Discussion

Mitochondria are the primary producers of ATP and play a critical role in the regulation of cellular metabolism. Therefore, the maintenance of the mitochondria is crucial for the health of the cell. The mitochondrial population is maintained through the interplay of three main processes: mitophagy, fission, and fusion. In this study, we have chosen to focus on the regulation of fission. However, each process plays an important role, and future work should incorporate these elements. In response to metabolic or environmental stressors, mitochondria can enter a state of hyperfission which can impair ATP production and lead to programmed cell death. Therefore, studying the mechanisms and regulation of fission is critical to understand mitochondrial homeostasis.

In this work, we examine the mathematical model for Drp1-dependent mitochondrial fission from Leinheiser et al. (2024) and develop an sdDDE model for mitochondrial fission. Whilst developing the sdDDE, we generated a simplified model by removing the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k-, disallowing the disassembly of oligomers on the mitochondrial membrane. The homogenization of this model leads to an advection PDE with non-local interactions via the boundary and velocity. One of the two model solutions is a stalled wave of middle sized oligomers which never reach a sufficient size to induce fission. With the initial conditions from Leinheiser et al., the solution for the simplified model is in the basin of attraction for this fission-free equilibrium. We show this stalled wave also exists in the original Leinheiser et al. model when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k- is set to zero. This suggests that a method for dissassembly of the Drp1 oligomers is necessary to reduce the potentiality of a fission-free equilibrium. In all three models, the stability of each equilibrium is left as another future direction, as the theorems present in this work consider only specific initial conditions and not the basins of attraction for each equilibrium.

Since a mechanism which allows a stable fission-free equilibrium would be detrimental to mitochondrial homeostasis, we reincorporated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k- into our mechanism with a bidirectional model. After homogenization, the bidirectional model also generates an advection PDE with non-local interactions via the boundary and velocity. Interestingly, even with the re-addition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k-, the solutions for our homogenized bidirectional system persist in the basin of attraction of the stable, fission-free equilibrium. Note that in numerical simulations of the Leinheiser et al. model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k- greater than zero is enough to escape this basin of attraction. As part of the homogenization, we chose an approximation which omitted the diffusion term. We conjecture that an approximation that is an advection–diffusion model could destabilize the fission-free equilibrium consistent with the original Leinheiser et al. simulations. However, we chose to pursue a form of the PDE which preserves the delay dynamics since our focus is understanding the inherent delay-like behavior observed in the oscillatory numerical solutions.

We therefore propose an atomization model which includes an atomization term b instead of a dissociation term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-$$\end{document}k-. This atomization term allows oligomers to return to size one at any point in the building process. With this mechanism, we are able to derive an sdDDE of threshold type which eliminates the fission-free equilibrium.

Another possible strategy would be to explore other models for the building of Drp1 oligomers. The atomization model presented here requires oligomers to build one block at a time. This choice is consistent with the Leinheiser et al. model as well as the suggested mechanism described in Michalska et al. (2018) for Drp1. This Becker-Döring type mechanism has also been adopted in other models of cellular polymerization (for example Edelstein-Keshet and Ermentrout 1998). However, Strack et al. (2013) hypothesizes that larger size oligomers may also be able to combine on the mitochondrial surface. In that case, we would arrive at a Smoluchowski coagulation (or coagulation-fragmentation) model with fission-driven non-local feedback on the boundary condition. A straightforward calculation shows that such a model also has no fission-free equilibrium since middle size oligomers can continue to combine even when the pool of monomers is depleted.

The homogenized atomization model allows another look at the Hopf bifurcation observed in Leinheiser et al. Analogous to that model, the total number of building blocks (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document}M and Q respectively) is a bifurcation parameter. This qualitatively similar Hopf bifurcation recovered in the sdDDE corroborates that the oscillatory behavior of the Leinheiser et al. model is due to underlying delay dynamics. This suggests that mitochondrial fission may display functional oscillations due to the intrinsic delay in Drp1-oligomerization. We do not know of any studies which have directly measured the time course of fission rate under cellular stress with sufficient temporal resolution to observe such oscillations. Thus, their physiological relevance remains an important open question. While several quantities related to mitochondrial function have been observed to oscillate (for example, Aon et al. 2008; Porat-Shliom et al. 2014; Neufeld-Cohen et al. 2016 and Yu et al. (2021)), it is unclear whether the oscillations observed here could occur under physiological conditions. The models discussed in this paper suggest that the material pool for oligomers as well as the oligomerization kinetics are key parameters to explore experimentally in order to investigate oscillatory fission behavior.

Beyond the application to mitochondrial fission, the derivation of this sdDDE model provides insight into homogenization techniques. A homogenized PDE model with nonlocal information on the boundary condition and the velocity term is difficult to analyze. Further, a homogenization with diffusion disallows the use of the method of characteristics. Thus, the introduction of the atomization parameter served to circumvent the issues with diffusion terms while retaining the qualitative behavior of the original discrete system. Further analysis of sdDDEs of threshold type as approximations of comparable ODE systems could reveal the differences between systems with a dissociation term and systems with an atomization term.

These results provide further evidence that the timing and regulation of oligomer assembly are central determinants of mitochondrial dynamics. Since excessive or unregulated mitochondrial fission is implicated in cardiovascular disease, metabolic dysfunction, neurodegeneration, and cancer, understanding the temporal mechanisms governing oligomerization provides insight into how mitochondrial populations transition between healthy and pathological states. The atomization framework introduced here highlights the importance of regulatory mechanisms that continually reset or redistribute oligomer populations.

More broadly, this study demonstrates how homogenization techniques and sdDDE reductions can uncover latent delay structure in large mechanistic systems. The resulting models provide analytically tractable descriptions of nonlocal transport processes while preserving key biological dynamics that may otherwise be obscured in high-dimensional ODE systems. Beyond mitochondrial fission dynamics, the framework developed here may prove useful for understanding other intracellular assembly processes in which threshold formation times and delayed feedback play a fundamental role.