# Modeling the Effects of Calcium Overload on Mitochondrial Ultrastructual Remodeling
## 1 Introduction  
Mitochondria are small organelles that provide the bulk of the energy requirements for nearly all cells in most eukaryotic organisms. They convert chemical energy in foodstuffs into a chemical potential called the ATP hydrolysis potential. Cells use this potential to synthesize proteins, power mechanical motion, sustain homeostasis, and perform other life-supporting functions. In the heart, the essential importance of mitochondria is evidenced by the fact that they constitute up to 30% of the total cardiomyocyte volume. For example, in the absence of functional mitochondria, the heart will exhaust all of its ATP reserves in less than a minute [1], so even brief interruptions in mitochondrial ATP production have devastating consequences on energy homeostasis. Such interruptions in ATP production are caused by a myriad of diseases and conditions, with calcium overload being one of the more prominent causes. While low matrix-free calcium (\(<5\) uM) activates matrix dehydrogenases and improves oxidative phosphorylation [2; 3; 4], calcium overloaddecreases ATP synthesis in a variety of ways [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. In extreme cases of calcium overload, there is a complete loss of mitochondrial ATP production caused by mitochondrial rupture and fragmentation [23; 24; 25]. The precise mechanisms responsible for these effects are unknown but appear to be associated with ultrastructural changes caused by calcium overload [14; 25]. Mitochondria consist of two compartments formed by two separate membranes: the outer membrane and the inner membrane. The outer membrane encases the inner membrane and serves as the gateway between the cytosolic and mitochondrial environments. This membrane forms transient connections with extra-mitochondrial organelles, such as the endoplasmic reticulum, ribosomes, nucleus, and inner membrane. The intermembrane space (IMS) compartment is between the outer and inner membrane, and the matrix compartment is encapsulated by the inner membrane. The inner membrane surface area is approximately five to ten times that of the outer membrane and forms compartments called cristae that fit within the envelope of the outer membrane. Cristae are invaginations of the inner membrane that contain the bulk of the metabolic enzymes required to produce and deliver ATP to the cell. Both membranes are under the active control of fission and fusion proteins that react to external stimuli and energy demand cues to coordinate ATP production [26; 27; 28]. While the major fission and fusion proteins of the outer membrane are known, much less is known about the key proteins involved in inner membrane dynamics. Even less is known about regulation of the inner membrane structures, such as cristae and cristae junctions. While we still lack a complete picture of cristae formation and how they are regulated, some of the key proteins involved have been identified [29]. Cristae are not static structures [30]; they are dynamically formed in a manner dependent on a variety of known and unknown factors which control ATP synthesis rates [31]. That said, the main proteins that exert some control over the cristae's shape are the mitochondrial contact site and cristae organizing system (MICOS) [32; 33; 34], optic atrophy factor 1 (OPA1) [33; 35; 36], and ATP synthase [37; 38; 39; 32]. The MICOS proteins are necessary to form proper cristae junctions, a region in mitochondria that connects the inner boundary membrane to the cristae lumen. OPA1 is a protein with several isoforms and proteolytic variants that serves multiple roles [40]. These proteins coordinate with the MICOS proteins to form the cristae junction and help shape the cristae's morphology. ATP synthase dimers contribute to cristae morphology by inducing inner membrane curvature. Cardiolipin also plays an important role in regulating the cristae shape and membrane curvature [41].