The Atomization Model

In order to guarantee the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^*$$\end{document}t∗ for any _ℓ ℓ, we propose the atomization model. The atomization model once again removes \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- and introduces a new parameter b, the atomization parameter. The atomization process does not deconstruct an oligomer one building block at a time, but instead releases all building blocks all at once (Fig. 6). This atomization process is separate from a fission event.

31a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_1&= a\sum _{i = \ell }^\infty i C_i - k \left( C_1^2 + \sum _{i=1}^{\infty } C_i C_1 \right) + b \sum _{i=2}^{\ell - 1}i C_i , \end{aligned}$$\end{document}C˙1=a∑i=ℓ∞iCi-kC12+∑i=1∞CiC1+b∑i=2ℓ-1iCi,

31b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_i&= k C_1 C_{i - 1} - k C_1 C_i - f(i) C_i, \quad i \ge 2, \end{aligned}$$\end{document}C˙i=kC1Ci-1-kC1Ci-f(i)Ci,i≥2,

31c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(i)&= {\left\{ \begin{array}{ll} b, & i < \ell , \\ a, & i \ge \ell , \end{array}\right. } \end{aligned}$$\end{document}f(i)=b,i<ℓ,a,i≥ℓ,

31d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1(0)&=Q, \quad C_i(0) = 0, \quad \text {for} \quad i \ge 2. \end{aligned}$$\end{document}C1(0)=Q,Ci(0)=0,fori≥2.

The total count of size 1 oligomers is conserved:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i=1}^\infty i C_i = C_1(0) = Q. \end{aligned}$$\end{document}∑i=1∞iCi=C1(0)=Q.

Fig. 6: The reaction schematic for the atomization model of mitochondrial fission. Fission complexes combine into oligomers at rate k and atomize back to size 1 at rate b. Oligomers cause a fission event at rate a once they reach size ℓℓ or larger

Homogenizing the Atomization Model

We homogenize in an analogous way to the two previous models.

32a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= a \int _\ell ^\infty u \Theta (u,t) du - k C\left( C+ \int _1^\infty \Theta (u,t) du \right) + b \int _1 ^\ell u \Theta (u,t) du, \end{aligned}$$\end{document}C˙=a∫ℓ∞uΘ(u,t)du-kCC+∫1∞Θ(u,t)du+b∫1ℓuΘ(u,t)du,

32b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \Theta }{\partial t}&= -k C \frac{\partial \Theta }{\partial s} - f(s) \Theta (s,t) ,\quad s > 1, \end{aligned}$$\end{document}∂Θ∂t=-kC∂Θ∂s-f(s)Θ(s,t),s>1,

with conserved quantityNote that Q is finite which implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _1^\infty u \Theta (u,t) du$$\end{document}∫1∞uΘ(u,t)du and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _1^\infty \Theta (u,t)du$$\end{document}∫1∞Θ(u,t)du are bounded. This is biologically sound.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C + \int _1^\infty u \Theta (u,t) du = Q. \end{aligned}$$\end{document}C+∫1∞uΘ(u,t)du=Q.

Note that System (32) no longer has a fission-free steady state. The only steady state isThe total fission rate of this equilibrium is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\ell ^\infty a \Theta (u,t)du =k C^2 e^{\frac{b}{kC}(1-\ell )} $$\end{document}∫ℓ∞aΘ(u,t)du=kC2ebkC(1-ℓ), and gives us confidence that the atomization model has removed the fission-free, stalled wave equilibrium present in the previous models.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q&= C + k C^2 e^{\frac{-b}{kC}(\ell - 1)}\left( \frac{\ell }{a} - \frac{\ell }{b} + \frac{k C}{a^2} - \frac{k C}{b^2} \right) + \frac{k C^2}{b} \left( \frac{k C}{b} + 1 \right) , \\ \Theta (s)&= C e^{\frac{-b}{k C}(s - 1)}, \quad \quad \quad \quad s < \ell , \\ \Theta (s)&= Ce^{\frac{1}{kC}(b - b\ell + a\ell - as)}, \quad s \ge \ell . \end{aligned}$$\end{document}Q=C+kC2e-bkC(ℓ-1)ℓa-ℓb+kCa2-kCb2+kC2bkCb+1,Θ(s)=Ce-bkC(s-1),s<ℓ,Θ(s)=Ce1kC(b-bℓ+aℓ-as),s≥ℓ.

Discontinuities in the Fission Function and Initial Conditions

Similar to previous sections, we have a discontinuous fission function given in Equation (31c), and we choose the initial conditions to be Equation (8). Any solution with these initial conditions solves System (32) in the weak sense. We define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^$$\end{document}s∗ analogously to Sect. (2.1.1) and define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ as the solution to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^{t^*} k C(\xi ) d\xi = \ell - 1$$\end{document}∫0∗kC(ξ)dξ=ℓ-1.

Due to the atomization term in Equation (32a), we need new variables: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n(t) = \int _{1}^\infty \Theta (u,t)du$$\end{document}n(t)=∫1∞Θ(u,t)du and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(t) = \int _{1}^\infty u \Theta (u,t) du$$\end{document}m(t)=∫1∞uΘ(u,t)du. Assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^*(t) < \ell $$\end{document}s∗(t)<ℓ,

33a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}(t)&= k C^2 - bn, \end{aligned}$$\end{document}n˙(t)=kC2-bn,

33b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{m}(t)&= kC (C + n) - bm. \end{aligned}$$\end{document}m˙(t)=kC(C+n)-bm.

Next we show that a solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^*$$\end{document}t∗ can always be found.

Stalled Wave is Eliminated in the Homogenized Atomization Model

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^*(t) < \ell $$\end{document}s∗(t)<ℓ, System (32), Equations (33), conserved quantity _C + m = Q_C+m=Q, and initial conditions (8) create the new ODE system:

34a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= - k C^2 - k C n - b C + bQ, \end{aligned}$$\end{document}C˙=-kC2-kCn-bC+bQ,

34b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}&= k C^2 - bn. \end{aligned}$$\end{document}n˙=kC2-bn.

We showed above that System (32) does not have a fission-free steady state. This gives us confidence we can always solve for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗. To prove this, we begin by showing C(t) in System (34) is not near 0 for any t. If this is true, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^{\infty } k C(\xi )d\xi $$\end{document}∫0∞kC(ξ)dξ will grow without bound and a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ can be found for any _ℓ _ℓ.

Theorem 3

If _b > 0_b>0, _k > 0_k>0, and _Q > 0_Q>0 then the flow of System (34) is within the trapping region defined by _n = 0_n=0, _n = Q_n=Q, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = - C + \sqrt{\frac{bQ}{k}} + Q$$\end{document}n=-C+bQk+Q, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = \hat{C}$$\end{document}C=C^, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{C}$$\end{document}C^ is the positive solution to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-k C^2 - k C Q - b C + bQ = 0$$\end{document}-kC2-kCQ-bC+bQ=0.

Proof

Consider the trapezoid in the C, n-plane defined by _n = 0_n=0, _n = Q_n=Q, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = - C + \sqrt{\frac{bQ}{k}} + Q$$\end{document}n=-C+bQk+Q, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = \hat{C}$$\end{document}C=C^.

We proceed by showing the flow of the system cannot leave the trapezoid through any of its edges. Along the _n = 0_n=0 edge, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{n} = k C^2 > 0$$\end{document}n˙=kC2>0. Along the _n = Q_n=Q edge of the trapezoid, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C < \sqrt{\frac{bQ}{k}}$$\end{document}C<bQk, therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{n} = k C^2 - b Q < 0$$\end{document}n˙=kC2-bQ<0. Along the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = \hat{C}$$\end{document}C=C^ edge, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C} = k \hat{C} (Q - n) > 0$$\end{document}C˙=kC^(Q-n)>0 (because _n < Q_n<Q on that edge).

The dynamics along the sloped edge of the trapezoid always point inward. We show this by using a dot product with a vector normal to the edge: (-1,-1)(-1,-1).Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{C} < Q$$\end{document}C^<Q and therefore our initial condition (Q, 0) is within the trapping region.

35\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\dot{C}, \dot{n}) \cdot (-1,-1)&= k C^2 + k C n + b C - b Q - k C^2 + bn\nonumber \\&= k C n + b C - b Q + b(-C + \sqrt{\frac{bQ}{k}} + Q)\nonumber \\&= k C n + b \sqrt{\frac{bQ}{k}} > 0. \end{aligned}$$\end{document}(C˙,n˙)·(-1,-1)=kC2+kCn+bC-bQ-kC2+bn=kCn+bC-bQ+b(-C+bQk+Q)=kCn+bbQk>0.

Therefore the flow of the system cannot leave the trapezoid and C(t) can never approach 0. _□ _□

With proof that a solution exists for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ in the homogenized atomization model, we finally consider the dynamics of System (32) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > t^$$\end{document}t>t∗. We define partial moments, recalling the derivative of _Θ (s,t)_Θ(s,t) may not exist at size _ℓ _ℓ and size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^* > \ell $$\end{document}s∗>ℓ. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_1(t) = \int _1^\ell \Theta (u,t) du$$\end{document}n1(t)=∫1ℓΘ(u,t)du, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_2(t) = \int _\ell ^{s^} \Theta (u,t) du$$\end{document}n2(t)=∫ℓ∗Θ(u,t)du, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1(t) = \int _1^\ell u \Theta (u,t) du$$\end{document}m1(t)=∫1ℓuΘ(u,t)du, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_2(t) = \int _\ell ^{s^} u \Theta (u,t) du$$\end{document}m2(t)=∫ℓ∗uΘ(u,t)du. We note that taking the derivatives of these and using integration by parts results in the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (\ell ,t)$$\end{document}Θ(ℓ,t) appearing. We denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (\ell ,t) = \Theta _\ell $$\end{document}Θ(ℓ,t)=Θℓ for the remainder of the section. Using these partial moments in System (32) yields with conserved quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + m_1 + m_2 = Q$$\end{document}C+m1+m2=Q.

36a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= a m_2 - k C (C + n_1 + n_2) + b m_1, \end{aligned}$$\end{document}C˙=am2-kC(C+n1+n2)+bm1,

36b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}_1&= -k C (\Theta _\ell - C) - b n_1, \end{aligned}$$\end{document}n˙1=-kC(Θℓ-C)-bn1,

36c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{m}_1&= -k C (\ell \Theta _\ell - C - n_1) - b m_1, \end{aligned}$$\end{document}m˙1=-kC(ℓΘℓ-C-n1)-bm1,

36d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}_2&= k C \Theta _\ell - a n_2, \end{aligned}$$\end{document}n˙2=kCΘℓ-an2,

36e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{m}_2&= k C (\ell \Theta _\ell + n_2) - a m_2, \end{aligned}$$\end{document}m˙2=kC(ℓΘℓ+n2)-am2,

36f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(0)&= Q, \quad n_1(0) = m_1(0) = n_2(0) = m_2(0) = 0, \end{aligned}$$\end{document}C(0)=Q,n1(0)=m1(0)=n2(0)=m2(0)=0,

Delay Dynamics in the Homogenized Atomization Model

We evaluate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _\ell $$\end{document}Θℓ using Equations (32b), (31c) and the method of characteristics. For _s < ℓ _s<ℓ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \Theta }{\partial t} + k C \frac{\partial \Theta }{\partial s} = - b \Theta (s,t). \end{aligned}$$\end{document}∂Θ∂t+kC∂Θ∂s=-bΘ(s,t).

Fig. 7: Numerical solutions for the homogenized atomization model. (A) Numerical simulation of the Θ (s,t)Θ(s,t) surface created by simulating C(t) using the sdDDE form of the atomization model and then using the resulting C(t) as a boundary condition for PDE Equation (32b). The sdDDE was solved using parameter set: ℓ = 15ℓ=15, a = 5a=5, k = 1k=1, b = 0.02b=0.02, and Q = 12Q=12. Three characteristics are graphed on top of the surface. Note the perspective has large time and large oligomer sizes in the foreground to avoid the largest peak obscuring the others. (B) The same surface from a side-on perspective. From this perspective, the exponential decay behavior in time along the characteristics is clear. (C) The same surface from a top-down perspective. At time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_i$$\end{document}ti one can see the corresponding characteristic intersects size ℓ =15ℓ=15. Following the characteristics back through size and time, the associated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (t_i)$$\end{document}τ(ti) is revealed as the time in the past where the characteristic intersects size 1

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d s(t)}{dt} = k C(t)$$\end{document}ds(t)dt=kC(t), then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d \Theta }{dt} = -b \Theta (s(t),t)$$\end{document}dΘdt=-bΘ(s(t),t). Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (s(t),t) = Ae^{-b t}$$\end{document}Θ(s(t),t)=Ae-bt for some A along characteristics. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t < t^$$\end{document}t<t∗ any characteristic that intersects (ℓ ,t)(ℓ,t) also intersects (s, 0) for _s > 0_s>0, where the initial condition is _Θ (s,0) = 0_Θ(s,0)=0. Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = 0 = \Theta _\ell $$\end{document}A=0=Θℓ for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t < t^$$\end{document}t<t∗. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > t^$$\end{document}t>t∗, we can integrate along the characteristics to define a delay term _τ (t)_τ(t).Intuitively, this delay can be thought of as the time it takes a cohort of oligomers to grow from size 1 to size _ℓ _ℓ. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > t^$$\end{document}t>t∗, the characteristics will intersect both (ℓ ,t)(ℓ,t) and (1,t - τ (t))(1,t-τ(t)) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (1,t - \tau (t)) = C(t - \tau (t))$$\end{document}Θ(1,t-τ(t))=C(t-τ(t)). Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t - \tau (t)) = A e^{-b(t - \tau (t))}$$\end{document}C(t-τ(t))=Ae-b(t-τ(t)) which implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = C(t - \tau (t))e^{b (t - \tau (t))}$$\end{document}A=C(t-τ(t))eb(t-τ(t)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _\ell = C(t - \tau (t))e^{-b \tau (t)}$$\end{document}Θℓ=C(t-τ(t))e-bτ(t) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > t^*$$\end{document}t>t∗. An example of these characteristic curves on a _Θ (s,t)_Θ(s,t) surface can be seen in Fig. 7.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{t - \tau (t)}^t k C(\xi )d\xi = \ell - 1 \end{aligned}$$\end{document}∫t-τ(t)tkC(ξ)dξ=ℓ-1

Finally, we have shown the homogenized atomization model is equivalent to a state-dependent delay-differential Equation (sdDDE) of threshold-type, with threshold value _ℓ - 1_ℓ-1: with conserved quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + m_1 + m_2 = Q$$\end{document}C+m1+m2=Q.

37a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= a m_2 - k C (C + n_1 + n_2) + b m_1, \end{aligned}$$\end{document}C˙=am2-kC(C+n1+n2)+bm1,

37b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}_1&= -k C (C(t - \tau )e^{-b \tau } - C) - bn_1, \end{aligned}$$\end{document}n˙1=-kC(C(t-τ)e-bτ-C)-bn1,

37c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{m}_1&= - k C (\ell C(t - \tau )e^{-b \tau } - C - n_1) - bm_1, \end{aligned}$$\end{document}m˙1=-kC(ℓC(t-τ)e-bτ-C-n1)-bm1,

37d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}_2&= k C C(t - \tau )e^{-b \tau } - a n_2, \end{aligned}$$\end{document}n˙2=kCC(t-τ)e-bτ-an2,

37e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{m}_2&= k C (\ell C(t - \tau )e^{-b \tau } + n_2) - a m_2, \end{aligned}$$\end{document}m˙2=kC(ℓC(t-τ)e-bτ+n2)-am2,

37f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{t - \tau (t)}^t&k C(\xi ) d\xi = \ell -1, \end{aligned}$$\end{document}∫t-τ(t)tkC(ξ)dξ=ℓ-1,

37g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(0)&= Q, \quad n_1(0) = m_1(0) = n_2(0) = m_2(0) = 0, \end{aligned}$$\end{document}C(0)=Q,n1(0)=m1(0)=n2(0)=m2(0)=0,

The existence of this sdDDE provides valuable insight into the process of mitochondrial fission. The solutions of the Leinheiser et al. model displayed oscillatory behavior, and they were numerically proved to be the result of a Hopf bifurcation. However, the reason a Drp1-dependent mitochondrial fission model undergoes a Hopf bifurcation is unclear from the ODE formulation in Leinheiser et al. The homogenized atomization model (which displays qualitatively similar behavior) contains delay terms which predispose the system to oscillatory behavior. Furthermore, the sdDDE formulation of Drp1-dependent mitochondrial fission highlights that the oscillations first observed in Leinheiser et al. are a consequence of the delay in oligomer construction on the outer mitochondrial membrane. Also, the sdDDE handles the discontinuous initial conditions without resorting to a weak formulation as we saw with the PDE version.

The Hopf Bifurcation of the Homogenized Atomization Model

Numerical solutions for the homogenized atomization model (System 37) showcase that for different values of Q, i.e. the material pool, the total fission rate exhibits damped oscillations that reach a steady state or stable oscillations that appear to not reach a steady state (Fig. 8). Note that while the scale of these solutions is different from Fig. 1, the qualitative structure of these solutions is the same. The structure indicates the homogenized atomization model produces an analogous Hopf bifurcation. To confirm this, we compute the eigenvalues of the system. Note, the steady state equations of System (37) are transcendental because of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-b \tau }$$\end{document}e-bτ terms. We continue with linearization knowing the steady states will be calculated numerically. The conserved quantity is used to eliminate the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{m}_2$$\end{document}m˙2 and so that we have a non-hyperbolic system. Note that one need not consider the state-dependent properties of _τ _τ near equilibrium, following the theorems from Cooke and Huang (1996).

Fig. 8: Numerical simulations (Shampine 2005) of the homogenized atomization model. The parameter values are a = 5a=5, k = 1k=1, b = 0.02b=0.02, and ℓ = 15ℓ=15 with various values of Q as indicated. Note, these numerical solutions are qualitatively similar to Fig. 1

The full process of checking the bifurcation structure of chosen parameters is as follows. First, a parameter set is chosen. Then the steady state values of each variable are numerically calculated using Newton’s Method. Those steady state values are plugged into the characteristic equation and the eigenvalues are calculated. The characteristic equation is also transcendental and therefore must be numerically calculated. Due to the complexity of this process, it is difficult to determine the exact value when bifurcation occurs. We chose a standard set of parameters within which a single parameter was varied and a range between which the bifurcation occurs is identified (Fynaardt 2026).

Our parameter set was chosen to be similar to the original Leinheiser et al. model. As a result, _a = 5_a=5, _k = 1_k=1, _b = 0.02_b=0.02, and _ℓ = 15_ℓ=15. For each calculation, these parameters were held constant while the parameter Q was varied over discrete values. The bifurcation is identified as the value for which the real part of the eigenvalue crosses the imaginary axis, following the canonical structure of a Hopf bifurcation (Fig. 9). The resulting bifurcation diagram appears in Fig. 10. In conclusion, the homogenized atomization model does indeed produce an analogous Hopf bifurcation to the Leinheiser et al. model. The biological implication is that the inherent delay behavior of oligomerization leads to oscillation in total fission rate.

Fig. 9: The eigenvalues with the largest real part of the characteristic equation of the homogenized atomization model with parameters a = 5a=5, k = 1k=1, b = 0.02b=0.02, and ℓ = 15ℓ=15, with varying Q. The figure shows the pairs of imaginary eigenvalues passing over the imaginary axis as parameter Q passes from 12 to 18, showing a Hopf bifurcation occurs at that point. We note the change in the real part of the eigenvalues from Q = 12Q=12 to Q = 18Q=18, where the bifurcation is predicted to occur, is approximately 5 thousandths

Fig. 10: A bifurcation diagram of the atomization model showing the change in the steady state of C as Q is varied. The structure of a pitchfork bifurcation is clear. We note this diagram shows the bifurcation occurs near Q = 13Q=13