Roles of four feedback loops in mitochondrial permeability transition pore opening induced by ${\mathrm{Ca}}^{2+}$ and reactive oxygen species

Transient or sustained permeability transition pore (PTP) opening is important in normal physiology or cell death, respectively. These are closely linked to ${\mathrm{Ca}}^{2+}$ and reactive oxygen species (ROS). The entry of ${\mathrm{Ca}}^{2+}$ into mitochondria regulates ROS production, and both ${\mathrm{Ca}}^{2+}$ and ROS trigger PTP opening. In addition to this feedforward loop, there exist four feedback loops in the ${\mathrm{Ca}}^{2+}$-ROS-PTP system. ROS promotes ${\mathrm{Ca}}^{2+}$ entering (F1) and induces further ROS generation (F2), forming two positive feedback loops. PTP opening results in the efflux of ${\mathrm{Ca}}^{2+}$ (F3) and ROS (F4) from the mitochondria, forming two negative feedback loops. Owing to these complexities, we construct a mathematical model to dissect the roles of these feedback loops in the dynamics of PTP opening. The qualitative agreement between simulation results and recent experimental observations supports our hypothesis that under physiological conditions the PTP opens in an oscillatory state, while under pathological conditions it opens in a high steady state. We clarify that the negative feedback loops are responsible for producing oscillations, wherein F3 plays a more prominent role than F4; whereas the positive feedback loops are beneficial for maintaining oscillation robustness, wherein F1 has a more dominant role than F2. Furthermore, we manifest that the proper increase in negative feedback strength or decrease in positive feedback strength not only facilitates the occurrence of oscillations and thus protects the system against a high steady state, but also assists in lowering the oscillation peak. This study may provide potential therapeutic strategies in treating neurodegenerative diseases due to PTP dysfunction.

Figure 1

Working hypothesis and schematic diagram. (a) Under physiological condition, a small amount of ${\mathrm{Ca}}^{2+}$ facilitates transient opening of PTP; while under pathological condition, large amount of ${\mathrm{Ca}}^{2+}$ promotes persistent opening of PTP. The transient opening mode and persistent opening mode correspond to the oscillatory state and high steady state, respectively. (b) The double-headed arrow for ${\mathrm{Ca}}^{2+}$ indicates the influx of ${\mathrm{Ca}}^{2+}$ into the mitochondria, for ROS indicates the basal synthesis of ROS. The dashed-line arrow for ${\mathrm{Ca}}^{2+}$ represents the efflux of ${\mathrm{Ca}}^{2+}$ from the mitochondria, for ROS represents the degradation of ROS. The regulatory effect of ${\mathrm{Ca}}^{2+}$ on ROS as well as the combined regulatory effect of ${\mathrm{Ca}}^{2+}$ and ROS on PTP are represented by the swallowtail arrow. Besides the regulatory effect, PTP opening occurs with a basal rate. PTP can switch between open state and closed state. The dotted-line arrows denote the four feedback loops. F1 and F2 are positive feedback loops, while F3 and F4 are negative feedback loops.

Figure 2

(a)–(l) Time course of 12 modes. For the schematic diagrams (left), solid lines indicate regulatory interactions that are present in all of the topologies, while dashed lines indicate the ones that maybe present or absent, depending on topology. Representative examples of time course of three variables (right). C, ${\mathrm{Ca}}^{2+}$; R, ROS; P, the fraction of open PTP. The unit is arbitrary.

Figure 3

Robustness properties of 12 modes. (a) Percentage of parameter sets that yielded sustained oscillations. (b) and (c) Probability of parameter sets that yielded sustained oscillations thanks to the presence of specified feedback loop(s). (d) The distribution of peak values of ten modes. Violin plots show mean, 25th and 75th percentiles; sample sizes are represented by the widths.

Figure 4

Bifurcation diagram of $P$ as a function of $I_1$. The intersection of solid and dashed lines is a bifurcation point. The solid and dashed line indicate, respectively, stable and unstable solutions of the ordinary differential equations corresponding to mode F3412. Each unstable solution is surrounded by a stable oscillation, the peak and trough of which are indicated by a green and brown filled circle, respectively. The unstable oscillations are represented by empty circles. The standard value of $I_1$ is marked by an asterisk. $I_1$, basal rate of ${\mathrm{Ca}}^{2+}$ uptake into mitochondria.

Figure 5

Bifurcation diagrams of P as a function of ${\mathrm{I}}_1$ under different value of ${\mathrm{F}}_{\mathrm{X}}$. ${\mathrm{F}}_{\mathrm{X}}$ ($\mathrm{X}=1$, 2, 3, 4), maximal value of the corresponding feedback term, can represent the strength of feedback loop of mode F3412. In each subfigure, the black bifurcation diagram is the same as in Fig. 4 and is shown for facilitating comparison; the red or blue bifurcation diagram, respectively, corresponds to the situation when ${\mathrm{F}}_{\mathrm{X}}$ is increased or decreased by 10%. The graphical notations are similar as in Fig. 4. ${\mathrm{I}}_1$, basal rate of ${\mathrm{Ca}}^{2+}$ uptake into mitochondria.

Figure 6

Bifurcation diagrams of P as a function of ${\mathrm{F}}_{\mathrm{X}}$. The graphical notations are the same as in Fig. 4. ${\mathrm{F}}_{\mathrm{X}}$ ($\mathrm{X}=1$, 2, 3, 4), maximal value of the corresponding feedback term, can represent the strength of feedback loop of mode F3412. The detailed meaning and standard value (marked by an asterisk) of ${\mathrm{F}}_{\mathrm{X}}$ are listed in Table S1.

Figure 7

Two-parameter bifurcation diagrams of the peak of P as a function of feedback strength, ${\mathrm{F}}_{\mathrm{X}}$ ($\mathrm{X}=1$, 2, 3, 4). Oscillations occur within the color region. In the white region, no oscillations occur. The color bar indicates the peak values of the oscillations derived from the corresponding parameter combinations of mode F3412.

Figure 8

Parameter sensitivity analysis. For each parameter, the circles depict the range of relative variation—expressed as a fold change of its standard value of mode F3412— in which oscillations are maintained.