Delay differential equation approach to oligomerization reveals the delay dynamics underlying oscillations in mitochondrial fission

Mitochondria can split and combine through processes called fission and fusion respectively. Disruptions in the balance of fission and fusion are implicated in a host of diseases including cardiovascular, metabolic, and neurodegenerative, as well as cancer. Leinheiser et al. [18] proposed a set of ordinary differential equations that model mitochondrial fission in which the protein dynamin related protein 1 (Drp1) attaches to the surface of the mitochondria and forms oligomers. In this work, we propose a homogenization of that model to a partial differential equation which we show is equivalent to a delay differential equation. The equivalence of this model reveals the intrinsic time delay present in the process of mitochondrial fission. In the development of this new set of delay equations, we generate a simplified version of the model which does not allow oligomers to shrink in size. However, the simplified equations do not capture the qualitative behavior of the Leinheiser et al. equations, as they predict no mitochondrial fission will happen. Therefore, we find it is crucial that oligomers be allowed to shrink in size. Even if we allow shrinking, the delay version of the equations still predict no fission will occur. So we allow oligomers to “atomize”, releasing all their building blocks at once. The model that includes atomization predicts qualitatively similar oscillations in the total fission rate as the Leinheiser et al. equations, and does so with a delay term included. This demonstrates that the behavior of the Leinheiser et al. model relies on the implicit delay present in mitochondrial fission.