Introduction

Mitochondria are organelles that are predominately known for their role in ATP production which is the main energy currency of cells. To maintain homeostasis, mitochondria undergo three main processes - mitophagy, fusion, and fission. This work focuses on a mathematical model of mitochondrial fission which is the process of a single mitochondrion splitting into two daughter mitochondria. In response to cellular stress, mitochondria rapidly fission, creating a large number of small mitochondria. This hyperfission state has been linked to neurodegenerative, cardiovascular, metabolic disease, and cancer (Youle and Bliek 2012; Ponce et al. 2020). For these reasons, understanding the process and regulation of mitochondrial fission is critical.

Leinheiser et al. proposes a model of mitochondrial fission focused on the oligomerization of dynamin related protein 1 (Drp1) (Leinheiser et al. 2024). In their model, Drp1-dependent fission begins when Drp1 binds to mitochondrial fission factor (Mff) on the outer mitochondrial membrane. These Drp1-Mff complexes bind one complex at a time to form oligomers. When an oligomer becomes long enough to wrap around the mitochondrion, it can constrict and facilitate a fission event. The Leinheiser et al. model is a high dimensional system of ordinary differential equations (ODEs) which tracks the concentration of each oligomer size.

As the total concentration of Mff varies, the total fission rate transitions from damped oscillations that reach an attracting steady state to a solution with stable oscillations, characteristic of a Hopf bifurcation (Fig. 1). Leinheiser et al. go on to confirm this Hopf bifurcation by numerical calculation of the eigenvalues. However, the underlying mechanism which causes these oscillations is still unclear.

In the Leinheiser et al. model, oligomers are built one building block at a time as in a Becker-Döring oligomerization model (Edelstein-Keshet and Ermentrout 1998; Slemrod 2000; Wattis 2006), and it imposes a maximum oligomer size of N. The concentrations of Drp1 in the cytosol and Mff on the mitochondrial membrane are denoted T and M respectively, with initial concentrations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document}T and \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 respectively. They associate at rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1$$\end{document}k1 to form the building blocks of oligomerization. These are viewed as oligomers of size 1 and therefore denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1$$\end{document}C1. An oligomer of size 1 can also dissociate at rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{-1}$$\end{document}k-1. The concentrations of oligomers of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i\le N$$\end{document}1≤i≤N are denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_i$$\end{document}Ci. Oligomers of size 1 can associate with an oligomer of size i at rate \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+ to create an oligomer of size i + 1_i+1. Any oligomer of size i can dissociate at rate \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 a size 1 oligomer and a size _i - 1_i-1 oligomer. The rate of mitochondrial fission is defined as a piecewise function, f(i), that depends on oligomer size. Any oligomer of size less than _ℓ _ℓ fissions at rate 0 while oligomers of size greater than or equal to _ℓ _ℓ fission at a constant rate a. This setup imposes a minimum oligomer size before fission can occur, consistent with the intuition that oligomers must be long enough to encircle the mitochondrion.

Drp1 is released from the endoplasmic reticulum following acute cellular stress (Osterlund et al. 2023). To simulate this sudden influx of oligomer building material, we choose the initial conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(0) = \mathcal {T}$$\end{document}T(0)=T, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(0) = \mathcal {M}$$\end{document}M(0)=M, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_i(0) = 0$$\end{document}Ci(0)=0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le i \le N$$\end{document}1≤i≤N. That is, there are no oligomers of any size on the mitochondrial surface, and there is an initial influx of Drp1 that can associate with the Mff present on the mitochondrial surface. This choice of initial conditions is also consistent with the simulations in Leinheiser et al. The full model is

1a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{T}&= \gamma \left( k_{-1} C_1 - k_1 T M + \sum _{i = 1}^N i f(i) C_i \right) , \end{aligned}$$\end{document}T˙=γk-1C1-k1TM+∑i=1Nif(i)Ci,

1b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{M}&= k_{-1} C_1 - k_1 T M + \sum _{i = 1}^N i f(i) C_i, \end{aligned}$$\end{document}M˙=k-1C1-k1TM+∑i=1Nif(i)Ci,

1c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_1&= k_1 T M - k_{-1} C_1 + k_- \left( 2 C_2 + \sum _{i = 3}^N C_i \right) - k_+ \left( 2 C_1^2 + \sum _{i = 2}^{N - 1} C_i C_1 \right) , \end{aligned}$$\end{document}C˙1=k1TM-k-1C1+k-2C2+∑i=3NCi-k+2C12+∑i=2N-1CiC1,

1d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_i&= k_+ C_{i - 1}C_1 + k_- C_{i + 1} - k_- C_i - k_+ C_i C_1 - f(i) C_i, \quad 2 \le i < N, \end{aligned}$$\end{document}C˙i=k+Ci-1C1+k-Ci+1-k-Ci-k+CiC1-f(i)Ci,2≤i<N,

1e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_N&= k_+ C_{N-1} C_1 - k_- C_N - f(i) C_N, \end{aligned}$$\end{document}C˙N=k+CN-1C1-k-CN-f(i)CN,

1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(i)&= {\left\{ \begin{array}{ll} 0, \quad 1 \le i \le \ell , \\ a, \quad \ell < i \le N, \end{array}\right. } \end{aligned}$$\end{document}f(i)=0,1≤i≤ℓ,a,ℓ<i≤N,

1g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T(0)&= \mathcal {T}, \quad M(0) = \mathcal {M}, \quad C_i(0) = 0 \quad 1 \le i \le N. \end{aligned}$$\end{document}T(0)=T,M(0)=M,Ci(0)=01≤i≤N.

Fig. 1: Numerical simulations of the total fission rate of mitochondria predicted by the Leinheiser et al model (Leinheiser et al. 2024). These simulations were run with the parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+ = 5$$\end{document}k+=5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- = 0.1$$\end{document}k-=0.1, ℓ = 25ℓ=25, a = 5a=5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1 = 1$$\end{document}k1=1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{-1} = 0.02$$\end{document}k-1=0.02, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T} = 20$$\end{document}T=20, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} = 15$$\end{document}M=15, γ = 0.01γ=0.01, and N = 30N=30. The total rate of fission transitions from a stable steady state to a stable oscillation as the pool of available fission proteins passes the Hopf bifurcation_

In this work, we propose an alternative state-dependent delay-differential equation (sdDDE) model of threshold type which reveals the inherent delay dynamics associated with mitochondrial fission. Our goal is to introduce a model which retains each of the qualitative behaviors of the original model while giving insight into the underlying mechanisms. In the development of this sdDDE model, we generate a simplified model which disallows oligomer disassembly on the mitochondrial membrane. Following homogenization, the simplified model in Sect. 2 has two steady states, one of which is dominated by medium-sized oligomers, and as a consequence, fission never occurs. We prove our initial conditions lie in the basin of attraction of this fission-free equilibrium. We confirm the existence of a stalled wave similar to the Leinheiser et al. model. Since a model without fission is not biologically accurate, we reincorporate oligomer disassembly in Sect. 3 with a bidirectional model. However, the stable fission-free equilibrium seen in the previous section persists after homogenization. To eliminate this fission-free equilibrium, we replace dissociation with an atomization term in Sect. 4. As a result, we eliminate the fission-free equilibrium and show the homogenized atomization model is equivalent to a sdDDE. This model has an analogous Hopf bifurcation to the original Drp1-dependent fission model.