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.