  In pancreatic $\beta$-cells, calcium oscillation signal is the core part of
glucose-stimulated insulin secretion. Intracellular calcium concentration
oscillates in response to the intake of glucose, which triggers the exocytosis
of insulin secretory granules. ATP plays a crucial part in this process. ATP
increases as the result of glucose intake, then ATP binds to ATP-sensitive
$K^+$ channels ($K_{ATP}$), depolarizes the cell and triggers calcium
oscillation, while the ion pumps on the cell membrane consumes the free energy
form ATP hydrolysis. Based on Betram et. al. 2004 model, we construct a kinetic
models to analyze the thermodynamic characteristics of this system, to reveal
how the ATP hydrolysis free energy affects the calcium oscillation. Our results
suggest that bifurcation point is sensitive to both the free energy level and
cellular ATP level, and the insufficient ATP energy supply would cause
dysfunction of calcium oscillation.
