Model Of Calcium Oscillations Research Articles

On the qualitative behavior and numerical analysis of a calcium oscillation model through Atangana–Baleanu operator in Caputo sense

In this research, we regard a miniature significant for many cell types and activates stimulants against external interventions, thereby causing calcium oscillation. We stage the dependent variables regarding the Atangana–Balenau derivative operator in terms of Caputo so that the model can be physically interpreted better. The study investigates the existence and uniqueness of solutions using fixed point theory, stability in terms of Hyers–Ulam, positivity and numerical solutions. In addition, spacetime, phase diagrams and three-dimensional graphic simulations are presented due to the Lagrange interpolation technique for parameter selections that match the existence and uniqueness condition of the solution.

Identification of [formula omitted] oscillations with the 0-1 test for periodic and chaotic time-series: One- and two-parameter bifurcations

This paper examines the oscillatory responses (periodic and chaotic) of a biosystem store model for bursting and complex Ca2+ oscillations in which three compartments have been taken into consideration: the cytosol, endoplasmic reticulum (ER) and mitochondria. The oscillatory model is used to examine the reliability of the 0-1 test for chaos in the bifurcation analysis of continuous signals obtained when the frequencies of oscillatory responses vary significantly with a relatively small changes of the bifurcation parameters. The illustrative examples in both the one- and two-parameter cases are designed to show that for a periodic time-series the test’s reliability may be questioned when a periodic series is classified as a chaotic one - the ’false-positive’ case. To prevent the incorrect result an additional computational work is needed to examine the frequency spectrum of the periodic time-series. The illustrative examples utilize an autonomous dynamical model of cytosolic calcium oscillations with three dynamical variables and sixteen parameters. The dynamical model is such that the frequency of oscillations may change by the factor of about 200, when a certain dynamical system’s parameter changes from its minimum to maximum values, making selection of the parameters in the 0-1 test extremely difficult. The extra computational work improves the test’s reliability and eliminates the ’false-positive’ outcomes of the test. The paper is focused on the computational aspects of the 0-1 test for periodic and chaotic oscillations rather than on the properties of the store model for bursting and complex Ca2+ oscillations.

Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams.

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the signal regulation.

Chaos in a calcium oscillation model via Atangana-Baleanu operator with strong memory

Calcium oscillations have a significant importance in gesturing messengers for a wide variety of cells because cells exhibit calcium oscillations in response to external stimulation. The significance of the calcium oscillations vividly attract researchers, this is due to the empirical, numerical or experimental analyses on calcium oscillations and the lack of modern fractional techniques. An analytic study of chaotic behavior in the calcium oscillation model is carried out through fractional and non-fractional techniques. The governing non-linear differential equations of calcium oscillations have been established through newly defined Atangana-Baleanu fractional operators. In order to analyze the dynamics of the fractional calcium oscillation model, powerful techniques are applied to the governing non-linear fractional differential equations namely the (Laplace transform and modified homotopy perturbation transform method MHATM). The newly defined Atangana-Baleanu fractional approach is examined for the parameters that show periodic doubling and route to chaos. Finally, several similarities and differences have been underlined due to the fractional-order effect for the calcium oscillations with chaotic region at different scale.

Limit Cycles in a Model of Olfactory Sensory Neurons

We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.

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 τh (representing relaxation time) indicates that with the increase of τh, 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 Vm1 (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 Vm1, calcium oscillations region shrinks gradually. Vm1 and μ both play a key role in regulating cell apoptosis. Only when Vm1 and μ are high enough can cells enter into programmed cell death and the higher Vm1 is, the lower the stimulus intensity required by cell apoptosis is.

Important role of MAMs in bifurcation and coherence resonance of calcium oscillations

MAM (mitochondria-associated endoplasmic reticulum membrane complex) is kind of complexes formed between the endoplasmic reticulum (ER) and mitochondria, and it plays an important role in calcium signal transduction of cell. Coherence resonance is cooperative effect of noise and nonlinear system. In the field of life science, coherence resonance reflects the rhythm of life, so studying it by using the theory and method of nonlinear stochastic dynamics has very important biological significance. On the basis of the calcium oscillation model considering the role of MAMs proposed by Szopa et al., and taking two important indexes - the maximal permeability of the Ca2+ channels in the ER membrane (kch) and the maximal permeability of MCU (mitochondrial calcium uniporter) at MAMs (kMAM) as the analysis parameters, this paper further studies the bifurcation characteristic of the system, including two-parameter bifurcation and one-parameter bifurcation, and theoretically expounds the reasons for the generation and disappearance of calcium oscillations. Then, on the basis of bifurcation analysis, we focus on the coherence resonance characteristic of calcium signals and its physiological significance when analysis parameters are affected by environmental noise.

Bifurcation Analysis and Transition Mechanism in a Modified Model of Ca2+ Oscillations**Supported by the National Natural Science Foundation of China under Grant Nos 11372017, 11572084 and 11472061, the Natural Science Foundation for the Higher Education Institutions of Anhui Province under Grant No KJ2016SD54, the Fundamental Research Funds for the Central Universities, and the Distinguished Young Professor Program of Donghua University

Some new elements are introduced into a mathematical model of intracellular calcium oscillations, which make it particularly suitable for the study of bifurcation. In addition to generating regular oscillations, such a modified model can be used to reproduce the burst discharges similar to those recorded in experiments and to describe two new types of oscillatory phenomena. By means of a fast/slow dynamical analysis, we explore the bifurcation and transition mechanisms associated with two types of bursters due to changes in the interaction of two slow variables with different timescales.

A computational model of intracellular calcium oscillations in urinary bladder smooth muscle cells.

This paper presents a computational model for calcium (Ca2+) oscillations in detrusor smooth muscle (DSM) cells. The proposed model simulates the temporal profile of oscillations by incorporating various cellular and subcellular components. The cellular components include calcium influx via membrane and the plasma membrane calcium ATPase (PMCA) pump. The subcellular components include ryanodine receptors (RyRs), inositol 1, 4, 5 trisphosphate receptors (IP3Rs) and the sarco/endoplasmic reticulum calcium ATPase (SERCA) pump. The use of both cellular and subcellular components provides a better estimation of the origin and factors affecting these oscillations. Moreover, our work correlates these computational findings with associated physiology of the smooth muscle cell that aids our understanding of intracellular calcium oscillations and its inception in DSM. A deeper insight into calcium signalling in DSM cells is expected to provide a firmer basis for understanding the mechanical contractile activity.

On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

ABSTRACTA canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.

Experimental Design for Stochastic Models of Nonlinear Signaling Pathways Using an Interval-Wise Linear Noise Approximation and State Estimation.

BackgroundComputational modeling is a key technique for analyzing models in systems biology. There are well established methods for the estimation of the kinetic parameters in models of ordinary differential equations (ODE). Experimental design techniques aim at devising experiments that maximize the information encoded in the data. For ODE models there are well established approaches for experimental design and even software tools. However, data from single cell experiments on signaling pathways in systems biology often shows intrinsic stochastic effects prompting the development of specialized methods. While simulation methods have been developed for decades and parameter estimation has been targeted for the last years, only very few articles focus on experimental design for stochastic models.MethodsThe Fisher information matrix is the central measure for experimental design as it evaluates the information an experiment provides for parameter estimation. This article suggest an approach to calculate a Fisher information matrix for models containing intrinsic stochasticity and high nonlinearity. The approach makes use of a recently suggested multiple shooting for stochastic systems (MSS) objective function. The Fisher information matrix is calculated by evaluating pseudo data with the MSS technique.ResultsThe performance of the approach is evaluated with simulation studies on an Immigration-Death, a Lotka-Volterra, and a Calcium oscillation model. The Calcium oscillation model is a particularly appropriate case study as it contains the challenges inherent to signaling pathways: high nonlinearity, intrinsic stochasticity, a qualitatively different behavior from an ODE solution, and partial observability. The computational speed of the MSS approach for the Fisher information matrix allows for an application in realistic size models.

Fractional Dynamics in Calcium Oscillation Model

The calcium oscillations have many important roles to perform many specific functions ranging from fertilization to cell death. The oscillation mechanisms have been observed in many cell types including cardiac cells, oocytes, and hepatocytes. There are many mathematical models proposed to describe the oscillatory changes of cytosolic calcium concentration in cytosol. Many experiments were observed in various kinds of living cells. Most of the experimental data show simple periodic oscillations. In certain type of cell, there exists the complex periodic bursting behavior. In this paper, we have studied further the fractional chaotic behavior in calcium oscillations model based on experimental study of hepatocytes proposed by Kummer et al. Our aim is to explore fractional-order chaotic pattern in this oscillation model. Numerical calculation of bifurcation parameters is carried out using modified trapezoidal rule for fractional integral. Fractional-order phase space and time series at fractional order are present. Numerical results are characterizing the dynamical behavior at different fractional order. Chaotic behavior of the model can be analyzed from the bifurcation pattern.

Modelling mechanism of calcium oscillations in pancreatic acinar cells.

We present a simple model for calcium oscillations in the pancreatic acinar cells. This model is based on the calcium release from two receptors, inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) through the process of calcium induced calcium release (CICR). In pancreatic acinar cells, when the Ca²⁺ concentration increases, the mitochondria uptake it very fast to restrict Ca(2+) response in the cell. Afterwards, a much slower release of Ca²⁺ from the mitochondria serves as a calcium supply in the cytosol which causes calcium oscillations. In this paper we discuss a possible mechanism for calcium oscillations based on the interplay among the three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosol. Our model predicts that calcium shuttling between ER and mitochondria is a pacemaker role in the generation of Ca²⁺ oscillations. We also consider the calcium dependent production and degradation of (1,4,5) inositol-trisphosphate (IP3), which is a key source of intracellular calcium oscillations in pancreatic acinar cells. In this study we are able to predict the different patterns of calcium oscillations in the cell from sinusoidal to raised-baseline, high frequency and low-frequency baseline spiking.

Wave instability of intercellular Ca2+ oscillations

Modulational instability is exclusively addressed in a minimal model for calcium oscillations in cells. The cells are considered to be coupled through paracrine signaling. The endoplasmic recticulum and cytosolic equations are reduced to a single differential-difference amplitude equation. The linear stability analysis of a plane wave is performed on the latter and the paracrine coupling parameter is shown to deeply influence the instability features. Our analytical expectations are confirmed by numerical simulations, as instability regions give rise to unstable wave patterns. We also discuss the possibility of perfect intercellular communication via the activation of modulational instability.

Linear noise approximation for stochastic oscillations of intracellular calcium

A stochastic model of intracellular calcium oscillations is analytically studied. The governing master equation is expanded under the linear noise approximation and a closed prediction for the power spectrum of fluctuations analytically derived. A peak in the obtained power spectrum profile signals the presence of stochastic, noise induced oscillations which extend also outside the region where a deterministic limit cycle is predicted to occur.

Different Types of Periodic Activities in a Calcium Oscillation Model

In this paper, a model proposed by Marhl et al. is considered to investigate the effect of several parameters on the calcium bursting oscillation behavior. Different types of bursting are presented . Fast-slow burster analysis and first return map are used to explain the mechanism of the four types of bursting. The results are instructive for understanding the role of these parameters played in complex dynamics in the Marhl-Haberichter calcium oscillation model.

HOPF BIFURCATION IN A CALCIUM OSCILLATION MODEL AND ITS CONTROL: FREQUENCY DOMAIN APPROACH

The Hopf bifurcation in a calcium oscillation model is theoretically analyzed by Hopf bifurcation theory in frequency domain. Approximation expressions for frequencies and amplitudes of periodic orbits arising from Hopf bifurcation are provided by using second-order harmonic balance method. In addition, a new method is proposed to control the amplitudes of the periodic orbits. Numerical simulations show the effectiveness of the method for suppressing periodic oscillations.

Hopf Bifurcation and Numerical Simulation in a Calcium Oscillation Model

Calcium oscillations play a very important role in providing the intracellular signaling, and many mathematical models have been proposed to describe calcium oscillations. The Shen-Larter model presented here is based on calcium-induced calcium release (CICR) and the inositol trisphosphate cross-coupling (ICC). Nonlinear dynamics of this model is investigated by using the centre manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with different parameter values. The results show that the appearance and disappearance of calcium oscillations are due to subcritical Hopf bifurcation of equilibria. The numerical simulations are performed in order to illustrate the correctness of our theoretical analysis, including the bifurcation diagram of fixed points, the phase diagram of the system in two dimensional space and time series.

Dynamical Analysis of a Calcium Oscillation Model in Non-Excitable Cells

The Borghans-Dupont model of calcium oscillations based on both the calcium-induced calcium release and calcium-activated inositol trisphosphate concentration degradation is considered. Dynamical effect of the stimulation level on the calcium oscillation behavior is studied. The qualitative theory of differential equations is used to explain the mechanism of these oscillations. We investigate the existence, types, stability and bifurcations of the equilibria by applying the centre manifold theorem, stability theory and bifurcation theory and prove that oscillations are due to supercritical Hopf bifurcation. Finally, we perform numerical simulations, including time courses, phase portraits and bifurcation diagram, to validate the correctness and the effectiveness of our theoretical analysis. These results may be instructive for understanding the role of the stimulation level played in complex dynamics in this model.

Complex Dynamics in the Kummer-Olsen Model of Calcium Oscillations

Oscillations of cytosolic calcium concentration, known as calcium oscillations, play a vital role in providing the intracellular signalling. These oscillations are explained with a model based on calcium-induced calcium release (CICR). The nonlinear dynamics of the Kummer-Olsen calcium oscillation model is discussed by using the center manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with parameter value. It is concluded that the appearance and disappearance of calcium oscillations in this system is due to supercritical Hopf bifurcation of equilibria. Finally, numerical simulations are carried out to support the theoretical analysis of the research. By combining the existing numerical results with the theoretical analysis results in this paper, a complete description of the dynamics of the Kummer-Olsen calcium oscillation model has now been obtained.