# Modeling the Effects of Calcium Overload on Mitochondrial Ultrastructual Remodeling
## 3 Results and Discussion
It is shown that in the simulation, the cristae could be observed, and the inner membrane and the fluid matrix could be separated. In the absence of calcium, the inner membrane and the fluid matrix were initially randomly and uniformly distributed in the circle domain. In Figure 5, a larger circle domain with radius of five is demonstrated. The parameter \(\gamma\) is one for the transition region length, which is indicated by the white, rim-like features in Figure 5. The red domain of Figure 5 is the fluid matrix, while the blue domain is the inner membrane part, which includes the cristae lumen. Compared with Figure 4, thinner cristae can be observed in Figure 5. This was caused by the high-frequency phase interactions introduced by imposing a larger domain. In the comparison with the experimental image, a similar pattern can be observed in Figure 5, in which cristae are randomly and uniformly distributed within the domain. Implementation of the CH model on a mitochondrial structure can help us understand a plausible mechanism that explains the dramatic and titratable decrease in mitochondrial ATP production as a function of calcium overloading. Figure 6 demonstrates the simulations of calcium load cases. In case one, a single calcium phosphate granule was loaded in the first quadrant. In the second case, and additional calcium phosphate granule was  
Figure 4: Calcium-free mitochondria simulation by a circle domain with a radius of four. The time step \(\Delta t=0.001\) for (**A**) \(t_{750}\), (**B**) \(t_{1250}\), (**C**) \(t_{1750}\), and (**D**) \(t_{2250}\). Merging and splitting of the matrix and intermembrane spaces are observed as the simulation time is lengthened. Figure 5: Calcium-free mitochondria simulation by a circle domain with a radius of five. The time step \(\Delta t=0.001\) for (**A**) \(t_{750}\), (**B**) \(t_{1250}\) (**C**) \(t_{1750}\), and (**D**) \(t_{2250}\). As shown in Figure 4, merging and splitting of the phase spaces are observed as the simulation time increases. loaded in the fourth quadrant. In each panel, the orange objects indicate the location of the calcium phosphate granules. In a fascinating similarity with the experimental observations, the simulations revealed a dramatic reorganization of the cristae membrane because of the presence of calcium phosphate granules. In addition, the matrix space around the granules dramatically expanded. In the simulations, these effects were a result of an energy minimization problem. The presence of these granules near cristae membranes acted as a repulsive force which led to matrix fluid filling the vacated space. Figure 7 shows the extension of the 2D simulations of the effects of calcium phosphate granules on the inner membrane topology to 3D. In this figure, colored surface plots as well as inside contour plots are presented. It can be seen from the inside contour plots in Figure 7B,D that the granule's location was at the center of the concave down domain. As the time increased from \(t_{500}\) to \(t_{1500}\), the concave down domain enlarged as a direct result of the presence of the calcium phosphate granule. Changing the geometric boundary conditions from circular and spheroid to elliptical and ellipsoid resulted in a similar inner membrane topology. These results are shown in Figure 2.0 Simulations of a 2D elliptical mitochondrion with a major axis of 10 and minor axis of 8 are given in Figure 8A,B. In Figure 8C,D, the 3D simulation results of an ellipsoid mitochondrion are presented. A similar pattern can be observed to that in the experimental image, which substantiates the use of the CH model and demonstrates that it can be used to characterize the internal ultrastructural features of mitochondrion with more irregular boundary conditions. Figure 2.1 The effect of calcium phosphate granules on the inner membrane morphology. (**A–D**) Simulations on the effect of a single calcium phosphate granule on the mitochondrial cristae network. The Cahn–Hilliard (CH) equation was solved on a circle domain with a radius of five.