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

Modelling spatio-temporal interactions between second messengers Ca 2 + and cAMP in a pancreatic β -cell

Calcium serves as a widespread second messenger in almost every human and animal cell. The regulation of various cellular processes, such as transcriptional control and the kinetics of membrane channels, is significantly influenced by intracellular calcium ions (Ca2 + ), and linkages between Ca2 + and other second messengers should activate signaling networks. The passage of ions across the cell membrane regulates Ca2 + levels in pancreatic β -cells and requires the coordinated interaction of various ion transport mechanisms and organelles. The signaling of Ca2 + in β -cells and its interactions with the intracellular dynamics of cyclic adenosine monophosphate (cAMP) is poorly understood. Therefore, the current investigation proposes a mathematical model to illustrate the spatiotemporal dynamical interaction between Ca2 + and cAMP. In order to construct a one-dimensional mathematical model, the fundamental initial and boundary conditions derived from the physiological characteristics of the β -cell are incorporated. The numerical results were obtained by MATLAB simulations using the finite element method and the Crank-Nicolson method. The current study aims to offer an update on regulation between Ca2 + and cAMP signaling circuits, with a focus on interactions that occur in localized areas of the β -cell. The model gives the individual effect of each parameter on the regulation of Ca2 + and cAMP profiles in a β -cell. Evidently, impairments in the regulation of messenger pathways contribute to the pathological conditions, as demonstrated by the results obtained.

Regulatory disturbances in the dynamical signaling systems of C a 2 + and NO in fibroblasts cause fibrotic disorders

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium ( C a 2 + ) and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of C a 2 + dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum C a 2 + -ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate ( I P 3 ) receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease.

Model of Calcium Dynamics Regulating [Formula: see text], ATP and Insulin Production in a Pancreatic [Formula: see text]-Cell

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

2D dynamic analysis of the disturbances in the calcium neuronal model and its implications in neurodegenerative disease

Ca2+ signaling is an essential function of neurons to control synaptic activity, memory formation, fertilization, proliferation, etc. Protein and voltage-dependent calcium channels (VDCCs) maintain an adequate level of calcium concentration ([Ca2+]). An alteration in [Ca2+] leads to the death of the neurons that start the primary symptoms of the disease. The present study deals with cell memory-based mathematical modeling of Ca2+ that is characterized by the presence of protein and VDCC. We developed a two-dimensional Ca2+ neuronal model to study the spatiotemporal behavior of the Ca2+ profile. All principal parameters like buffer concentration, diffusion coefficient, VDCC fluxes, etc. are incorporated in this model. Apposite initial and boundary conditions are applied to the physiology of the problem. We obtained an approximate Ca2+ profile by the fractional integral transform method. The application of obtained results is performed to provide its implications to estimate the [Ca2+] in neurodegenerative disease. It is observed that the protein and VDCC provide a significant impact in the presence of cell memory. The memory of cells shrinks the Ca2+ flow from elevation and provides better results to estimated Ca2+ flow in the disease state.

Study of disorders in regulatory spatiotemporal neurodynamics of calcium and nitric oxide

Experimental studies have reported the dependence of nitric oxide (NO) on the regulation of neuronal calcium ([Ca2+]) dynamics in neurons. But, there is no model available to estimate the disorders caused by various parameters in their regulatory dynamics leading to various neuronal disorders. A mathematical model to analyze the impacts due to alterations in various parameters like buffer, ryanodine receptor, serca pump, source influx, etc. leading to regulation and dysregulation of the spatiotemporal calcium and NO dynamics in neuron cells is constructed using a system of reaction-diffusion equations. The numerical simulation is performed with the finite element approach. The disturbances in the different constitutive processes of [Ca2+] and nitric oxide including source influx, buffer mechanism, ryanodine receptor, serca pump, IP3 receptor, etc. can be responsible for the dysregulation in the [Ca2+] and NO dynamics in neurons. Also, the results reveal novel information about the magnitude and intensity of disorders in response to a range of alterations in various parameters of this neuronal dynamics, which can cause dysregulation leading to neuronal diseases like Parkinson's, cerebral ischemia, trauma, etc.

To study the effect of ER flux with buffer on the neuronal calcium

Calcium signaling is decisive for cellular functions. This calcium random walk stipulates neuronal functions. Calcium concentration could provoke gene transcription, apoptosis, neuronal plasticity, etc. A malformation in calcium could change the neuron's intracellular behavior. Calcium concentration balancing is a complex cellular mechanism. This occurrence can be handled with the Caputo fractional reaction-diffusion equation. In this mathematical modeling, we have included the STIM-Orai mechanism and Endoplasmic Reticulum (ER) flux, Inositol Triphosphate Receptor (IPR), SERCA, plasma membrane flux, voltage-gated calcium entry, and different buffer interactions. A hybrid integral transform and Green's function approach were taken to solve the initial boundary problem. A closed-form solution of a Mittag-Leffler family function plotted using MATLAB software. Different parameters impact changes in the spatiotemporal behavior of the calcium concentration. Specific roles of organelles involved in Alzheimer's disease-affected neurons are computed. Ethylene glycol tetraacetic acid (EGTA), 1,2-bis(o-aminophenoxy)ethane N,N,N,N-tetraacetic acid (BAPTA), and S100B protein effects are also observed. In all simulations, we can say S100B and the STIM-Orai effect cannot be neglected. This model lights up the different approaches for calcium signaling pathway simulation. As a consequence, we determine that a generalized reaction-diffusion approach is a better fit realistic model.

Cellular nitric oxide synthesis is affected by disorders in the interdependent [Formula: see text] and [Formula: see text] dynamics during cystic fibrosis disease

Calcium ([Formula: see text]), inositol trisphosphate ([Formula: see text]), and nitric oxide (NO) signaling are essential to maintain the structural integrity and physiological activity of fibroblast cells. The accumulation of excess quantity of NO for longer periods can lead to a variety of fibrotic disorders, including heart disease, penile fibrosis in Peyronie's disease (PD), and cystic fibrosis. The dynamics of these three signaling processes and their interdependence in fibroblast cells are not clearly known to date. A systems biology model is proposed using reaction-diffusion equations for calcium, [Formula: see text], and calcium-dependent NO synthesis in fibroblast cells. The finite element method (FEM) is used to examine [Formula: see text], [Formula: see text], and NO regulation and dysregulation in cells. The results throw light on the conditions that disturb the coupled [Formula: see text] and [Formula: see text] dynamics and the influence of these factors on the levels of NO concentration in the fibroblast cell. The findings suggest that changes in source inflow, buffers, and diffusion coefficient might induce an increase or reduction in nitric oxide and [Formula: see text] synthesis, resulting in fibroblast cell diseases. Furthermore, the findings provide new information regarding the size and intensity of diseases in response to changes in several factors of their dynamics, which has been linked to the development of cystic fibrosis and cancer. This knowledge could be valuable for developing novel approaches to the diagnosis of diseases and therapies for various disorders of fibroblast cells.

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.

Effect of disturbances in neuronal calcium and IP3 dynamics on β-amyloid production and degradation

Overproduction and accumulation of β-amyloid and its improper clearance can cause neurotoxicity leading to Alzheimer's disease. The production and degradation of β-amyloid depend on the calcium ([Ca2+]) and IP3 dynamics in the nerve cells. Thus, there is a need to understand the impacts of disturbances in the processes of [Ca2+] and IP3 dynamics on β-amyloid production and its degradation. Here, a model is proposed to investigate the role of [Ca2+] and IP3 dynamics on β-amyloid production and degradation. The problem is formulated in terms of the initial boundary value problem involving the system of two reaction-diffusion equations respectively for [Ca2+] and IP3 in the nerve cell. The solution is obtained by employing the Finite element approach. The numerical results are used to analyze the impact of various mechanisms of calcium and IP3 dynamics on β-amyloid production and degradation in a neuron cell. The results indicate that disturbances in any of the constitutive processes of interdependent calcium and IP3 dynamics like source influx, buffering, serca pump, and IP3 dynamics, etc. can cause dynamic changes in β-amyloid production and degradation, which in turn can be the cause of neurotoxicity and neuronal disorders like Alzheimer's disease. Thus, the relationships obtained by the proposed model among various mechanisms can be useful in addressing the challenges of identifying specific constitutive processes causing neuronal disorders like Alzheimer's disease, etc., and developing the framework for their diagnosis and treatment.

Complex dynamics in an unexplored simple model of the peroxidase-oxidase reaction

A previously overlooked version of the so-called Olsen model of the peroxidase-oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern-Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.

Calcium imaging in intact mouse acinar cells in acute pancreas tissue slices

The physiology and pathophysiology of the exocrine pancreas are in close connection to changes in intra-cellular Ca²⁺ concentration. Most of our knowledge is based on in vitro experiments on acinar cells or acini enzymatically isolated from their surroundings, which can alter their structure, physiology, and limit our understanding. Due to these limitations, the acute pancreas tissue slice technique was introduced almost two decades ago as a complementary approach to assess the morphology and physiology of both the endocrine and exocrine pancreas in a more conserved in situ setting. In this study, we extend previous work to functional multicellular calcium imaging on acinar cells in tissue slices. The viability and morphological characteristics of acinar cells within the tissue slice were assessed using the LIVE/DEAD assay, transmission electron microscopy, and immunofluorescence imaging. The main aim of our study was to characterize the responses of acinar cells to stimulation with acetylcholine and compare them with responses to cerulein in pancreatic tissue slices, with special emphasis on inter-cellular and inter-acinar heterogeneity and coupling. To this end, calcium imaging was performed employing confocal microscopy during stimulation with a wide range of acetylcholine concentrations and selected concentrations of cerulein. We show that various calcium oscillation parameters depend monotonically on the stimulus concentration and that the activity is rather well synchronized within acini, but not between acini. The acute pancreas tissue slice represents a viable and reliable experimental approach for the evaluation of both intra- and inter-cellular signaling characteristics of acinar cell calcium dynamics. It can be utilized to assess many cells simultaneously with a high spatiotemporal resolution, thus providing an efficient and high-yield platform for future studies of normal acinar cell biology, pathophysiology, and screening pharmacological substances.

The Effect of Gap Junctional Coupling on the Spatiotemporal Patterns of Ca2+ Signals and the Harmonization of Ca2+-Related Cellular Responses

Calcium ions (Ca²⁺) are important mediators of a great variety of cellular activities e.g. in response to an agonist activation of a receptor. The magnitude of a cellular response is often encoded by frequency modulation of Ca²⁺ oscillations and correlated with the stimulation intensity. The stimulation intensity highly depends on the sensitivity of a cell to a certain agonist. In some cases, it is essential that neighboring cells produce a similar and synchronized response to an agonist despite their different sensitivity. In order to decipher the presumed function of Ca²⁺ waves spreading among connecting cells, a mathematical model was developed. This model allows to numerically modifying the connectivity probability between neighboring cells, the permeability of gap junctions and the individual sensitivity of cells to an agonist. Here, we show numerically that strong gap junctional coupling between neighbors ensures an equilibrated response to agonist stimulation via formation of Ca²⁺ phase waves, i.e. a less sensitive neighbor will produce the same or similar Ca²⁺ signal as its highly sensitive neighbor. The most sensitive cells within an ensemble are the wave initiator cells. The Ca²⁺ wave in the cytoplasm is driven by a sensitization wave front in the endoplasmic reticulum. The wave velocity is proportional to the cellular sensitivity and to the strength of the coupling. The waves can form different patterns including circular rings and spirals. The observed pattern depends on the strength of noise, gap junctional permeability and the connectivity probability between neighboring cells. Our simulations reveal that one highly sensitive region gradually takes the lead within the entire noisy system by generating directed circular phase waves originating from this region.

The calcium ion (Ca²⁺), a universal signaling molecule, is widely recognized to play a fundamental role in the regulation of various biological processes. Agonist–evoked Ca²⁺ signals often manifest as rhythmic changes in the cytosolic free Ca²⁺ concentration (c_cyt) called Ca²⁺ oscillations. Stimuli intensity was found to be proportional to the oscillation frequency and the evoked down-steam cellular response. Stochastic receptor expression in individual cells in a cell population inevitably leads to individually different oscillation frequencies and individually different Ca²⁺-related cellular responses. However, in many organs, the neighboring cells have to overcome their individually different sensitivity and produce a synchronized response. Gap junctions are integral membrane structures that enable the direct cytoplasmic exchange of Ca²⁺ ions and InsP₃ molecules between neighboring cells. By simulations, we were able to demonstrate how the strength of intercellular gap junctional coupling in relation to stimulus intensity can modify the spatiotemporal patterns of Ca²⁺ signals and harmonize the Ca²⁺-related cellular responses via synchronization of oscillation frequency. We demonstrate that the most sensitive cells are the wave initiator cells and that a highly sensitive region plays an important role in the determination of the Ca²⁺ phase wave direction. This sensitive region will then also progressively determine the global behavior of the entire system.