A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte

Periodic ER-plasma membrane junctions support long-range Ca2+ signal integration in dendrites

Neuronal dendrites must relay synaptic inputs over long distances, but the mechanisms by which activity-evoked intracellular signals propagate over macroscopic distances remain unclear. Here, we discovered a system of periodically arranged endoplasmic reticulum-plasma membrane (ER-PM) junctions tiling the plasma membrane of dendrites at ∼1 μm intervals, interlinked by a meshwork of ER tubules patterned in a ladder-like array. Populated with Junctophilin-linked plasma membrane voltage-gated Ca2+ channels and ER Ca2+-release channels (ryanodine receptors), ER-PM junctions are hubs for ER-PM crosstalk, fine-tuning of Ca2+ homeostasis, and local activation of the Ca2+/calmodulin-dependent protein kinase II. Local spine stimulation activates the Ca2+ modulatory machinery, facilitating signal transmission and ryanodine-receptor-dependent Ca2+ release at ER-PM junctions over 20 μm away. Thus, interconnected ER-PM junctions support signal propagation and Ca2+ release from the spine-adjacent ER. The capacity of this subcellular architecture to modify both local and distant membrane-proximal biochemistry potentially contributes to dendritic computations.

Computational model of the spatiotemporal synergetic system dynamics of calcium, IP3 and dopamine in neuron cells

The functioning of several cellular processes in neuron cells relies on the interplay between multiple systems, such as calcium ([Ca2+]), inositol 1, 4, 5-trisphosphate (IP3), and dopamine. But, their individual dynamics provide very little insight into the various regulatory and dysregulatory cellular processes. The interaction of two systems dynamics offers some useful information about cell functioning in neurons. But, no attempt has been noted in the literature about the cooperation of three systems dynamics of [Ca2+], IP3, and dopamine in neurons. A mathematical model was utilized to examine the dynamic interactions of [Ca2+], IP3, and dopamine in neurons, considering their spatiotemporal aspects. Numerical findings were obtained using the finite element technique in conjunction with the Crank-Nicholson scheme. The effects of different component events like IP3-receptor (IP3R), sodium-calcium exchanger (NCX), calbindin-D28K buffer, etc. on the synergetic calcium, IP3, and dopamine dynamics have been studied in neuronal cells. The present model offers novel insights into the effects of regulation and dysregulation in different mechanisms like IP3R, NCX, calbindin-D28K, etc. on the synergetic systems of [Ca2+], IP3 and dopamine in neurons and their association with multiple neurological disorders, including Alzheimer's disease and Parkinson's disease.

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.

Spleen Tyrosine Kinase (SYK) negatively regulates ITAM-mediated human NK cell signaling and CD19-CAR NK cell efficacy

NK cells express activating receptors that signal through ITAM-bearing adapter proteins. The phosphorylation of each ITAM creates binding sites for SYK and ZAP70 protein tyrosine kinases to propagate downstream signaling including the induction ofCa 2 + influx. While all immature and mature human NK cells co-express SYK and ZAP70, clonally driven memory or adaptive NK cells can methylate SYK genes and signaling is mediated exclusively using ZAP70. Here, we examined the role of SYK and ZAP70 in a clonal human NK cell line KHYG1 by CRISPR-based deletion using a combination of experiments and mechanistic computational modeling. Elimination of SYK resulted in more robustCa + + influx after cross-linking of the CD16 and NKp30 receptors and enhanced phosphorylation of downstream proteins, whereas ZAP70 deletion diminished these responses. By contrast, ZAP70 depletion increased proliferation of the NK cells. As immature T cells express both SYK and ZAP70 but mature T cells often express only ZAP70, we transduced the human Jurkat cell line with SYK and found that expression of SYK increased proliferation but diminished TCR-inducedCa 2 + flux and activation. We performed transcriptional analysis of the matched sets of variant Jurkat and KHYG1 cells and observed profound alterations caused by SYK expression. As depletion of SYK in NK cells increased their activation, primary human NK cells were transduced with a CD19-targeting CAR and were CRISPR edited to ablate SYK or ZAP70. Deletion of SYK resulted in more robust cytotoxic activity and cytokine production, providing a new therapeutic strategy of NK cell engineering for cancer immunotherapy.

Differential behaviors of calcium-induced calcium release in one dimensional dendrite by Nernst-Planck equation, cable model and pure diffusion model

The source and dynamics of calcium is the key factor that regulates dendritic integration. Apart from the voltage-gated and ligand-gated calcium influx, an important source of calcium is from inner store of endoplasmic reticulum with a regenerative process of calcium-induced calcium release (CICR). To trigger this process, inositol 1,4,5-trisphosphate (IP3) and calcium are needed to satisfy certain requirements. The aim of our paper is to investigate how the CICR depends on the dynamics of membrane potential. We utilize one dimensional dendritic model to calculate membrane potential by Nernst-Planck Equation (NPE) and cable model and Pure Diffusion (PD) model, computational simulations are carried out to inject the calcium influx by synaptic stimulation and to predict subsequent CICR and calcium wave propagation. Our results demonstrate that CICR initiation and calcium wave propagation have much difference between electro-diffusion process of NPE and cable model. We find that cable model has lower threshold of IP3 stimulation to trigger CICR but is more difficult for calcium propagation than NPE, PD model requires even higher threshold of IP3 to initiate CICR process and calcium duration is shorter than NPE; the regenerative calcium wave propagates with faster speed in NPE than that in cable model and in PD model. Our work addresses the important role of electro-diffusion dynamics of charged ions in regulating CICR process in dendritic structure; and provides theoretical predictions for neurological process which requires sustaining calcium for downstream signaling processes.

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.

On the Ca2＋ elevation in vascular endothelial cells due to inositol trisphosphate-sensitive store receptors activation: A data-driven modeling approach

Agonist-induced Ca2+ signaling is essential for the regulation of many vital functions in endothelial cells (ECs). A broad range of stimuli elevate the cytosolic Ca2＋ concentration by promoting a pathway mediated by inositol 1,4,5 trisphosphate (IP3) which causes Ca2＋ release from intracellular stores. Despite its importance, there are very few studies focusing on the quantification of such dynamics in the vascular endothelium. Here, by using data from isolated ECs, we established a minimalistic modeling framework able to quantitatively capture the main features (averaged over a cell population) of the cytosolic Ca2＋ response to different IP3 stimulation levels. A suitable description of Ca2＋-regulatory function of inositol 1,4,5 trisphosphate receptors (IP3Rs) and corresponding parameter space are identified by comparing the different model variants against experimental mean population data. The same approach is used to numerically assess the relevance of cytosolic Ca2＋ buffering, as well as Ca2＋ store IP3-sensitivity in the overall cell dynamics. The variability in the dynamics' features observed across the population can be explained (at least in part) through variation of certain model parameters (such as buffering capacity or Ca2＋ store sensitivity to IP3). The results, in terms of experimental fitting and validation, support the proposed minimalistic model as a reference framework for the quantification of the EC Ca2＋ dynamics induced by IP3Rs activation.

Mechanistic insights of neuronal calcium and IP3 signaling system regulating ATP release during ischemia in progression of Alzheimer's disease

The mechanisms of calcium ([Ca²⁺]) signaling in various human cells have been widely analyzed by scientists due to its crucial role in human organs like the heartbeat, muscle contractions, bone activity, brain functionality, etc. No study is reported for interdependent [Ca²⁺] and IP₃ mechanics regulating the release of ATP in neuron cells during Ischemia in Alzheimer's disease advancement. In the present investigation, a finite element method (FEM) is framed to explore the interdependence of spatiotemporal [Ca²⁺] and IP₃ signaling mechanics and its role in ATP release during Ischemia as well as in the advancement of Alzheimer's disorder in neuron cells. The results provide us insights of the mutual spatiotemporal impacts of [Ca²⁺] and IP₃ mechanics as well as their contributions to ATP release during Ischemia in neuron cells. The results obtained for the mechanics of interdependent systems differ significantly from the results of simple independent system mechanics and provide new information about the processes of the two systems. From this study, it is concluded that neuronal disorders cannot only be simply attributed to the disturbance caused directly in the processes of calcium signaling mechanics, but also to the disturbances caused in IP₃ regulation mechanisms impacting the calcium regulation in the neuron cell and ATP release.

Analysis of Network Models with Neuron-Astrocyte Interactions

Neural networks, composed of many neurons and governed by complex interactions between them, are a widely accepted formalism for modeling and exploring global dynamics and emergent properties in brain systems. In the past decades, experimental evidence of computationally relevant neuron-astrocyte interactions, as well as the astrocytic modulation of global neural dynamics, have accumulated. These findings motivated advances in computational glioscience and inspired several models integrating mechanisms of neuron-astrocyte interactions into the standard neural network formalism. These models were developed to study, for example, synchronization, information transfer, synaptic plasticity, and hyperexcitability, as well as classification tasks and hardware implementations. We here focus on network models of at least two neurons interacting bidirectionally with at least two astrocytes that include explicitly modeled astrocytic calcium dynamics. In this study, we analyze the evolution of these models and the biophysical, biochemical, cellular, and network mechanisms used to construct them. Based on our analysis, we propose how to systematically describe and categorize interaction schemes between cells in neuron-astrocyte networks. We additionally study the models in view of the existing experimental data and present future perspectives. Our analysis is an important first step towards understanding astrocytic contribution to brain functions. However, more advances are needed to collect comprehensive data about astrocyte morphology and physiology in vivo and to better integrate them in data-driven computational models. Broadening the discussion about theoretical approaches and expanding the computational tools is necessary to better understand astrocytes' roles in brain functions.

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.

Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme

Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum is governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard's existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in H ¹ norm is obtained. Numerical experiments illustrate the theoretical results.

Decoding the Bell-Shaped Calcium Spikes in Phosphorylation Cycles of Flagella

We investigate the messenger role of calcium ions implicated in the regulation of wave-like bending dynamics of flagella. The emphasis is on microtubules of flagellar axoneme serving as nonlinear transmission lines for bell-shaped spikes of calcium ions. The calcium sensitive proteins, such as calmodulin, exhibit activation dependence on the spike train frequency and amplitude. Here, we analyze a Ca²⁺ decoding module IDA-I1 whose activity is controlled by Ca²⁺ activated kinase. We find that trains of Ca²⁺ spikes are advantageous compared to a constant rise in Ca²⁺ concentration as being more efficient and much less prone to noisy fluctuations.

The role of calcium oscillations in the phenotype selection in endothelial cells

Angiogenesis is an important process in the formation and maintenance of tissues which is driven by a complex system of intracellular and intercellular signaling mechanisms. Endothelial cells taking part in early angiogenesis must select their phenotype as either a tip cells (leading, migratory) or a stalk cells (following). Recent experiments have demonstrated that rapid calcium oscillations within active cells characterize this phenotype selection process and that these oscillations play a necessary role in governing phenotype selection and eventual vessel architecture. In this work, we develop a mathematical model capable of describing these oscillations and their role in phenotype selection then use it to improve our understanding of the biological mechanisms at play. We developed a model based on two previously published and experimentally validated mathematical models of calcium and angiogenesis then use our resulting model to simulate various multi-cell scenarios. We are able to capture essential calcium oscillation dynamics and intercellular communication between neighboring cells. The results of our model show that although the late DLL4 (a transmembrane protein that activates Notch pathway) levels of a cell are connected with its initial IP3 (Inositol 1,4,5-trisphosphate) level, cell-to-cell communication determines its eventual phenotype.

Modeling of Ca2+ transients initiated by GPCR agonists in mesenchymal stromal cells

The integrative study that included experimentation and mathematical modeling was carried out to analyze dynamic aspects of transient Ca²⁺ signaling induced by brief pulses of GPCR agonists in mesenchymal stromal cells from the human adipose tissue (AD-MSCs). The experimental findings argued for IP₃/Ca²⁺-regulated Ca²⁺ release via IP₃ receptors (IP₃Rs) as a key mechanism mediating agonist-dependent Ca²⁺ transients. The consistent signaling circuit was proposed to formalize coupling of agonist binding to Ca²⁺ mobilization for mathematical modeling. The model properly simulated the basic phenomenology of agonist transduction in AD-MSCs, which mostly produced single Ca²⁺ spikes upon brief stimulation. The spike-like responses were almost invariantly shaped at different agonist doses above a threshold, while response lag markedly decreased with stimulus strength. In AD-MSCs, agonists and IP₃ uncaging elicited similar Ca²⁺ transients but IP₃ pulses released Ca²⁺ without pronounced delay. This suggested that IP₃ production was rate-limiting in agonist transduction. In a subpopulation of AD-MSCs, brief agonist pulses elicited Ca²⁺ bursts crowned by damped oscillations. With properly adjusted parameters of IP₃R inhibition by cytosolic Ca²⁺, the model reproduced such oscillatory Ca²⁺ responses as well. GEM-GECO1 and R-CEPIA1er, the genetically encoded sensors of cytosolic and reticular Ca²⁺, respectively, were co-expressed in HEK-293 cells that also responded to agonists in an "all-or-nothing" manner. The experimentally observed Ca²⁺ signals triggered by ACh in both compartments were properly simulated with the suggested signaling circuit. Thus, the performed modeling of the transduction process provides sufficient theoretical basis for deeper interpretation of experimental findings on agonist-induced Ca²⁺ signaling in AD-MSCs.

Nontrivial amplification below the threshold for excitable cell signaling

In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.

Cardiomyocyte Calcium Ion Oscillations-Lessons From Physics

We review a theoretical, coarse-grained description for cardiomyocytes calcium dynamics that is motivated by experiments on RyR channel dynamics and provides an analogy to other spontaneously oscillating systems. We show how a minimal model, that focuses on calcium channel and pump dynamics and kinetics, results in a single, easily understood equation for spontaneous calcium oscillations (the Van-der-Pol equation). We analyze experiments on isolated RyR channels to quantify how the channel dynamics depends both on the local calcium concentration, as well as its temporal behavior (“adaptation”). Our oscillator model analytically predicts the conditions for spontaneous oscillations, their frequency and amplitude, and how each of those scale with the small number of relevant parameters related to calcium channel and pump activity. The minimal model is easily extended to include the effects of noise and external pacing (electrical or mechanical). We show how our simple oscillator predicts and explains the experimental observations of synchronization, “bursting” and reduction of apparent noise in the beating dynamics of paced cells. Thus, our analogy and theoretical approach provides robust predictions for the beating dynamics, and their biochemical and mechanical modulation.

Physics of Spontaneous Calcium Oscillations in Cardiac Cells and Their Entrainment

Mechanical contraction in muscle cells requires Ca to allow myosin binding to actin. Beating cardiomyocytes contain internal Ca stores whose cytoplasmic concentration oscillates. Our theory explains observed single channel dynamics as well as cellular oscillations in spontaneously beating cardiomyocytes. The Ca dependence of channel activity responsible for Ca release includes positive feedback with a delayed response. We use this to predict a dynamical equation for global calcium oscillations with only a few physically relevant parameters. The theory accounts for the observed entrainment of beating to an oscillatory electric or mechanical field.

Intercellular calcium waves integrate hormonal control of glucose output in the intact liver

Key points

Sympathetic outflow and circulating glucogenic hormones both regulate liver function by increasing cytosolic calcium, although how these calcium signals are integrated at the tissue level is currently unknown.

We show that stimulation of hepatic nerve fibres or perfusing the liver with physiological concentrations of vasopressin only will evoke localized cytosolic calcium oscillations and modest increases in hepatic glucose production.

The combination of these stimuli acted synergistically to convert localized and asynchronous calcium responses into co‐ordinated intercellular calcium waves that spread throughout the liver lobule and elicited a synergistic increase in hepatic glucose production.

The results obtained in the present study demonstrate that subthreshold levels of one hormone can create an excitable medium across the liver lobule, which allows global propagation of calcium signals in response to local sympathetic innervation and integration of metabolic regulation by multiple hormones. This enables the liver lobules to respond as functional units to produce full‐strength metabolic output at physiological levels of hormone.

Abstract

Glucogenic hormones, including catecholamines and vasopressin, induce frequency‐modulated cytosolic Ca²⁺ oscillations in hepatocytes, and these propagate as intercellular Ca²⁺ waves via gap junctions in the intact liver. We investigated the role of co‐ordinated Ca²⁺ waves as a mechanism for integrating multiple endocrine and neuroendocrine inputs to control hepatic glucose production in perfused rat liver. Sympathetic nerve stimulation elicited localized Ca²⁺ increases that were restricted to hepatocytes in the periportal zone. During perfusion with subthreshold vasopressin, sympathetic stimulation converted asynchronous Ca²⁺ signals in a limited number of hepatocytes into co‐ordinated intercellular Ca²⁺ waves that propagated across entire lobules. A similar synergism was observed between physiological concentrations of glucagon and vasopressin, where glucagon also facilitated the recruitment of hepatocytes into a Ca²⁺ wave. Hepatic glucose production was significantly higher with intralobular Ca²⁺ waves. We propose that inositol 1,4,5‐trisphosphate (IP₃)‐dependent Ca²⁺ signalling gives rise to an excitable medium across the functional syncytium of the hepatic lobule, co‐ordinating and amplifying the metabolic responses to multiple hormonal inputs.

Sympathetic outflow and circulating glucogenic hormones both regulate liver function by increasing cytosolic calcium, although how these calcium signals are integrated at the tissue level is currently unknown.

We show that stimulation of hepatic nerve fibres or perfusing the liver with physiological concentrations of vasopressin only will evoke localized cytosolic calcium oscillations and modest increases in hepatic glucose production.

The combination of these stimuli acted synergistically to convert localized and asynchronous calcium responses into co‐ordinated intercellular calcium waves that spread throughout the liver lobule and elicited a synergistic increase in hepatic glucose production.

The results obtained in the present study demonstrate that subthreshold levels of one hormone can create an excitable medium across the liver lobule, which allows global propagation of calcium signals in response to local sympathetic innervation and integration of metabolic regulation by multiple hormones. This enables the liver lobules to respond as functional units to produce full‐strength metabolic output at physiological levels of hormone.

A simple mechanochemical model for calcium signalling in embryonic epithelial cells

Calcium signalling is one of the most important mechanisms of information propagation in the body. In embryogenesis the interplay between calcium signalling and mechanical forces is critical to the healthy development of an embryo but poorly understood. Several types of embryonic cells exhibit calcium-induced contractions and many experiments indicate that calcium signals and contractions are coupled via a two-way mechanochemical feedback mechanism. We present a new analysis of experimental data that supports the existence of this coupling during apical constriction. We then propose a simple mechanochemical model, building on early models that couple calcium dynamics to the cell mechanics and we replace the hypothetical bistable calcium release with modern, experimentally validated calcium dynamics. We assume that the cell is a linear, viscoelastic material and we model the calcium-induced contraction stress with a Hill function, i.e. saturating at high calcium levels. We also express, for the first time, the "stretch-activation" calcium flux in the early mechanochemical models as a bottom-up contribution from stretch-sensitive calcium channels on the cell membrane. We reduce the model to three ordinary differential equations and analyse its bifurcation structure semi-analytically as two bifurcation parameters vary-the [Formula: see text] concentration, and the "strength" of stretch activation, [Formula: see text]. The calcium system ([Formula: see text], no mechanics) exhibits relaxation oscillations for a certain range of [Formula: see text] values. As [Formula: see text] is increased the range of [Formula: see text] values decreases and oscillations eventually vanish at a sufficiently high value of [Formula: see text]. This result agrees with experimental evidence in embryonic cells which also links the loss of calcium oscillations to embryo abnormalities. Furthermore, as [Formula: see text] is increased the oscillation amplitude decreases but the frequency increases. Finally, we also identify the parameter range for oscillations as the mechanical responsiveness factor of the cytosol increases. This work addresses a very important and not well studied question regarding the coupling between chemical and mechanical signalling in embryogenesis.

Causality Analysis and Cell Network Modeling of Spatial Calcium Signaling Patterns in Liver Lobules

Dynamics as well as localization of Ca²⁺ transients plays a vital role in liver function under homeostatic conditions, repair, and disease. In response to circulating hormonal stimuli, hepatocytes exhibit intracellular Ca²⁺ responses that propagate through liver lobules in a wave-like fashion. Although intracellular processes that control cell autonomous Ca²⁺ spiking behavior have been studied extensively, the intra- and inter-cellular signaling factors that regulate lobular scale spatial patterns and wave-like propagation of Ca²⁺ remain to be determined. To address this need, we acquired images of cytosolic Ca²⁺ transients in 1300 hepatocytes situated across several mouse liver lobules over a period of 1600 s. We analyzed this time series data using correlation network analysis, causal network analysis, and computational modeling, to characterize the spatial distribution of heterogeneity in intracellular Ca²⁺ signaling components as well as intercellular interactions that control lobular scale Ca²⁺ waves. Our causal network analysis revealed that hepatocytes are causally linked to multiple other co-localized hepatocytes, but these influences are not necessarily aligned uni-directionally along the sinusoids. Our computational model-based analysis showed that spatial gradients of intracellular Ca²⁺ signaling components as well as intercellular molecular exchange are required for lobular scale propagation of Ca²⁺ waves. Additionally, our analysis suggested that causal influences of hepatocytes on Ca²⁺ responses of multiple neighbors lead to robustness of Ca²⁺ wave propagation through liver lobules.

A Model for the Acrosome Reaction in Mammalian Sperm

The acrosome reaction is a complex, calcium-dependent reaction that results in an exocytotic event required for successful fertilization of the egg. It has long been thought that the acrosome reaction occurs upon sperm binding to the zona pellucida, a viscoelastic layer surrounding the oocyte. Recent studies have suggested that the reaction may even occur before the sperm encounters the zona, perhaps mediated by progesterone or some other agonist. It has been particularly difficult to understand differences between progesterone-induced and zona-induced reactions experimentally and whether one substance is the more biologically relevant trigger. Until this present work, there has been little effort to mathematically model the acrosome reaction in sperm as a whole. Instead, attention has been paid to modeling portions of the pathways involved in other cell types. Here we present a base model for the acrosome reaction which characterizes the known biochemical reactions and behaviors of the system. Our model allows us to analyze several pathways that may act as a stabilizing mechanism for avoiding sustained oscillatory calcium responses often observed in other cell types. Such an oscillatory regime might otherwise prevent acrosomal exocytosis and therefore inhibit fertilization. Results indicate that the acrosome reaction may rely upon multiple redundant mechanisms to avoid entering an oscillatory state and instead maintain a high resting level of calcium, known to be required for successful acrosomal exocytosis and, ultimately, fertilization of the oocyte.

Calcium as a signal integrator in developing epithelial tissues

Decoding how tissue properties emerge across multiple spatial and temporal scales from the integration of local signals is a grand challenge in quantitative biology. For example, the collective behavior of epithelial cells is critical for shaping developing embryos. Understanding how epithelial cells interpret a diverse range of local signals to coordinate tissue-level processes requires a systems-level understanding of development. Integration of multiple signaling pathways that specify cell signaling information requires second messengers such as calcium ions. Increasingly, specific roles have been uncovered for calcium signaling throughout development. Calcium signaling regulates many processes including division, migration, death, and differentiation. However, the pleiotropic and ubiquitous nature of calcium signaling implies that many additional functions remain to be discovered. Here we review a selection of recent studies to highlight important insights into how multiple signals are transduced by calcium transients in developing epithelial tissues. Quantitative imaging and computational modeling have provided important insights into how calcium signaling integration occurs. Reverse-engineering the conserved features of signal integration mediated by calcium signaling will enable novel approaches in regenerative medicine and synthetic control of morphogenesis.

Dynamic analysis on the calcium oscillation model considering the influences of mitochondria

Based on the model considering the influences of mitochondria, a further theoretical study on the dynamic behaviors of calcium signals is made. First of all, the reason for the generation and disappearance of calcium oscillations is verified in theory. Second, an analysis on the model considering the influences of mitochondria and the model neglecting the influences of mitochondria is carried out. Third, β (representing calcium leak) is introduced and it can be found that with the increase of β, the Hopf bifurcation points of system move towards the decreasing direction of μ (representing stimulus intensity) and calcium oscillations region gradually decreases. Forth, the study on τ_h (representing relaxation time) indicates that with the increase of τ_h, the second Hopf bifurcation point of system moves towards the increasing direction of μ and calcium oscillations region gradually increases. Under certain stimulus intensity, when relaxation time increases, calcium oscillation peak rises rapidly and the period increases obviously. Fifth, two-parameter bifurcation diagram of V_m1 (representing mitochondria activity) and μ contains three regions: stable region, oscillation region and unstable region. When the parameters fall in the unstable region Ca²⁺ gather towards mitochondria and further lead to cell apoptosis. With the increase of V_m1, calcium oscillations region shrinks gradually. V_m1 and μ both play a key role in regulating cell apoptosis. Only when V_m1 and μ are high enough can cells enter into programmed cell death and the higher V_m1 is, the lower the stimulus intensity required by cell apoptosis is.

Reproducibility and Comparability of Computational Models for Astrocyte Calcium Excitability

The scientific community across all disciplines faces the same challenges of ensuring accessibility, reproducibility, and efficient comparability of scientific results. Computational neuroscience is a rapidly developing field, where reproducibility and comparability of research results have gained increasing interest over the past years. As the number of computational models of brain functions is increasing, we chose to address reproducibility using four previously published computational models of astrocyte excitability as an example. Although not conventionally taken into account when modeling neuronal systems, astrocytes have been shown to take part in a variety of in vitro and in vivo phenomena including synaptic transmission. Two of the selected astrocyte models describe spontaneous calcium excitability, and the other two neurotransmitter-evoked calcium excitability. We specifically addressed how well the original simulation results can be reproduced with a reimplementation of the models. Additionally, we studied how well the selected models can be reused and whether they are comparable in other stimulation conditions and research settings. Unexpectedly, we found out that three of the model publications did not give all the necessary information required to reimplement the models. In addition, we were able to reproduce the original results of only one of the models completely based on the information given in the original publications and in the errata. We actually found errors in the equations provided by two of the model publications; after modifying the equations accordingly, the original results were reproduced more accurately. Even though the selected models were developed to describe the same biological event, namely astrocyte calcium excitability, the models behaved quite differently compared to one another. Our findings on a specific set of published astrocyte models stress the importance of proper validation of the models against experimental wet-lab data from astrocytes as well as the careful review process of models. A variety of aspects of model development could be improved, including the presentation of models in publications and databases. Specifically, all necessary mathematical equations, as well as parameter values, initial values of variables, and stimuli used should be given precisely for successful reproduction of scientific results.

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.

IP3, still on the move but now in the slow lane

In this issue of Science Signaling, Dickinson et al show that the intracellular messenger inositol 1,4,5-trisphosphate (IP₃), which triggers the release of calcium (Ca²⁺) from the endoplasmic reticulum, is a slowly diffusing local signal, rather than a rapidly diffusing global one. These findings have implications for the understanding of the mechanisms of Ca²⁺ wave propagation, especially long-range, cell-to-cell propagating Ca²⁺ waves.

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.

Modeling of TRPV₄-C₁ -mediated calcium signaling in vascular endothelial cells induced by fluid shear stress and ATP

The calcium signaling plays a vital role in flow-dependent vascular endothelial cell (VEC) physiology. Variations in fluid shear stress and ATP concentration in blood vessels can activate dynamic responses of cytosolic-free [Formula: see text] through various calcium channels on the plasma membrane. In this paper, a novel dynamic model has been proposed for transient receptor potential vanilloid 4 [Formula: see text]-mediated intracellular calcium dynamics in VECs induced by fluid shear stress and ATP. Our model includes [Formula: see text] signaling pathways through P2Y receptors and [Formula: see text] channels (indirect mechanism) and captures the roles of the [Formula: see text] compound channels in VEC [Formula: see text] signaling in response to fluid shear stress (direct mechanism). In particular, it takes into account that the [Formula: see text] compound channels are regulated by intracellular [Formula: see text] and [Formula: see text] concentrations. The simulation studies have demonstrated that the dynamic responses of calcium concentration produced by the proposed model correlate well with the existing experimental observations. We also conclude from the simulation studies that endogenously released ATP may play an insignificant role in the process of intracellular [Formula: see text] response to shear stress.

A deterministic model predicts the properties of stochastic calcium oscillations in airway smooth muscle cells

The inositol trisphosphate receptor () is one of the most important cellular components responsible for oscillations in the cytoplasmic calcium concentration. Over the past decade, two major questions about the have arisen. Firstly, how best should the be modeled? In other words, what fundamental properties of the allow it to perform its function, and what are their quantitative properties? Secondly, although calcium oscillations are caused by the stochastic opening and closing of small numbers of , is it possible for a deterministic model to be a reliable predictor of calcium behavior? Here, we answer these two questions, using airway smooth muscle cells (ASMC) as a specific example. Firstly, we show that periodic calcium waves in ASMC, as well as the statistics of calcium puffs in other cell types, can be quantitatively reproduced by a two-state model of the , and thus the behavior of the is essentially determined by its modal structure. The structure within each mode is irrelevant for function. Secondly, we show that, although calcium waves in ASMC are generated by a stochastic mechanism, stochasticity is not essential for a qualitative prediction of how oscillation frequency depends on model parameters, and thus deterministic models demonstrate the same level of predictive capability as do stochastic models. We conclude that, firstly, calcium dynamics can be accurately modeled using simplified models, and, secondly, to obtain qualitative predictions of how oscillation frequency depends on parameters it is sufficient to use a deterministic model.

The inositol trisphosphate receptor () is one of the most important cellular components responsible for calcium oscillations. Over the past decade, two major questions about the have arisen. Firstly, what fundamental properties of the allow it to perform its function? Secondly, although calcium oscillations are caused by the stochastic properties of small numbers of is it possible for a deterministic model to be a reliable predictor of calcium dynamics? Using airway smooth muscle cells as an example, we show that calcium dynamics can be accurately modeled using simplified models, and, secondly, that deterministic models are qualitatively accurate predictors of calcium dynamics. These results are important for the study of calcium dynamics in many cell types.

Modelling the transition from simple to complex Ca²⁺ oscillations in pancreatic acinar cells

A mathematical model is proposed which systematically investigates complex calcium oscillations in pancreatic acinar cells. This model is based on calcium-induced calcium release via inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) and includes calcium modulation of inositol (1,4,5) trisphosphate (IP3) levels through feedback regulation of degradation and production. In our model, the apical and the basal regions are separated by a region containing mitochondria, which is capable of restricting Ca2+ responses to the apical region. We were able to reproduce the observed oscillatory patterns, from baseline spikes to sinusoidal oscillations. The model predicts that calcium-dependent production and degradation of IP3 is a key mechanism for complex calcium oscillations in pancreatic acinar cells. A partial bifurcation analysis is performed which explores the dynamic behaviour of the model in both apical and basal regions.

Exploring oscillations in a point model of the intracellular Ca²⁺ concentration

Intracellular calcium (Ca²⁺) oscillations are a key signaling mechanism in most cell types. A prominent approach to modeling intracellular Ca²⁺ oscillations is the use of ordinary differential equations (ODEs), which treat the intracellular Ca²⁺ concentration as spatially homogenous. Although ODEs cannot account for the interaction of Ca²⁺ microdomains to form cell-wide Ca²⁺ patterns, modelers still choose ODEs because (a) the study of ODEs is computationally cheap, and a large body of techniques is available to investigate ODEs in great detail, or (b) sufficient experimental data are not available to develop a spatially extended model. Irrespective of the reason, analyzing ODEs is a key instrument in the toolbox of modelers. In this protocol, we look at a well-known model for Ca²⁺ oscillations, the De Young-Keizer model, along with the Li-Rinzel approximation of the De Young-Keizer model. The main emphasis of this protocol is the use of the open source software package XPPAUT to numerically study ODEs. The knowledge gained here can be directly transferred to other ODE systems and therefore may serve as a template for future studies.

A computational kinetic model of diffusion for molecular systems

Regulation of biomolecular transport in cells involves intra-protein steps like gating and passage through channels, but these steps are preceded by extra-protein steps, namely, diffusive approach and admittance of solutes. The extra-protein steps develop over a 10-100 nm length scale typically in a highly particular environment, characterized through the protein's geometry, surrounding electrostatic field, and location. In order to account for solute energetics and mobility of solutes in this environment at a relevant resolution, we propose a particle-based kinetic model of diffusion based on a Markov State Model framework. Prerequisite input data consist of diffusion coefficient and potential of mean force maps generated from extensive molecular dynamics simulations of proteins and their environment that sample multi-nanosecond durations. The suggested diffusion model can describe transport processes beyond microsecond duration, relevant for biological function and beyond the realm of molecular dynamics simulation. For this purpose the systems are represented by a discrete set of states specified by the positions, volumes, and surface elements of Voronoi grid cells distributed according to a density function resolving the often intricate relevant diffusion space. Validation tests carried out for generic diffusion spaces show that the model and the associated Brownian motion algorithm are viable over a large range of parameter values such as time step, diffusion coefficient, and grid density. A concrete application of the method is demonstrated for ion diffusion around and through the Eschericia coli mechanosensitive channel of small conductance ecMscS.

Mathematical modeling of calcium waves induced by mechanical stimulation in keratinocytes

Recent studies have shown that the behavior of calcium in the epidermis is closely related to the conditions of the skin, especially the differentiation of the epidermal keratinocytes and the permeability barrier function, and therefore a correct understanding of the calcium dynamics is important in explaining epidermal homeostasis. Here we report on experimental observations of in vitro calcium waves in keratinocytes induced by mechanical stimulation, and present a mathematical model that can describe the experimentally observed wave behavior that includes finite-range wave propagation and a ring-shaped pattern. A mechanism of the ring formation hypothesized by our model may be related to similar calcium propagation patterns observed during the wound healing process in the epidermis. We discuss a possible extension of our model that may serve as a tool for investigating the mechanisms of various skin diseases.

Modeling of progesterone-induced intracellular calcium signaling in human spermatozoa

Calcium ion is a secondary messenger of mammalian spermatozoa. The dynamic change of its concentration plays a vital role in the process of sperm motility, capacitation, acrosome and fertilization. Progesterone released by the cumulus cells, as a potent stimulator of fertilization, can activate the calcium channels on the plasma membrane, which in turn triggers the dynamic change of intracellular calcium concentration. In this paper, a mathematical model of calcium dynamic response in mammalian spermatozoa induced by progesterone is proposed and numerical simulation of the dynamic model is conducted. The results show that the dynamic response of calcium concentration predicted by the model is in accordance with experimental evidence. The proposed dynamic model can be used to explain the phenomena observed in the experiments and predict new phenomena to be revealed by experimental investigations, which will provide the basis to quantitatively investigate the fluid mechanics and biochemistry for the sperm motility induced by progesterone.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods

This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.

Chimeras of sperm PLCζ reveal disparate protein domain functions in the generation of intracellular Ca2+ oscillations in mammalian eggs at fertilization

Phospholipase C-zeta (PLCζ) is a sperm-specific protein believed to cause Ca(2+) oscillations and egg activation during mammalian fertilization. PLCζ is very similar to the somatic PLCδ1 isoform but is far more potent in mobilizing Ca(2+) in eggs. To investigate how discrete protein domains contribute to Ca(2+) release, we assessed the function of a series of PLCζ/PLCδ1 chimeras. We examined their ability to cause Ca(2+) oscillations in mouse eggs, enzymatic properties using in vitro phosphatidylinositol 4,5-bisphosphate (PIP2) hydrolysis and their binding to PIP2 and PI(3)P with a liposome interaction assay. Most chimeras hydrolyzed PIP2 with no major differences in Ca(2+) sensitivity and enzyme kinetics. Insertion of a PH domain or replacement of the PLCζ EF hands domain had no deleterious effect on Ca(2+) oscillations. In contrast, replacement of either XY-linker or C2 domain of PLCζ completely abolished Ca(2+) releasing activity. Notably, chimeras containing the PLCζ XY-linker bound to PIP2-containing liposomes, while chimeras containing the PLCζ C2 domain exhibited PI(3)P binding. Our data suggest that the EF hands are not solely responsible for the nanomolar Ca(2+) sensitivity of PLCζ and that membrane PIP2 binding involves the C2 domain and XY-linker of PLCζ. To investigate the relationship between PLC enzymatic properties and Ca(2+) oscillations in eggs, we have developed a mathematical model that incorporates Ca(2+)-dependent InsP3 generation by the PLC chimeras and their levels of intracellular expression. These numerical simulations can for the first time predict the empirical variability in onset and frequency of Ca(2+) oscillatory activity associated with specific PLC variants.

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.

Cell-type-specific modelling of intracellular calcium signalling: a urothelial cell model

Calcium signalling plays a central role in regulating a wide variety of cell processes. A number of calcium signalling models exist in the literature that are capable of reproducing a variety of experimentally observed calcium transients. These models have been used to examine in more detail the mechanisms underlying calcium transients, but very rarely has a model been directly linked to a particular cell type and experimentally verified. It is important to show that this can be achieved within the general theoretical framework adopted by these models. Here, we develop a framework designed specifically for modelling cytosolic calcium transients in urothelial cells. Where possible, we draw upon existing calcium signalling models, integrating descriptions of components known to be important in this cell type from a number of studies in the literature. We then add descriptions of several additional pathways that play a specific role in urothelial cell signalling, including an explicit ionic influx term and an active pumping mechanism that drives the cytosolic calcium concentration to a target equilibrium. The resulting one-pool model of endoplasmic reticulum (ER)-dependent calcium signalling relates the cytosolic, extracellular and ER calcium concentrations and can generate a wide range of calcium transients, including spikes, bursts, oscillations and sustained elevations in the cytosolic calcium concentration. Using single-variate robustness and multivariate sensitivity analyses, we quantify how varying each of the parameters of the model leads to changes in key features of the calcium transient, such as initial peak amplitude and the frequency of bursting or spiking, and in the transitions between bursting- and plateau-dominated modes. We also show that, novel to our urothelial cell model, the ionic and purinergic P2Y pathways make distinct contributions to the calcium transient. We then validate the model using human bladder epithelial cells grown in monolayer cell culture and show that the model robustly captures the key features of the experimental data in a way that is not possible using more generic calcium models from the literature.

Firing rate propagation through neuronal-astrocytic network

Understanding the underlying mechanism of the propagation of neuronal activities within the brain is a fundamental issue in neuroscience. Traditionally, communication and information processing have been exclusively considered as the province of synaptic coupling between neurons. Astrocytes, however, have recently been acknowledged as active partners in neuronal information processing. So, it is more reasonable and accurate to study the nature of neuronal signal propagation with the participation of astrocytes. In this paper, we first propose a feedforward neuronal-astrocytic network (FNAsN), which includes the mutual neuron-astrocyte interaction. Besides, we also consider the unreliability of both the synaptic transmission between neurons and the coupling between neurons and astrocytes. Then, the performance of firing rate propagation through the proposed FNAsN is studied through a series of simulations. Results show that the astrocytes can mediate neuronal activities, and consequently improve the performance of firing rate propagation, especially in a weak and noisy environment. From this point of view, astrocytes can be regarded as a realistic internal source of noise, which collaborates with an externally applied weak noise to prevent synchronous neuron firing within the same layer and thus to ensure reliable transmission.

Modeling of the kinetics of vitamin D(3) in osteoblastic cells

A one-dimensional model for the transport of vitamin D(3) in an osteoblast cell is proposed, from its entry through the membrane to its activation of RANKL synthesis in the nucleus. In the membrane and cytoplasm, the transport of D(3) and RANKL is described by a diffusion process, while their interaction in the nucleus is modeled by a reaction-diffusion process. For the latter, an integral equation involving the boundary conditions, as well as an asymptotic solution in the regime of small concentrations, are derived. Numerical simulations are also performed to investigate the kinetics of D(3) and RANKL through the entire cell. Comparison between the asymptotics and numerics in the nucleus shows an excellent agreement. To our knowledge, this is the first time, albeit using a simple model, a description of the complete passage of D(3) through the cell membrane, the cytoplasm, into the cell nucleus, and finally the production of RANKL with its passage to the exterior of the cell, has been modeled.

Role of microprojections in myoendothelial feedback--a theoretical study

We investigated the role of myoendothelial projections (MPs) in endothelial cell (EC) feedback response to smooth muscle cell (SMC) stimulation using mathematical modelling. A previously developed compartmental EC-SMC model is modified to include MPs as subcellular compartments in the EC. The model is further extended into a 2D continuum model using a finite element method (FEM) approach and electron microscopy images to account for MP geometry. The EC and SMC are coupled via non-selective myoendothelial gap junctions (MEGJs) which are located on MPs and allow exchange of Ca(2+), K(+), Na(+) and Cl(-) ions and inositol 1,4,5-triphosphate (IP3). Models take into consideration recent evidence for co-localization of intermediate-conductance calcium-activated potassium channels (IKCa) and IP3 receptors (IP3Rs) in the MPs. SMC stimulation causes an IP3-mediated Ca(2+) transient in the MPs with limited global spread in the bulk EC. A hyperpolarizing feedback generated by the localized IKCa channels is transmitted to the SMC via MEGJs. MEGJ resistance (Rgj) and the density of IKCa and IP3R in the projection influence the extent of EC response to SMC stimulation. The predicted Ca(2+) transients depend also on the volume and geometry of the MP. We conclude that in the myoendothelial feedback response to SMC stimulation, MPs are required to amplify the SMC initiated signal. Simulations suggest that the signal is mediated by IP3 rather than Ca(2+) diffusion and that a localized rather than a global EC Ca(2+) mobilization is more likely following SMC stimulation.

Modeling the contributions of Ca2+ flows to spontaneous Ca2+ oscillations and cortical spreading depression-triggered Ca2+ waves in astrocyte networks

Astrocytes participate in brain functions through Ca(2+) signals, including Ca(2+) waves and Ca(2+) oscillations. Currently the mechanisms of Ca(2+) signals in astrocytes are not fully clear. Here, we present a computational model to specify the relative contributions of different Ca(2+) flows between the extracellular space, the cytoplasm and the endoplasmic reticulum of astrocytes to the generation of spontaneous Ca(2+) oscillations (CASs) and cortical spreading depression (CSD)-triggered Ca(2+) waves (CSDCWs) in a one-dimensional astrocyte network. This model shows that CASs depend primarily on Ca(2+) released from internal stores of astrocytes, and CSDCWs depend mainly on voltage-gated Ca(2+) influx. It predicts that voltage-gated Ca(2+) influx is able to generate Ca(2+) waves during the process of CSD even after depleting internal Ca(2+) stores. Furthermore, the model investigates the interactions between CASs and CSDCWs and shows that the pass of CSDCWs suppresses CASs, whereas CASs do not prevent the generation of CSDCWs. This work quantitatively analyzes the generation of astrocytic Ca(2+) signals and indicates different mechanisms underlying CSDCWs and non-CSDCWs. Research on the different types of Ca(2+) signals might help to understand the ways by which astrocytes participate in information processing in brain functions.