The Bidirectional Model

To remedy the stalled wave phenomenon discussed in Sect. 2.2, we reintroduce the dissociation 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- from the Leinheiser et al. model (Fig. 5). This produces the bidirectional model:

22a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_1&= a \sum _{i = \ell }^\infty i C_i - k_+ \left( C_1^2 + \sum _{i = 1}^{\infty } C_i C_1 \right) + k_- \left( C_2 + \sum _{i = 2}^{\infty } C_i \right) , \end{aligned}$$\end{document}C˙1=a∑i=ℓ∞iCi-k+C12+∑i=1∞CiC1+k-C2+∑i=2∞Ci,

22b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_i&= k_+ C_1 C_{i - 1} - k_+ C_1 C_i - k_- C_i + k_- C_{i + 1} - f(i) C_i, \quad i \ge 2, \end{aligned}$$\end{document}C˙i=k+C1Ci-1-k+C1Ci-k-Ci+k-Ci+1-f(i)Ci,i≥2,

22c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(i)&= {\left\{ \begin{array}{ll} 0, & i < \ell , \\ a, & i \ge \ell , \end{array}\right. } \end{aligned}$$\end{document}f(i)=0,i<ℓ,a,i≥ℓ,

22d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1(0)&= Q, \quad C_i(0) = 0, \quad i>1. \end{aligned}$$\end{document}C1(0)=Q,Ci(0)=0,i>1.

The total number of size 1 oligomers is still 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. 5: The reaction schematic for a bidirectional model of mitochondrial fission. Oligomers of size 1 (\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) combine into larger oligomers at the 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+ and separate at the 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-. Once oligomers reach size ℓℓ or larger, they can initiate a fission event at rate a_

Homogenizing the Bidirectional Model

Similar to the simplified model (System (2)), the bidirectional model (System (22)) becomes unwieldy as _ℓ _ℓ becomes large. We homogenize following the same procedure described in Sect. 2.1. The resulting homogenized bidirectional model is

23a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= a \int _\ell ^\infty u \Theta (u,t)du - (k_+ C - k_-) \left( C + \int _1^\infty \Theta (u,t) du \right) , \end{aligned}$$\end{document}C˙=a∫ℓ∞uΘ(u,t)du-(k+C-k-)C+∫1∞Θ(u,t)du,

23b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \Theta }{\partial t}&= -(k_+ C - k_-) \frac{\partial \Theta }{\partial s} - f(s) \Theta (s,t), \quad s > 1. \end{aligned}$$\end{document}∂Θ∂t=-(k+C-k-)∂Θ∂s-f(s)Θ(s,t),s>1.

with the 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.

We note here a critical step that was taken when homogenizing Equation (22b). We chose to use both a forward and backward difference approximation for the partial derivative:This choice was made because using only forward, backward, or center difference when homogenizing produces a diffusion term that disallows the use of the method of characteristics in later steps.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_{i+1} - C_i \approx \frac{\partial \Theta }{\partial s} \approx C_i - C_{i - 1}. \end{aligned}$$\end{document}Ci+1-Ci≈∂Θ∂s≈Ci-Ci-1.

System (23) has two steady states, one is a fission-free equilibrium: where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h: \mathbb {R} \rightarrow \mathbb {R}$$\end{document}h:R→R is any function such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(1) = \frac{k_-}{k_+}$$\end{document}h(1)=k-k+ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C + \int _1^{\ell } s , h(s) ds=Q$$\end{document}C+∫1ℓsh(s)ds=Q. The other steady state is

24a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C&= \frac{k_-}{k_+}, \end{aligned}$$\end{document}C=k-k+,

24b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= h(s),&s < \ell , \end{aligned}$$\end{document}Θ(s)=h(s),s<ℓ,

24c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= 0,&s \ge \ell . \end{aligned}$$\end{document}Θ(s)=0,s≥ℓ.

25a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C&= \frac{\left( \frac{k_-}{a} - \frac{\ell ^2}{2} - \frac{1}{2}\right) + \sqrt{\left( \frac{\ell ^2}{2} + \frac{1}{2} - \frac{k_-}{a}\right) ^2 + 4 \frac{k_+ Q}{a}}}{2 \frac{k_+}{a}}, \end{aligned}$$\end{document}C=k-a-ℓ22-12+ℓ22+12-k-a2+4k+Qa2k+a,

25b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= C,&s < \ell , \end{aligned}$$\end{document}Θ(s)=C,s<ℓ,

25c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= C e^{\frac{-a}{k_+ C - k_-}(s - \ell )},&s \ge \ell . \end{aligned}$$\end{document}Θ(s)=Ce-ak+C-k-(s-ℓ),s≥ℓ.

The 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}$$\int \ell ^\infty a \Theta (u,t)du = C(k+ C - k_-)$$\end{document}∫ℓ∞aΘ(u,t)du=C(k+C-k-). We will determine if our initial conditions once again lie in the basin of attraction for the fission-free equilibrium.

Discontinuities in the Fission Function and Initial Conditions

We choose the fission function to be Equation (7), and we choose the initial conditions to be Equation (8). Any solution with these initial conditions solves System (23) 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∗ and \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∗ analogously to Sect. 2.1.1. Then, 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, we defineand find

26\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n(t) = \int _1^\infty \Theta (u,t)du = \int _1^{s^*} \Theta (u,t)du \end{aligned}$$\end{document}n(t)=∫1∞Θ(u,t)du=∫1∗Θ(u,t)du

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

A Stalled Wave persists in the Homogenized Bidirectional 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 (23), with Equation (27), and initial conditions (8) create the new ODE system: This system can be solved for C and used in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^{t^} k_+ C(\xi ) - k_- d\xi = \ell - 1$$\end{document}∫0∗k+C(ξ)-k-dξ=ℓ-1 to 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∗. The following theorem shows there is no solution for reasonable _ℓ _ℓ.

28a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= -(k_+ C - k_-)(C + n), \end{aligned}$$\end{document}C˙=-(k+C-k-)(C+n),

28b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}&= (k_+ C - k_-)C, \end{aligned}$$\end{document}n˙=(k+C-k-)C,

28c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(0)&= Q, \quad n(0) = 0. \end{aligned}$$\end{document}C(0)=Q,n(0)=0.

Theorem 2

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_+ > 0$$\end{document}k+>0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- > 0$$\end{document}k->0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \ge \frac{k_-}{k_+} $$\end{document}Q≥k-k+, and C(t) is a solution to System (28), then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^\infty k_+ C(\xi ) - k_- d\xi \le \ln \left( \frac{Q k_+}{k_-} \right) . \end{aligned}$$\end{document}∫0∞k+C(ξ)-k-dξ≤lnQk+k-.

Proof

Let C(t) and n(t) be solutions to System (28). Note that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t) = \frac{k_-}{k_+}$$\end{document}C(t)=k-k+ for any t, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C}(t) = 0$$\end{document}C˙(t)=0, and therefore C cannot cross the horizontal line at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{k_-}{k_+}$$\end{document}k-k+. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(0) = Q \ge \frac{k_-}{k_+}$$\end{document}C(0)=Q≥k-k+, that means \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(t) > \frac{k_-}{k_+}$$\end{document}C(t)>k-k+ for t \ge 0_t≥0. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{n} = (k+C - k_-)C$$\end{document}n˙=(k+C-k-)C and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{k_-}{k_+} > 0$$\end{document}k-k+>0, n(t) is increasing. We seek to define a new ODE with a closed-form solution that is guaranteed to be larger than C(t) for all t. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C} = -(k_+ C - k_-)(C + n_1) \le -(k_+ C - k_-)C$$\end{document}C˙=-(k+C-k-)(C+n1)≤-(k+C-k-)C for _t \ge 0_t≥0. We can use this fact to define dynamics on a new ODE that has a derivative always larger than C(t) (but still negative).

Define A(t) to be the solution to

29a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{A}&= -(k_+A - k_-)A, \end{aligned}$$\end{document}A˙=-(k+A-k-)A,

29b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(0)&= Q. \end{aligned}$$\end{document}A(0)=Q.

Since C(t) and A(t) have the same initial conditions and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C}(t) \le \dot{A}(t)$$\end{document}C˙(t)≤A˙(t) for _t \ge 0_t≥0, we know _C(t) łe A(t)_C(t)≤A(t) for _t \ge 0_t≥0. The solution is:Thus,By continuity of the integral,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(t) = \frac{k_- Q e^{k_- t}}{k_- + k_+ Q (e^{k_- t} - 1)}. \end{aligned}$$\end{document}A(t)=k-Qek-tk-+k+Q(ek-t-1).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(t) \le \frac{k_- Q e^{k_- t}}{k_- + k_+ Q (e^{k_- t} - 1)}, \quad \text {for} \ t \ge 0. \end{aligned}$$\end{document}C(t)≤k-Qek-tk-+k+Q(ek-t-1),fort≥0.

30a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^\infty k_+ C(\xi ) - k_- d\xi&\le \int _0^\infty k_+ A(\xi ) - k_- d\xi \end{aligned}$$\end{document}∫0∞k+C(ξ)-k-dξ≤∫0∞k+A(ξ)-k-dξ

30b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&= \lim _{t \rightarrow \infty } \ln {\left( \frac{k_+ Q (e^{ t k_-} - 1) + k_-}{k_-} \right) } - t k_- \end{aligned}$$\end{document}=limt→∞lnk+Q(etk--1)+k-k--tk-

30c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&= \ln \left( \frac{Q k_+}{k_-} \right) . \end{aligned}$$\end{document}=lnQk+k-.

_□ _□

At the nominal values from Leinheiser et al. (2024), Q = 20_Q=20, \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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ln \left( \frac{Q k_+}{k_-}\right) \approx 6.9$$\end{document}lnQk+k-≈6.9. Thus we prove the wave stalls for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 8$$\end{document}ℓ≥8. Numerical evidence finds the wave actually stalls for _ℓ _ℓ as small as 2.

The form of the bound in this theorem gives the impression that small \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- or even \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- = 0$$\end{document}k-=0 may cause the integral defining \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 grow without bound. However, this theorem is not bidirectional. The bound being larger does not imply the integral will reach that bound, or even grow larger at all. In fact, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- = 0$$\end{document}k-=0 case is equivalent to the previous section: the simplified model, which also had a stalled wave. This discourages the idea that small \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- removes the stalled-wave in the bidirectional model. Note that because the wave speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_+ C - k_-$$\end{document}k+C-k- includes a negative term, the above argument needed only one estimate of the solution C(t), unlike the proof of Theorem (1) which needed two estimates for C(t) and an estimate for n(t).

Therefore, the stalled wave persists in the homogenized bidirectional model. Note that in 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}$$k_- > 0$$\end{document}k->0 remedies the stalled wave issue. Yet the 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- in the bidirectional model does not have the same effect. We conjecture this inconsistency is from our choice in homogenizing Equation (22b), where a “hybrid" between the backward and forward difference approximations was used to remove diffusion from the resulting homogenization. We further conjecture that a homogenized version of the bidirectional model which includes diffusion may also avoid the stalled wave equilibrium for the given initial conditions. However, with diffusion included in the homogenized bidirectional system, there is no obvious analog to using the method of characteristics to track the discontinuity of the initial conditions. Therefore, we leave it to future work to investigate a homogenized bidirectional model which includes diffusion. Instead, we continue with a change to the model that removes the fission-free equilibrium altogether, and recovers an analogous Hopf bifurcation to the Leinheiser et al. model.