Systematical bifurcation analysis of an intracellular calcium oscillation model

A simple mechanochemical model for calcium signalling in embryonic epithelial cells

Calcium signalling is one of the most important mechanisms of information propagation in the body. In embryogenesis the interplay between calcium signalling and mechanical forces is critical to the healthy development of an embryo but poorly understood. Several types of embryonic cells exhibit calcium-induced contractions and many experiments indicate that calcium signals and contractions are coupled via a two-way mechanochemical feedback mechanism. We present a new analysis of experimental data that supports the existence of this coupling during apical constriction. We then propose a simple mechanochemical model, building on early models that couple calcium dynamics to the cell mechanics and we replace the hypothetical bistable calcium release with modern, experimentally validated calcium dynamics. We assume that the cell is a linear, viscoelastic material and we model the calcium-induced contraction stress with a Hill function, i.e. saturating at high calcium levels. We also express, for the first time, the "stretch-activation" calcium flux in the early mechanochemical models as a bottom-up contribution from stretch-sensitive calcium channels on the cell membrane. We reduce the model to three ordinary differential equations and analyse its bifurcation structure semi-analytically as two bifurcation parameters vary-the [Formula: see text] concentration, and the "strength" of stretch activation, [Formula: see text]. The calcium system ([Formula: see text], no mechanics) exhibits relaxation oscillations for a certain range of [Formula: see text] values. As [Formula: see text] is increased the range of [Formula: see text] values decreases and oscillations eventually vanish at a sufficiently high value of [Formula: see text]. This result agrees with experimental evidence in embryonic cells which also links the loss of calcium oscillations to embryo abnormalities. Furthermore, as [Formula: see text] is increased the oscillation amplitude decreases but the frequency increases. Finally, we also identify the parameter range for oscillations as the mechanical responsiveness factor of the cytosol increases. This work addresses a very important and not well studied question regarding the coupling between chemical and mechanical signalling in embryogenesis.

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 τ_h (representing relaxation time) indicates that with the increase of τ_h, 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 V_m1 (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 V_m1, calcium oscillations region shrinks gradually. V_m1 and μ both play a key role in regulating cell apoptosis. Only when V_m1 and μ are high enough can cells enter into programmed cell death and the higher V_m1 is, the lower the stimulus intensity required by cell apoptosis is.