• The model is divided into two sub domains: the apical cytosolic sub domain, and the mitochondrial sub domain. However, these two sub domains are connected with the Ca2+. • The proposed model is Class II model. It includes the dynamics of IP3, we take into account the Ca2+ modulation of IP3 generation through PLC, and Ca2+ activated IP3 removal by IP3K. • The biphasic regulation of mitochondrial Ca2+ uptake by apical Ca2+ concentration, and the rapid mode of Ca2+ uptake through mitochondria are considered. • In this model the three types of mechanism by Ca2+ have been well considered. Firstly, the time dependent gating of IPR by Ca2+. Secondly, the Ca2+ enhances the production of IP3. Subsequently, due to the 3- kinase stimulated by Ca2+, degradation of IP3. Thirdly, the gating of uniporter by Ca2+: i.e. Ca2+ activated CU uptake, and subsequently it inactivates the uniporter for further up taking Ca2+. Thus, a [Ca2+] concentration is required for the activation and the inactivation of CU. • How these two sub domain: apical sub domain and the mitochondrial sub domain combine together into a well- mixed model. • The bifurcation analysis is performed to identify the oscillatory structures depending on the different agonist strength to produce complex dynamics in the apical region of the cell. The complex bifurcation points like hopf, period doubling, torus, saddle node bifurcation are studied. The pancreatic acinar cells display wide varieties of Ca 2+ oscillations within the cell. The study of these calcium signals/oscillations are of great interest because they are physiologically important. The Ca 2+ release from the endoplasmic reticulum via inositol trisphosphate receptors (IPR) and the ryanodine receptors (RyR) creates a high cytosolic Ca 2+ signal in the acinar cell. The pancreatic acinar cells are highly polarized cells, and the Ca 2+ response is quite extensive in the apical (peri-granular) region than the basal region of the cell. The Ca 2+ signals which originated in the apical region control the exocytosis process and triggered peri-granular mitochondria (Mito) to uptake Ca 2+ . The peri-granular mitochondria play a critical role in confining cytosolic Ca 2+ elevation by acting as a Ca 2+ buffer barrier. They also participate in shaping the Ca 2+ response by restricting the Ca 2+ signals into the granular region of the acinar cells. They sequestered Ca 2+ very rapidly. Then they release Ca 2+ very slowly to provide Ca 2+ supply to cause Ca 2+ oscillations in the apical region. Thus, herein we proposed a Class II compartmental mathematical model which systematically simulates the complex Ca 2+ oscillations in the apical region, and the peri-granular mitochondria matrix of the pancreatic acinar cells. The model assumes the functional coupling between the IPR and the RyR through the mechanism of calcium induced calcium release (CICR). In the proposed compartmental model of Ca 2+ dynamics, the Ca 2+ dependent production and degradations of (1,4,5) inositol-trisphosphate (IP 3 ) along with the apical parameters, apical endoplasmic reticulum (ER), and the peri-granular mitochondria are considered. The bifurcation analysis of the model is performed which explores the intricate dynamic behaviour of the apical region of the acinar cells, and hence derive the complex Ca 2+ oscillations in both the apical region and the peri-granular mitochondria. The model predicts the different patterns of Ca 2+ oscillation from the regular to sinusoidal to raised-baseline to sustained spikes in the apical region of the acinar cells.
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