Mathematical modelling of calcium wave propagation in mammalian airway epithelium: evidence for regenerative ATP release

A metabolic reaction-diffusion model for PKCα translocation via PIP2 hydrolysis in an endothelial cell

Hydrolysis of the phospholipid phosphatidylinositol 4,5-bisphosphate (PIP2) at the cell membrane induces the release of inositol 1,4,5-trisphosphate (IP3) into the cytoplasm and diffusion of diacylglycerol (DAG) through the membrane, respectively. Release of IP3 subsequently increases Ca2+ levels in the cytoplasm, which results in activation of protein kinase C α (PKCα) by Ca2+ and DAG, and finally the translocation of PKCα from the cytoplasm to the membrane. In this study, we developed a metabolic reaction-diffusion framework to simulate PKCα translocation via PIP2 hydrolysis in an endothelial cell. A three-dimensional cell model, divided into membrane and cytoplasm domains, was reconstructed from confocal microscopy images. The associated metabolic reactions were divided into their corresponding domain; PIP2 hydrolysis at the membrane domain resulted in DAG diffusion at the membrane domain and IP3 release into the cytoplasm domain. In the cytoplasm domain, Ca2+ was released from the endoplasmic reticulum, and IP3, Ca2+, and PKCα diffused through the cytoplasm. PKCα bound Ca2+ at, and diffused through, the cytoplasm, and was finally activated by binding with DAG at the membrane. Using our model, we analyzed IP3 and DAG dynamics, Ca2+ waves, and PKCα translocation in response to a microscopic stimulus. We found a qualitative agreement between our simulation results and our experimental results obtained by live-cell imaging. Interestingly, our results suggest that PKCα translocation is dominated by DAG dynamics. This three-dimensional reaction-diffusion mathematical framework could be used to investigate the link between PKCα activation in a cell and cell function.

Transmission of Mechanical Information by Purinergic Signaling

The skeleton constantly interacts and adapts to the physical world. We have previously reported that physiologically relevant mechanical forces lead to small repairable membrane injuries in bone-forming osteoblasts, resulting in release of ATP and stimulation of purinergic (P2) calcium responses in neighboring cells. The goal of this study was to develop a theoretical model describing injury-related ATP and ADP release, their extracellular diffusion and degradation, and purinergic responses in neighboring cells. After validation using experimental data for intracellular free calcium elevations, ATP, and vesicular release after mechanical stimulation of a single osteoblast, the model was scaled to a tissue-level injury to investigate how purinergic signaling communicates information about injuries with varying geometries. We found that total ATP released, peak extracellular ATP concentration, and the ADP-mediated signaling component contributed complementary information regarding the mechanical stimulation event. The total amount of ATP released governed spatial factors, such as the maximal distance from the injury at which purinergic responses were stimulated. The peak ATP concentration reflected the severity of an individual cell injury, allowing to discriminate between minor and severe injuries that released similar amounts of ATP because of differences in injury repair, and determined temporal aspects of the response, such as signal propagation velocity. ADP-mediated signaling became relevant only in larger tissue-level injuries, conveying information about the distance to the injury site and its geometry. Thus, we identified specific features of extracellular ATP and ADP spatiotemporal signals that depend on tissue mechanoresilience and encode the severity, scope, and proximity of the mechanical stimulus.

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.

Three-dimensional model of intracellular and intercellular Ca2+ waves propagation in endothelial cells

Intracellular and intercellular Ca²⁺ waves play key roles in cellular functions, and focal stimulation triggers Ca²⁺ wave propagation from stimulation points to neighboring cells, involving localized metabolism reactions and specific diffusion processes. Among these, inositol 1,4,5-trisphosphate (IP₃) is produced at membranes and diffuses into the cytoplasm to release Ca²⁺ from endoplasmic reticulum (ER). In this study, we developed a three-dimensional (3D) simulation model for intercellular and intracellular Ca²⁺ waves in endothelial cells (ECs). 3D model of 2 cells was reconstructed from confocal microscopic images and was connected via gap junctions. Cells have membrane and cytoplasm domains, and metabolic reactions were divided into each domain. Finally, the intracellular and intercellular Ca²⁺ wave propagations were induced using microscopic stimulation and were compared between numerical simulations and experiments. The experiments showed that initial sharp increases in intracellular Ca²⁺ occurred approximately 0.3 s after application of stimuli. In addition, Ca²⁺ wave speeds remained constant in cells, with intracellular and intercellular speeds of approximately 35 and 15 μm/s, respectively. Simulations indicated initial increases in Ca²⁺ concentrations at points of stimulation, and these were then propagated across stimulated and neighboring cells. In particular, initial rapid increases in intracellular Ca²⁺ were delayed and subsequent intracellular and intercellular Ca²⁺ wave speeds were approximately 25 and 12 μm/s, respectively. Simulation results were in agreement with those from cell culture experiments, indicating the utility of our 3D model for investigations of intracellular and intercellular messaging in ECs.

Spatiotemporal Characteristics of Intercellular Calcium Wave Communication in Micropatterned Assemblies of Single Cells

Micropatterned substrates offer a unique possibility to define and control spatial organization of biological cells at the microscale, which greatly facilitates investigations of the cell-to-cell communication in vitro. Here, we developed a simple micropatterning strategy to resolve various spatiotemporal characteristics of intercellular calcium wave (ICW) communication among isolated BV-2 microglial cells. By using a single-ring assembly, we found that the direction of the initial transmitter secretion was strongly correlated with the site of the cell at which the mechanical stimulus triggering the ICWs was imposed. By using multiring assemblies, we observed that the response ratio of the same outmost cells 160 μm away from the center increased from 0% in the single-ring assembly to 9.6% in the four-ring assembly. This revealed that cells located in the interring acted as regenerative amplifiers for the ICWs generated by the central cell. By using a special oval-type micropattern, we found that calcium mobilization in lamellipodia of a fusiform BV-2 microglia cell occurred 2.9 times faster than that in the middle part of the cell, demonstrating a higher region-specific sensitivity of lamellipodia to the transmitter. Taken together, our micropatterning strategy opened up new experimental prospects to study ICWs and revealed novel spatiotemporal characteristics of ICW communication including stimulation site-dependent secretion, regenerative propagation, and region-specific cell sensitivity.

Inverted battery design as ion generator for interfacing with biosystems

In a lithium-ion battery, electrons are released from the anode and go through an external electronic circuit to power devices, while ions simultaneously transfer through internal ionic media to meet with electrons at the cathode. Inspired by the fundamental electrochemistry of the lithium-ion battery, we envision a cell that can generate a current of ions instead of electrons, so that ions can be used for potential applications in biosystems. Based on this concept, we report an 'electron battery' configuration in which ions travel through an external circuit to interact with the intended biosystem whereas electrons are transported internally. As a proof-of-concept, we demonstrate the application of the electron battery by stimulating a monolayer of cultured cells, which fluoresces a calcium ion wave at a controlled ionic current. Electron batteries with the capability to generate a tunable ionic current could pave the way towards precise ion-system control in a broad range of biological applications.

Wound-induced Ca2+ wave propagates through a simple release and diffusion mechanism

Damage-associated molecular patterns (DAMPs) are critical mediators of information concerning tissue damage from damaged cells to neighboring healthy cells. ATP acts as an effective DAMP when released into extracellular space from damaged cells. Extracellular ATP receptors monitor tissue damage and activate a Ca²⁺ wave in the surrounding healthy cells. How the Ca²⁺ wave propagates through cells after a wound is unclear. Ca²⁺ wave activation can occur extracellularly via external receptors or intracellularly through GAP junctions. Three potential mechanisms to propagate the Ca²⁺ wave are source and sink, amplifying wave, and release and diffusion. Both source and sink and amplifying wave regulate ATP levels using hydrolysis or secretion, respectively, whereas release and diffusion relies on dilution. Here we systematically test these hypotheses using a microfluidics assay to mechanically wound an epithelial monolayer in combination with direct manipulation of ATP hydrolysis and release. We show that a release and diffusion model sufficiently explains Ca²⁺-wave propagation after an epithelial wound. A release and diffusion model combines the benefits of fast activation at short length scales with a self-limiting response to prevent unnecessary inflammatory responses harmful to the organism.

Model for calcium-mediated reduction of structural fluctuations in epidermis

We propose a reaction-advection-diffusion model of epidermis consisting of two variables, the degree of differentiation and the calcium ion concentration, where calcium ions enhance differentiation. By analytically and numerically investigating this system, we show that a calcium localization layer formed beneath the stratum corneum helps reduce spatiotemporal fluctuations of the structure of the stratum corneum. In particular, spatially or temporally small-scale fluctuations in the lower structure are suppressed and do not affect the upper structure, due to acceleration of differentiation by calcium ions. Analytical expressions for the reduction rate of fluctuation amplitudes are shown.

Computational Model of Ca2+ Wave Propagation in Human Retinal Pigment Epithelial ARPE-19 Cells

Objective

Computational models of calcium (Ca²⁺) signaling have been constructed for several cell types. There are, however, no such models for retinal pigment epithelium (RPE). Our aim was to construct a Ca²⁺ signaling model for RPE based on our experimental data of mechanically induced Ca²⁺ wave in the in vitro model of RPE, the ARPE-19 monolayer.

Methods

We combined six essential Ca²⁺ signaling components into a model: stretch-sensitive Ca²⁺ channels (SSCCs), P₂Y₂ receptors, IP₃ receptors, ryanodine receptors, Ca²⁺ pumps, and gap junctions. The cells in our epithelial model are connected to each other to enable transport of signaling molecules. Parameterization was done by tuning the above model components so that the simulated Ca²⁺ waves reproduced our control experimental data and data where gap junctions were blocked.

Results

Our model was able to explain Ca²⁺ signaling in ARPE-19 cells, and the basic mechanism was found to be as follows: 1) Cells near the stimulus site are likely to conduct Ca²⁺ through plasma membrane SSCCs and gap junctions conduct the Ca²⁺ and IP³ between cells further away. 2) Most likely the stimulated cell secretes ligand to the extracellular space where the ligand diffusion mediates the Ca²⁺ signal so that the ligand concentration decreases with distance. 3) The phosphorylation of the IP₃ receptor defines the cell’s sensitivity to the extracellular ligand attenuating the Ca²⁺ signal in the distance.

Conclusions

The developed model was able to simulate an array of experimental data including drug effects. Furthermore, our simulations predict that suramin may interfere ligand binding on P₂Y₂ receptors or accelerate P₂Y₂ receptor phosphorylation, which may partially be the reason for Ca²⁺ wave attenuation by suramin. Being the first RPE Ca²⁺ signaling model created based on experimental data on ARPE-19 cell line, the model offers a platform for further modeling of native RPE functions.

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.

Calcium (Ca2+) waves data calibration and analysis using image processing techniques

Background

Calcium (Ca2+) propagates within tissues serving as an important information carrier. In particular, cilia beat frequency in oviduct cells is partially regulated by Ca2+ changes. Thus, measuring the calcium density and characterizing the traveling wave plays a key role in understanding biological phenomena. However, current methods to measure propagation velocities and other wave characteristics involve several manual or time-consuming procedures. This limits the amount of information that can be extracted, and the statistical quality of the analysis.

Results

Our work provides a framework based on image processing procedures that enables a fast, automatic and robust characterization of data from two-filter fluorescence Ca2+ experiments. We calculate the mean velocity of the wave-front, and use theoretical models to extract meaningful parameters like wave amplitude, decay rate and time of excitation.

Conclusions

Measurements done by different operators showed a high degree of reproducibility. This framework is also extended to a single filter fluorescence experiments, allowing higher sampling rates, and thus an increased accuracy in velocity measurements.

A dynamic model for intracellular calcium response in fibroblasts induced by electrical stimulation

Regulation of intracellular calcium ion concentration ([Ca(2+)]in) in fibroblasts induced by exogenous electrical stimulation could be applied to control gene expressions selectively which in turn modulate the function of the fibroblasts. Regarding the mechanism for electric-field-induced Ca(2+) influx via voltage-gated Ca(2+) channels and/or stretch-activated cation channels in the fibroblasts, a dynamic mathematical model is proposed to quantify the [Ca(2+)]in dynamics in response to direct current or alternating current electric fields. Simulation results demonstrate that the changes in [Ca(2+)]in predicted by our dynamic model are consistent with the experimental data in the literature. The proposed dynamic model could provide not only more insights into the electric-field-induced intracellular Ca(2+) response but also a quantitative way to regulate the [Ca(2+)]in dynamics by controlling the external electrical stimulation.

A mechanochemical model for auto-regulation of lung airway surface layer volume

We develop a proof-of-principle model for auto-regulation of water volume in the lung airway surface layer (ASL) by coupling biochemical kinetics, transient ASL volume, and homeostatic mechanical stresses. The model is based on the hypothesis that ASL volume is sensed through soluble mediators and phasic stresses generated by beating cilia and air drag forces. Model parameters are fit based on the available data on human bronchial epithelial cell cultures. Simulations then demonstrate that homeostatic volume regulation is a natural consequence of the processes described. The model maintains ASL volume within a physiological range that modulates with phasic stress frequency and amplitude. Next, we show that the model successfully reproduces the responses of cell cultures to significant isotonic and hypotonic challenges, and to hypertonic saline, an effective therapy for mucus hydration in cystic fibrosis patients. These results compel an advanced airway hydration model with therapeutic value that will necessitate detailed kinetics of multiple molecular pathways, feedback to ASL viscoelasticity properties, and stress signaling from the ASL to the cilia and epithelial cells.

Intercellular Ca(2+) waves: mechanisms and function

Intercellular calcium (Ca(2+)) waves (ICWs) represent the propagation of increases in intracellular Ca(2+) through a syncytium of cells and appear to be a fundamental mechanism for coordinating multicellular responses. ICWs occur in a wide diversity of cells and have been extensively studied in vitro. More recent studies focus on ICWs in vivo. ICWs are triggered by a variety of stimuli and involve the release of Ca(2+) from internal stores. The propagation of ICWs predominately involves cell communication with internal messengers moving via gap junctions or extracellular messengers mediating paracrine signaling. ICWs appear to be important in both normal physiology as well as pathophysiological processes in a variety of organs and tissues including brain, liver, retina, cochlea, and vascular tissue. We review here the mechanisms of initiation and propagation of ICWs, the key intra- and extracellular messengers (inositol 1,4,5-trisphosphate and ATP) mediating ICWs, and the proposed physiological functions of ICWs.

Ciliary beat co-ordination by calcium

Motile cilia in the airway epithelium are the engine for mucociliary clearance, the mechanism responsible for cleaning the airways from inhaled particles. Human airway epithelial cilia appear to have a slow constitutive rate of beating, driven by inherent and spontaneous dynein ATPase activity. Additionally, cilia can increase their beating frequency by activation of several different control mechanisms. One of these controllers is calcium. Its intracellular concentration is regulated by purinergic and acetylcholine receptors. Besides the rate regulatory effect of calcium on ciliary beat, calcium is also involved in synchronizing the beat among cilia of one single cell as well as between cilia on different cells. This article gives an overview of the complex effects of calcium on the beating of motile cilia in the airways.

The effect of intracellular calcium oscillations on fluid secretion in airway epithelium

Airway epithelium has been shown to elicit fluid secretion after a rise in intracellular calcium. This rise in intracellular calcium has been shown to display complex oscillations in many species after the binding of particular agonists to extracellular receptors. Fluid secreted by the airway epithelium is used to maintain the depth of the periciliary liquid (PCL) above the apical membrane of the epithelial cells lining the bronchial airways. Previous mathematical models have been published which separately consider the electrophysiology involved in regulating periciliary liquid depth, and the transmission of intracellular calcium waves in airway epithelial tissue. In this paper we present a mathematical model that combines these previous models and allows the effect of oscillations in intracellular calcium on fluid secretion by airway epithelial cells to be investigated. We show that an oscillatory calcium response produces different fluid secretion properties to that elicited by a tonic rise in intracellular calcium. These differences are shown to be due to saturation of the Ca(2+) activated ion channels.

The role of airway epithelium in replenishment of evaporated airway surface liquid from the human conducting airways

This article presents a multi-scale computational model describing the transport of water vapor and heat within the human conducting airways and its interaction with cellular fluid transport kinetics. This tight coupling between the cell and the evaporative flux allows the periciliary liquid (PCL) depth to be investigated within the context of a geometric framework of the human conducting airways with spatial and temporal variations. Within the in vivo airway, the epithelium is not the only source of fluid available for hydration of the PCL, and fluid may also be supplied from submucosal glands (SMGs) or via axial transport of the PCL. The model predicts that without fluid supplied by either SMGs or via PCL transport, significant dehydration would occur under normal breathing conditions. Previous studies have suggested that PCL transport from the periphery to the trachea would require absorption of the fluid by the epithelium; here we show that this can theoretically be sustained by the evaporative load under normal breathing conditions. SMGs could also provide a significant supply of fluid for airway hydration, a hypothesis which is corroborated by comparing the distribution of SMGs as a function of airway generation with the distribution of airway evaporative flux.

A dynamic model of saliva secretion

We construct a mathematical model of the parotid acinar cell with the aim of investigating how the distribution of K(+) and Cl(-) channels affects saliva production. Secretion of fluid is initiated by Ca(2+) signals acting on Ca(2+) dependent K(+) and Cl(-) channels. The opening of these channels facilitates the movement of Cl(-) ions into the lumen which water follows by osmosis. We use recent results into both the release of Ca(2+) from internal stores via the inositol (1,4,5)-trisphosphate receptor (IP(3)R) and IP(3) dynamics to create a physiologically realistic Ca(2+) model which is able to recreate important experimentally observed behaviours seen in parotid acinar cells. We formulate an equivalent electrical circuit diagram for the movement of ions responsible for water flow which enables us to calculate and include distinct apical and basal membrane potentials to the model. We show that maximum saliva production occurs when a small amount of K(+) conductance is located at the apical membrane, with the majority in the basal membrane. The maximum fluid output is found to coincide with a minimum in the apical membrane potential. The traditional model whereby all Cl(-) channels are located in the apical membrane is shown to be the most efficient Cl(-) channel distribution.