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

Identification of Ca2+ 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, Ls 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 Ca2+ signal regulation.

Effects of store-operated and receptor-operated calcium channels on synchronization of calcium oscillations in astrocytes

Intercellular calcium signaling allows cells to communicate with each other and to interact with adjacent cells. Gap junction is the most common and important way for cellular communication. Recently, mathematical models have been widely used to gain a precise and quantitative understanding of the dynamics of intracellular calcium ions (Ca^2＋). In this paper, we establish a mathematical model considering the gap junction permeable to Ca^2＋ and to IP₃ for describing the calcium oscillations in coupled astrocytes. Store-operated calcium entry (SOCE) is viewed as the main process which controls the non-excitable cells, hence, we focus on the effect of store-operated calcium channel (SOCC) and receptor-operated calcium channel (ROCC) on the intercellular synchronization, respectively. By employing bifurcation analysis on this model, the dynamic behaviors of the coupled system with different physiological state cells is obtained with changes in the maximum capacity of the SOCC and the ROCC. The synchronization boundaries for different conditions are gained in the two parameters space of the channel parameters and the coupling strength. The results suggest that the variation of the maximum flow for different calcium channels determines the stable oscillations of the coupled system, as well as for the frequency and amplitude of oscillations. The SOCC has an expected effect on the change of the oscillatory interval while the ROCC demonstrated the influence on the amplitude modulation. Furthermore, the coupling strength and channel parameters could induce 1:1 locking of intercellular Ca^2＋ oscillations and the synchronization region like Arnol'd tongue is found.

4-Hexylresorcinol Exhibits Different Characteristics to Estrogen

4-Hexylresorcinol (4HR) has been used as a food additive and antiseptic. The aim of this study was to evaluate whether the application of 4HR in breast cancer cells and ovariectomized rats showed estrogen-like effects. MCF-7 and SK-BR-3 cells were treated by solvent, 1–100 µM bisphenol-A (BPA), or 1–100 µM 4HR, respectively. 3-(4, 5-Dimethylthiazole-2-yl)- 2,5-diphenyltetrazolium bromide (MTT) assay and Western blot for extracellular signal-regulated kinase-1/-2 (ERK1/2), phosphorylated ERK1/2 (p-ERK1/2), estrogen receptor-α (ERα), and ERβ were done. As an in vivo study, ovariectomized rats (n = 15) received solvent, 125 mg/kg of 4HR, or 10 µg of 17-β estradiol via daily subcutaneous injection for 7 days. Blood samples were obtained for evaluation of prolactin levels. Pituitary glands and uteruses were biopsied for histological evaluation and Western blot analysis. Compared with the control group, the application of 4HR decreased the proliferation of MCF-7 and SK-BR-3 cells, while the application of BPA increased (p < 0.05). The application of BPA increased the expression of ERα, ERβ, and p-ERK1/2, but 4HR did not change the expression of ERα, ERβ, or p-ERK1/2 in MCF-7 cells. In an animal model, the 4HR group showed similar levels of ERα, ERβ, and prolactin expression in the pituitary gland compared to the solvent only group, while the estradiol group showed higher levels. Serum prolactin levels were similar between the 4HR and solvent only groups. Taken together, 1–100 µM 4HR did not show BPA-like behavior in MCF-7 cells, and 125 mg/kg of 4HR daily subcutaneous injection for 7 days did not demonstrate estradiol-like effects in ovariectomized rats. Collectively, 4HR has no estrogen-like effects on both ERα-positive cells and estrogen-deficient rat models.

Modeling the role of endoplasmic reticulum-mitochondria microdomains in calcium dynamics

Upon inositol trisphosphate (IP₃) stimulation of non-excitable cells, including vascular endothelial cells, calcium (Ca²⁺) shuttling between the endoplasmic reticulum (ER) and mitochondria, facilitated by complexes called Mitochondria-Associated ER Membranes (MAMs), is known to play an important role in the occurrence of cytosolic Ca²⁺ concentration ([Ca²⁺]_Cyt) oscillations. A mathematical compartmental closed-cell model of Ca²⁺ dynamics was developed that accounts for ER-mitochondria Ca²⁺ microdomains as the µd compartment (besides the cytosol, ER and mitochondria), Ca²⁺ influx to/efflux from each compartment and Ca²⁺ buffering. Varying the distribution of functional receptors in MAMs vs. the rest of ER/mitochondrial membranes, a parameter called the channel connectivity coefficient (to the µd), allowed for generation of [Ca²⁺]_Cytoscillations driven by distinct mechanisms at various levels of IP₃ stimulation. Oscillations could be initiated by the transient opening of IP₃ receptors facing either the cytosol or the µd, and subsequent refilling of the respective compartment by Ca²⁺ efflux from the ER and/or the mitochondria. Only under conditions where the µd became the oscillation-driving compartment, silencing the Mitochondrial Ca²⁺ Uniporter led to oscillation inhibition. Thus, the model predicts that alternative mechanisms can yield [Ca²⁺]_Cyt oscillations in non-excitable cells, and, under certain conditions, the ER-mitochondria µd can play a regulatory role.