# Modeling the Effects of Calcium Overload on Mitochondrial Ultrastructual Remodeling
## 1 Introduction
A layer of regulation above this includes proteases and chaperones that process these structural proteins, such as OMA1 [42], YME1L [42], and prohibitions [43; 44], which are also intricately involved in a signaling network that controls cristae morphology. However, the precise biophysical mechanisms that govern the shape of the inner membrane, including inner membrane fission and cristae structure, remain elusive. A few modeling studies have explored the potential role of mitochondrial proteins in the inner membrane architecture and metabolic function [45; 46; 47; 48; 49; 50; 51; 52; 53; 54]. These studies investigate a range of phenomena, including how proteins and effectors influence inner membrane curvature and intramatrix diffusion anomalies. However, none have looked at the dynamics of the IMS and matrix volumes under healthy and diseased conditions. In addition, there are no computational studies focusing on how the inner membrane morphology influences ATP synthesis rates. Therefore, we propose a novel computational modeling approach capable of predicting the intermembrane and matrix space geometry under physiological and pathophysiological conditions. The spontaneous pattern formation and evolution of a two-component system can be described by a number of models, including the Canham-Helfrich curvature functional model, the Ginzburg-Landau theory, and the Cahn-Hilliard (CH) free energy [55; 56; 57; 58]. Among them, the Canham-Helfrich model is typically used to describe cell membrane pattern formation in a water-based environment, driven by the curvature energy per unit area of the closed lipid bilayer, osmotic pressure, and surface tension [59]. This model captures cell membrane patterns without the molecular dynamics of the lipid bilayer. The Ginzburg-Landau theory describes the phase transition of type-I superconductors, the superconducting state, and the normal state without examining their microscopic electronic and magnetic properties. In the setting of a complex vector bundle over a compact Riemannian manifold, the Ginzburg-Landau theory interconnects differential geometry, quantum field theory, Yang-Mills-Higgs equations, and string theory, among others [60,61]. The CH model is widely used in material science, chemistry, and biology to describe the phase separation of a binary mixture [62]. It is the simplest model among the aforementioned theories. The CH model can be derived from the classic conservation law and free energy minimization, with the free energy function determining the final composition of the binary system [63,64]. The pattern and morphology of the binary system is further determined by the domain size and geometric shape [65,66]. The CH model can be modified to describe the nucleation dynamics and process of a specific species in the mixture [67,68]. Additionally, multicomponent CH equations have been proposed to account for the phase separation of a system consisting of more than two species [69,70]. These variations of the CH model enable researchers to develop CH-based theories and algorithms to understand the chemistry, physics, biology, and material sciences in a complex system that is intractable with detailed microscopic descriptions. For this study, we analyzed a set of cryo-electron tomograms of healthy and calcium-overloaded mitochondria. These data show that the mitochondrial ultrastructural morphology undergoes dramatic changes in the presence of mitochondrial calcium phosphate granules. However, it is unclear how mitochondrial calcium phosphate granules regulate the mitochondrial ultrastructure. The tomographic data, combined with the respiratory data presented herein, support the concept that such morphological changes lead to a significant reduction in the mitochondrial ATP production rate. Based on these data, we speculate that the mitochondrial ATP production rate correlates with the types and numbers of cristae and cristae junctions.