# Modeling the Effects of Calcium Overload on Mitochondrial Ultrastructual Remodeling
## 2 Materials and Methods
Images were collected in linear mode on a Falcon 3 EC direct electron detector with an electron dose of \(\sim\)2e\({}^{-}\)/A\({}^{2}\) per tilt image. The images were collected at 22,000\(\times\) magnification (4.7 A/pixel). The total electron dose was approximately 100 e\({}^{-}\)/A\({}^{2}\) for each tomogram. Tomogram Reconstruction and Modeling. Motion correction was performed for the individual micrographs and binned by an index factor of 6 using Motioncor2 v1.2.6. Tilt series alignment was performed using IMOD v4.9.12 and standard tomographic reconstruction particles, using the Simultaneous Iterative Reconstruction Technique (SIRT) strategy. Key features of the tomograms were traced using the drawing tools functionality in IMOD (3 dmod) [71]. Statistics. The Shapiro-Wilks test was used to confirm data normality. All data were analyzed and plotted using MATLAB 2020a (Mathworks, Inc., Natick, MA, USA). The data in Figure 1 (n = 3-4) are presented as a mean standard deviation. An unpaired Student's \(t\)-test was used to compare the calcium treatment with the control group. A value of \(p<0.05\) was assumed to be statistically significant. Modeling. The dimensionless form of the Cahn-Hilliard equation is given by  
\[\frac{\partial\Psi}{\partial t}=\frac{1}{2}\
abla^{2}\Big{(}-\gamma\
abla^{2}\Psi -\Psi+\Psi^{3}\Big{)} \tag{1}\]  
where \(\sqrt{\gamma}\) is the length of the transition regions between the domain and \(\Psi(\mathbf{r},t)\), \((\Psi(\mathbf{r},t)\vee\leq 1)\) is a scalar order parameter chosen to be the difference between the local densities of the inner membrane \([\Psi=-1]\) and the fluid matrix \([\Psi=1]\). The Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation (PDE), and that does not admit a general analytical solution. Numerical simulation is the major approach to obtaining the solution of the Cahn-Hilliard equation. However, some theoretical analysis can offer a basic understanding of its behavior and underlying physics. First, it is important to note that Equation (1) can be cast in its conversational form:  
\[\frac{\partial\Psi}{\partial t}=-\
abla\cdot\mathbf{J} \tag{2}\]  
where \(\mathbf{J}=-\frac{1}{2}\
abla\mu\) is a flux vector and the chemical potential of the binary system is expressed as  
\[\mu=\Big{(}\Psi^{2}-1-\gamma\
abla^{2}\Big{)}\Psi \tag{3}\]  
This potential-driven flow has a trivial solution at \(\Psi=0\). Additionally, two other asymptotic solutions are given by \(\Psi^{2}-1=0\) (i.e., \(\Psi=\pm 1\)), which represents two phases. Since the amplitude of the leading term in Equation (3) is one, the characteristic length of the crista is one in the present setting. Therefore, the domain size should be chosen accordingly for a given set of experimental data. The boundary condition, meanwhile, is the usual zero-flux condition for the chemical potential \(\mu\) and the scale order parameter \(\Psi\):  
\[\mathbf{n}\cdot\
abla\mu(\mathbf{r},t)=0\text{ and }\mathbf{n}\cdot\
abla\Psi( \mathbf{r},t)=0,\mathbf{r}\in\Gamma \tag{4}\]  
Figure 1: Oxidative phosphorylation decreasing proportionally with the calcium load. (**A**) Representative mitochondrial respiration profiles in the presence of the calcium chelator EGTA (–4 \(\upmu\)M residual calcium (ResCa)) and after a 25 \(\upmu\)M or 50 \(\upmu\)M bolus of calcium. The mitochondrial calcium loads for these conditions were 0, 40, 250, and 500 nmol calcium/mg mitochondria, respectively. Mitochondria were energized with 5 mM sodium pyruvate and 1 mM L-malate. (**B**) The average maximum ADP-stimulated respiration rate decreased dramatically at higher calcium doses. Maximal ADP-stimulated respiration was induced following the addition of a 500 \(\upmu\)M ADP bolus. Data are presented as the mean \(\pm\) standard deviation for a sample size of five biological replicates. Here, a * and *** represent \(p\) values < 0.01 and <0.001, respectively. where \(n\) is normal to the boundary and \(\Gamma\) is the boundary of the geometrical shape.