Apart from replicating through mitogenesis and being degraded through processes such as mitophagy, mitochondria are constantly undergoing fission and fusion in order to adapt their morphology to the cellular environment and energetic demands [2]. That is, mitochondrial dynamics are governed by two sets of opposing processes: mitochondrial fusion and fission, as well as mitochondrial biogenesis and degradation; with other processes such as intracellular transport also playing an important role. Overall, mitochondrial dynamics processes maintain homeostasis, regulate mitochondrial form, volume, and function, and are increasingly recognized as critical components of the cellular stress response [16].

Mitochondrial fusion is the physical merging of two or more previously distinct mitochondria's outer and then inner mitochondrial membranes. Fusion results in networks with continuous membranes and a matrix lumen. As a result, each network's constituents share solutes, metabolites, and proteins [3]. Recent in vitro fluorescence microscopy studies [3] have also shown that fusion preferentially involves mitochondria with high electrochemical potential across their inner membrane (Ψ_m). All of this suggests that fusion is a mechanism by which intact mitochondria can supplement a damaged one and possibly restore its activity, preserving metabolic efficiency.

Mitochondrial fission is the splitting of a single mitochondrial body into two. According to in vitro studies, approximately 30% of fission events resulted in a statistically significant difference in the membrane potentials of the two daughter cells [3]. The mechanism underlying this asymmetry is unknown, but it has a significant impact on the relative capacities for ATP and reactive oxygen species (ROS) production of the two daughter mitochondria [17].

Mitophagy is the selective degradation of mitochondrial by autophagy. It frequently follows damage or stress events because it targets depolarized mitochondria for digestion and elimination [18].

Mitogenesis, also known as mitochondrial biogenesis, is the process by which new mitochondria are formed through the growth and division of existing mitochondria [19]. This process is strictly governed by intracellular or extracellular signals that communicate an energy imbalance caused by increased energy demand and/or decreased energy production.

According to Twig et al. [3], mitochondrial fusion-fission and autophagy form a quality-control axis that maintains mitochondrial health by segregating dysfunctional mitochondria into a separate pool of mitochondria that are unable to fuse and undergo autophagy.

Based on the previous discussion, we propose the following conceptual model for mitochondrial dynamics. First, we assume that the total mitochondrial volume density (the percentage of muscle fibber volume occupied by mitochondria) can be classified into four different groups: healthy or little damaged (H), moderately damaged (M), severely damaged (S) and fused (F) mitochondria. Given that cell division is energetically demanding, dysfunctional (severely damaged) mitochondria replicate at a much smaller rate than functional ones, and the persistence or their damaged genome can cause severe problems. However, we assume for the sake of simplicity that only healthy and moderately damaged mitochondria can proliferate. Concomitantly, the model takes into consideration that the volume density of healthy and moderately damaged mitochondria increases due to mitogenesis. The Lotka-Volterra model for interspecific competition between species, in which all four mitochondria subpopulations compete for resources, is employed to model mitogenesis. Mitochondria are subject to constant damage due to multiple factors, like oxidative stress. This process is modeled as linear fluxes from compartment H to compartment M, and from compartment M to compartment S. Severely damaged mitochondria are degraded via mitophagy; a process that is also assumed to be linear. Given that fusion is Ψ_m-dependent (with Ψ_m the potential across the mitochodrial inner membrane), we assume that only healthy and moderately damaged mitochondria may fuse. The rate at which fusion takes place is assumed to be proportional to the mitochondrial volume density. To account for the fact that mitochondria randomly exchange material while fused, we assume that fission gives rise to equal amounts of healthy, moderately damaged and severely damaged mitochondria. The rate at which fission occurs is assumed to be directly proportional to the volume density of fussed mitochondria.

The mitochondrial dynamics clockwork depicted in the formed paragraph (which is schematically represented in Figure 1) can be translated into the following set of Ordinary Differential Equations (ODEs):

With N = H + M + S + F. In the equations above, H, M, S, and F, respectively, stand for the volume density of healthy or little damaged, moderately damaged, severely damaged, and fused mitochondria, while N denotes the total mitochondrial volume density. The first terms in the right hand side of Equations (1), (2) represent mitogenesis flows into the populations of healthy and moderately damaged mitochondria. The expressions for these flows implicitly assume that mitochondrial growth is logistic and that all mitochondrial subpopulations compete for resources with the same efficiency—see for instance [20], Section 3.5. Accordingly, parameter a represents the intrinsic mitogenesis rate (i.e. the rate of mitogenesis when resources are abundant), while K is the volume-density carrying capacity (the maximum total mitochondrial volume density cells can sustain). Finally, taking into consideration that the rest of the flows accounted for in Equations (1)–(4) are linear, the meaning of the corresponding parameters is as follows. Parameter b is the mitochondrial damaging rate, c is the fusion rate for healthy and moderately damaged mitochondria, d is the fission rate, and e is the mitophagy rate.

It is convenient to work with non-dimensional variables and parameters to simplify the analysis of the model dynamic behavior. To this end, define the following normalized variables:

and parameters:

With this, Equations (1)–(4) can be rewritten as

Where x′ denotes the derivative of variable x with respect to time t′. Notice that variables h, m, s, and f correspond to the volume density of healthy, moderately damaged, severely damaged and fused mitochondria, normalized to the carrying capacity, K. Moreover, parameters β, γ, δ and ϵ, respectively, stand for the damaging, fusion, fission and mitophagy rates, normalized to the intrinsic mitogenesis rate.

It is straightforward to verify that (h,m,s,f)=0→ is a steady state of the ordinary differential equation system defined in Equations (5)–(8).

To find other positive fixed points, let us start analyzing two specific cases: no fission (δ = 0) and no mitophagy (ϵ = 0).

In the no fission case, Equations (5)–(8) become

Let (h^*, m^*, d^*, f^*) denote a stationary solution. Then, it must satisfy:

Under the assumption that h^* > 0, Equation (9) implies that 1 − n^* = β + γ. However, when this last result is substituted into Equation (10), it yields h^* = 0. Furthermore, substitution into Equations (12), (11) demonstrates that m^* = s^* = 0, while f can take arbitrary values that are ultimately determined by the initial condition. In practice, having only a population of fussed mitochondria is unachievable and so the trivial steady state is the only one available for the system in the no fission case.

Consider now the no mitophagy case (ϵ = 0). Under this assumption, the ordinary differential equation (ODE) system given by Equations (5)–(8) becomes

The equations determining the steady states in this case are

Since all model parameters are positive, it follows from Equation (15) that m^*, f^* = 0. These results, together with Equation (16) further imply that h^* = 0, while the value of s^* is determined by the initial condition. In practice, solely having severely damaged mitochondria is physiologically unattainable. Hence no other steady state than the trivial one exists in the no mitophagy case.

We have seen that in the absence of fission and/or mitophagy, the only available steady state in practice is the trivial one. This suggests that such processes are required for the maintenance of a functional mitochondial network. To gain a better understanding of the significance of these parameters, let us look for positive fixed points in the general normalized model.

The algebraic equations determining the stationary solution of the ODE system given by Equations (5)–(8) are:

By solving for f^* in Equation (20) and substituting into Equations (17), (18) we get a couple of equations involving variables h^*, m^* and f^*. If, for the sake of algebraic manipulation n^* is regarded as a constant parameter, such equations result to be a complete set of linear homogeneous equations for variable h^* and m^*. Hence, nontrivial solutions only exist when the determinant of the coefficient matrix equals zero. By doing the corresponding algebra, this last constrains yields a equation for n^* whose positive solution is:

Furthermore, by taking into consideration that the nontrivial solutions of the previously described homogeneous equation system are of the form

we get:

By substituting this last result into Equation (20) and solving for f^* we obtain:

Finally, substitution of this last result into Equation (19) allows us to solve for s^* to get:

If we now substitute Equations (21)–(24) into the definition of n^* (n^* = h^* + m^* + s^* + f^*), we can solve for h^* to obtain an expression for this variable in terms of the model parameters. Further substitution into Equations (22)–(24) allows us to do the same for variables m^*, s^* and f^*. The resulting algebraic expressions are omitted for the sake of brevity. Nonetheless the can be straightforwardly obtained and they constitute a nontrivial positive solution for the complete model normalized ODE system—Equations (5)–(8).

It is not hard to verify that this solution is consistent with the special cases studied in the former subsection. That is, h^* = m^* = s^* = 0 in the absence of fission (δ = 0), whereas f^* can take any value. However, one can argue from biological considerations that is has to be f^* = 0 (no fused mitochondria can be obtained without healthy of moderately damaged ones). Similarly, h^* = m^* = f^* = 0 without mitophagy (ϵ = 0), while the value of s^* is arbitrary from a mathematical standpoint, but null from biological considerations.

Another interesting result is that, according to Equation (21), the total mitochondrial volume density is only a function of the damaging (β) and fusion (γ) rates (normalized to the mitogenesis rates). In other words, it is independent of fission and mitophagy. Thus, our model predicts that modifying the rates of fission (δ) or mitophagy (ϵ) would change the way mitochondria distribute among the various populations, without affecting the total amount (see Figure 3).

The way n^* depends on parameters β and γ is illustrated in Figure 2. Notice that n^* is a decreasing function of β, while in general the curves n^* vs. β are concave with a single maximum. Further observe that n^* attains positive values in the interval [0, 3] when β = 0—see Equation (21), but this interval shrinks as the value of β increases.

To simplify the parametric analysis of the model, we make the assumption that, given the value of β, parameter γ takes on the value that maximizes the total mitochondrial volume density, n^*. By performing the necessary mathematical calculations, we can straightforwardly show that this assumption imposes the following restriction:

With this, n^* is only a function of β, while the value of variables h^*, m^*, s^* and f^* depends on parameters β, δ and ϵ.

Figure 3 depicts the impact of fission and mitophagy on the distribution of mitochondrial volume density across compartments. Specifically, the steady-state values of variables h, m, f, and s (normalized to the total volume density, n) are plotted against parameters ϵ and δ for a specific value of β. Observe that when the fission rate (δ) or the mitophagy rate (ϵ) increases, there is an increase in the volume densities of healthy (h^*) and moderately damaged (m^*) mitochondria. Further note that this effect is more significant for moderately damaged mitochondria. On the other hand, it can be observed that the volume density of fused mitochondria remains almost constant regardless of the mitophagy rate, but it rapidly decreases with increasing fission rate. In contrast, the volume density of severely damaged mitochondria remains almost constant regardless of δ, but rapidly decreases with increasing ϵ.

Of particular interest are the values of h^* and m^* because these variables represent the volume density of functional mitochondria. In this study, we do not regard fused mitochondria as functional, as they represent a temporary state involving the exchange of mitochondrial components. Since changes of parameters δ and ϵ do not affect the value of n^* but only the way the total mitochondrial volume density distributes among different subpopulations, is it important to investigate the effect these parameters have upon the fractions of healthy, h^*/n^*, and moderately damaged mitochondria, h^*/n^*. It is possible to prove that both quantities are monotonic decreasing functions of β and monotonic increasing functions of δ and ϵ, but for simplicity we refer the reader to Figure 4 to discuss this behavior. There, we plot the fraction of functional mitochondrial volume density, which defined as Φ = (h^* + m^*)/n^*, as a function of parameters δ and ϵ, for a couple of different values of parameter β.

Note in Figure 4 that Φ = 0 when either δ = 0 or ϵ = 0, in agreement with previous results. This confirms that, according to our model, mitochondrial fission and mitophagia are essential for maintaining a functional mitochondrial volume density. Further observe that Φ decreases as β increases, for all values of δ = 0 and ϵ = 0. That is, increasing the rate of mitochondrial damaging not only causes a decrement of the total mitochondrial volume density, but also diminishes the fraction of functional mitochondria. Finally, the value of Φ augments when δ and/or ϵ, are increased. Indeed, by increasing the rates of mitochondrial fission and/or mitophagy, it is possible to at least partially revert the decrease of ϕ originated by an increment of the mitochondrial damaging rate.