Model of Calcium Dynamics Regulating \(IP_{3}\), ATP and Insulin Production in a Pancreatic \(\beta\)-Cell

The calcium signals regulate the production and secretion of many signaling molecules like inositol trisphosphate (\(IP_{3}\)) and adenosine triphosphate (ATP) in various cells including pancreatic \(\beta\)-cells. The calcium signaling mechanisms regulating \(IP_{3}\), ATP and insulin responsible for various functions of \(\beta\)-cells are still not well understood. Any disturbance in these mechanisms can alter the functions of \(\beta\)-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 \(IP_{3}\), ATP-production and insulin secretion with initial and boundary conditions. The model incorporates the temporal dependence of \(IP_{3}\)-production and degradation, ATP production and insulin secretion on calcium dynamics in a \(\beta\)-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 \(IP_{3}\), ATP and insulin levels in a \(\beta\)-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 \(IP_{3}\), 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 \(IP_3\), ATP and insulin levels in pancreatic cells, which can be helpful for devising protocols for diagnosis and treatment of various metabolic diseases.