Based on experimental results, we tune a single-compartment mathematical model of IP₃-dependent Ca²⁺ responses in astrocytes and use that model to study response heterogeneity. Using information from the experimental data and the underlying bifurcation structure of our mathematical model, we categorize all astrocyte Ca²⁺ responses into four general types based on their temporal characteristics: Single-Peak, Multi-Peak, Plateau, and Long-Lasting responses. We find that the distribution of experimentally-recorded response types depends on the location within an astrocyte, with somatic responses dominated by Single-Peak (SP) responses and large and small processes generating more Multi-Peak responses. On the other hand, response kinetics differ more between cells and trials than with location within a given astrocyte. We use the computational model to elucidate possible sources of Ca²⁺ response variability: (1) temporal dynamics of IP₃, and (2) relative flux rates through Ca²⁺ channels and pumps. Our model also predicts the effects of blocking Ca²⁺ channels/pumps; for example, blocking store-operated Ca²⁺ (SOC) channels in the model eliminates Plateau and Long-Lasting responses (consistent with previous experimental observations). Finally, we propose that observed differences in response type distributions between astrocyte somas and processes can be attributed to systematic differences in IP₃ rise durations and Ca²⁺ flux rates.

[Mg2+]o/[Ca2+]o determines Ca2+ response at fertilization: tuning of adult phenotype?

Alteration of the postnatal phenotype has sparked great concern about the developmental impact of culture media used at fertilization. However, the mechanisms and compounds involved are yet to be determined. Here, we used the Ca²⁺ responses from mouse eggs fertilized by ICSI as a dynamic and quantitative marker to understand the role of compounds in egg functioning and establish possible correlations with adult phenotypes. We computed 134 Ca²⁺ responses from the first to the last oscillation in media with specific formulations. Analyses demonstrate that eggs generated two times as many Ca²⁺ oscillations in KSOM as in M16 media (18.8 ± 7.0 vs 9.2 ± 2.5). Moreover, the time increment of the delay between two consecutive oscillations, named TIbO, is the most sensitive coefficient characterizing the mechanism that paces Ca²⁺ oscillations once the egg has been fertilized. Neither doubling external free Ca²⁺ nor dispermic fertilization increased significantly the total number of Ca²⁺ oscillations. In contrast, removing Mg²⁺ from the M16 boosted Ca²⁺ oscillations to 54.0 ± 35.2. Hence, [Mg²⁺]_o/[Ca²⁺]_o appears to determine the number, duration and frequency of the Ca²⁺ oscillations. These changes were correlated with long-term effects. The rate of female's growth was impacted with the 'KSOM' females having only half the fat deposit of 'M16' females. Moreover, adult animals issued from M16 had significantly smaller brain weight vs 'KSOM' and 'control' animals. TIbO is a new Ca²⁺ coefficient that gauges the very early functional impact of culture media. It offers the possibility of establishing correlations with postnatal consequences according to IVF medium formulation.Free French abstract: A French translation of this abstract is freely available at http://www.reproduction-online.org/content/154/5/675/suppl/DC2.

A mathematical model of calcium dynamics in HSY cells

Saliva is an essential part of activities such as speaking, masticating and swallowing. Enzymes in salivary fluid protect teeth and gums from infectious diseases, and also initiate the digestion process. Intracellular calcium (Ca2+) plays a critical role in saliva secretion and regulation. Experimental measurements of Ca2+ and inositol trisphosphate (IP3) concentrations in HSY cells, a human salivary duct cell line, show that when the cells are stimulated with adenosine triphosphate (ATP) or carbachol (CCh), they exhibit coupled oscillations with Ca2+ spike peaks preceding IP3 spike peaks. Based on these data, we construct a mathematical model of coupled Ca2+ and IP3 oscillations in HSY cells and perform model simulations of three different experimental settings to forecast Ca2+ responses. The model predicts that when Ca2+ influx from the extracellular space is removed, oscillations gradually slow down until they stop. The model simulation of applying a pulse of IP3 predicts that photolysis of caged IP3 causes a transient increase in the frequency of the Ca2+ oscillations. Lastly, when Ca2+-dependent activation of PLC is inhibited, we see an increase in the oscillation frequency and a decrease in the amplitude. These model predictions are confirmed by experimental data. We conclude that, although concentrations of Ca2+ and IP3 oscillate, Ca2+ oscillations in HSY cells are the result of modulation of the IP3 receptor by intracellular Ca2+, and that the period is modulated by the accompanying IP3 oscillations.

On the dynamical structure of calcium oscillations

Oscillations in the concentration of free cytosolic Ca²⁺ are an important and ubiquitous control mechanism in many cell types. It is thus correspondingly important to understand the mechanisms that underlie the control of these oscillations and how their period is determined. We show that Class I Ca²⁺ oscillations (i.e., oscillations that can occur at a constant concentration of inositol trisphosphate) have a common dynamical structure, irrespective of the oscillation period. This commonality allows the construction of a simple canonical model that incorporates this underlying dynamical behavior. Predictions from the model are tested, and confirmed, in three different cell types, with oscillation periods ranging over an order of magnitude. The model also predicts that Ca²⁺ oscillation period can be controlled by modulation of the rate of activation by Ca²⁺ of the inositol trisphosphate receptor. Preliminary experimental evidence consistent with this hypothesis is presented. Our canonical model has a structure similar to, but not identical to, the classic FitzHugh-Nagumo model. The characterization of variables by speed of evolution, as either fast or slow variables, changes over the course of a typical oscillation, leading to a model without globally defined fast and slow variables.

New insights into mammalian signaling pathways using microfluidic pulsatile inputs and mathematical modeling

Temporally modulated input mimics physiology. This chemical communication strategy filters the biochemical noise through entrainment and phase-locking. Under laboratory conditions, it also expands the observability space for downstream responses. A combined approach involving microfluidic pulsatile stimulation and mathematical modeling has led to deciphering of hidden/unknown temporal motifs in several mammalian signaling pathways and has provided mechanistic insights, including how these motifs combine to form distinct band-pass filters and govern fate regulation under dynamic microenvironment. This approach can be utilized to understand signaling circuit architectures and to gain mechanistic insights for several other signaling systems. Potential applications include synthetic biology and biotechnology, in developing pharmaceutical interventions, and in developing lab-on-chip models.

Amplitude-Modulated Bursting: A Novel Class of Bursting Rhythms

We report on the discovery of a novel class of bursting rhythms, called amplitude-modulated bursting (AMB), in a model for intracellular calcium dynamics. We find that these rhythms are robust and exist on open parameter sets. We develop a new mathematical framework with broad applicability to detect, classify, and rigorously analyze AMB. Here we illustrate this framework in the context of AMB in a model of intracellular calcium dynamics. In the process, we discover a novel family of singularities, called toral folded singularities, which are the organizing centers for the amplitude modulation and exist generically in slow-fast systems with two or more slow variables.

The Inositol Trisphosphate/Calcium Signaling Pathway in Health and Disease

Many cellular functions are regulated by calcium (Ca2+) signals that are generated by different signaling pathways. One of these is the inositol 1,4,5-trisphosphate/calcium (InsP3/Ca2+) signaling pathway that operates through either primary or modulatory mechanisms. In its primary role, it generates the Ca2+ that acts directly to control processes such as metabolism, secretion, fertilization, proliferation, and smooth muscle contraction. Its modulatory role occurs in excitable cells where it modulates the primary Ca2+ signal generated by the entry of Ca2+ through voltage-operated channels that releases Ca2+ from ryanodine receptors (RYRs) on the internal stores. In carrying out this modulatory role, the InsP3/Ca2+ signaling pathway induces subtle changes in the generation and function of the voltage-dependent primary Ca2+ signal. Changes in the nature of both the primary and modulatory roles of InsP3/Ca2+ signaling are a contributory factor responsible for the onset of a large number human diseases.

Communication shapes sensory response in multicellular networks

Collective sensing by interacting cells is observed in a variety of biological systems, and yet, a quantitative understanding of how sensory information is collectively encoded is lacking. Here, we investigate the ATP-induced calcium dynamics of monolayers of fibroblast cells that communicate via gap junctions. Combining experiments and stochastic modeling, we find that increasing the ATP stimulus increases the propensity for calcium oscillations, despite large cell-to-cell variability. The model further predicts that the oscillation propensity increases with not only the stimulus, but also the cell density due to increased communication. Experiments confirm this prediction, showing that cell density modulates the collective sensory response. We further implicate cell-cell communication by coculturing the fibroblasts with cancer cells, which we show act as "defects" in the communication network, thereby reducing the oscillation propensity. These results suggest that multicellular networks sit at a point in parameter space where cell-cell communication has a significant effect on the sensory response, allowing cells to simultaneously respond to a sensory input and the presence of neighbors.

Calcium Dynamics of Ex Vivo Long-Term Cultured CD8+ T Cells Are Regulated by Changes in Redox Metabolism

T cells reach a state of replicative senescence characterized by a decreased ability to proliferate and respond to foreign antigens. Calcium release associated with TCR engagement is widely used as a surrogate measure of T cell response. Using an ex vivo culture model that partially replicates features of organismal aging, we observe that while the amplitude of Ca2+ signaling does not change with time in culture, older T cells exhibit faster Ca2+ rise and a faster decay. Gene expression analysis of Ca2+ channels and pumps expressed in T cells by RT-qPCR identified overexpression of the plasma membrane CRAC channel subunit ORAI1 and PMCA in older T cells. To test whether overexpression of the plasma membrane Ca2+ channel is sufficient to explain the kinetic information, we adapted a previously published computational model by Maurya and Subramaniam to include additional details on the store-operated calcium entry (SOCE) process to recapitulate Ca2+ dynamics after T cell receptor stimulation. Simulations demonstrated that upregulation of ORAI1 and PMCA channels is not sufficient to explain the observed alterations in Ca2+ signaling. Instead, modeling analysis identified kinetic parameters associated with the IP3R and STIM1 channels as potential causes for alterations in Ca2+ dynamics associated with the long term ex vivo culturing protocol. Due to these proteins having known cysteine residues susceptible to oxidation, we subsequently investigated and observed transcriptional remodeling of metabolic enzymes, a shift to more oxidized redox couples, and post-translational thiol oxidation of STIM1. The model-directed findings from this study highlight changes in the cellular redox environment that may ultimately lead to altered T cell calcium dynamics during immunosenescence or organismal aging.

Modeling calcium waves in an anatomically accurate three-dimensional parotid acinar cell

We construct a model of calcium waves in a three-dimensional anatomically accurate parotid acinar cell, constructed from experimental data. Gradients of inositol trisphosphate receptor (IPR) density are imposed, with the IPR density being greater closer to the lumen, which has a branched structure, and inositol trisphosphate (IP₃) is produced only at the basal membrane. We show (1) that IP₃ equilibrates so quickly across the cell that it can be assumed to be spatially homogeneous; (2) spatial separation of the sites of IP₃ action and IP₃ production does not preclude the formation of stable oscillatory Ca²⁺ waves. However, these waves are not waves in the mathematical sense of a traveling wave with fixed profile. They result instead from a time delay between the Ca²⁺ rise in the apical and basal regions; (3) the ryanodine receptors serve to reinforce the Ca²⁺ wave, but are not necessary for the wave to exist; (4) a spatially independent model is not sufficient to study saliva secretion, although a one-dimensional model might be sufficient. Our results here form the first stages of the construction of a multiscale and multicellular model of saliva secretion in an entire acinus.

The Phase Lag between Agonist-Induced Oscillatory Ca2+ and IP3 Signals Does Not Imply Causality (December 2015)

Activated phospholipase C (PLC*) generates 1,4,5-triphosphate (IP₃) and diacylglycerol (DAG) from phosphatidyl inositol (PIP₂). The DAG remains in the plasma membrane and co-activates conventional protein kinase C (PKC) with Ca²⁺. We have developed a mathematical model for the activation of the Ca²⁺-dependent PKC and its negative feedback on phospholipase C (PLC) and coupled it to the De Young-Keizer model for IP₃ mediated Ca²⁺ oscillations. The model describes the cascade of reactions for the translocation of PKC to plasma membrane, and simulates activation of Ca²⁺ and diacylglycerol (DAG) oscillations. The model demonstrates that oscillations in Ca²⁺ and DAG are possible with or without a positive Ca²⁺ feedback on phospholipase C consistent with experiment. In many experimental studies, the timing of the peaks of the Ca²⁺ and IP₃ oscillations have been used to suggest causality, i.e. that the IP₃ oscillations cause the Ca²⁺ oscillations. The model is used to explore this question. To this end, the positive and negative feedback between Ca²⁺ and IP₃ production are modulated, resulting in changes to the phase lag between the peaks in [Ca²⁺]_cyt and [IP]_cyt. The model simulates a possible experimental protocol that can be used to differentiate whether or not the positive feedback of Ca²⁺ on PLC is needed for the oscillations.

Effect of sound on gap-junction-based intercellular signaling: Calcium waves under acoustic irradiation

We present a previously unrecognized effect of sound waves on gap-junction-based intercellular signaling such as in biological tissues composed of endothelial cells. We suggest that sound irradiation may, through temporal and spatial modulation of cell-to-cell conductance, create intercellular calcium waves with unidirectional signal propagation associated with nonconventional topologies. Nonreciprocity in calcium wave propagation induced by sound wave irradiation is demonstrated in the case of a linear and a nonlinear reaction-diffusion model. This demonstration should be applicable to other types of gap-junction-based intercellular signals, and it is thought that it should be of help in interpreting a broad range of biological phenomena associated with the beneficial therapeutic effects of sound irradiation and possibly the harmful effects of sound waves on health.