What is the function of mitochondrial networks? A theoretical assessment of hypotheses and proposal for future research

Summary

Mitochondrial quality control is the process that maintains a healthy mitochondrial population by identifying and degrading dysfunctional mitochondria 24, 72, degrading damaged mitochondrial components 73 and transporting damaged components out of the mitochondrion 74, 75. Three mechanisms contributing to this control are fusion, fission and mitophagy. To ensure that a degradation bias is present (dysfunctional mitochondria should be more likely to be degraded than healthy mitochondria), selection occurring in one of these mechanisms is sufficient. We argue that selective fusion combined with non-selective fission and non-selective mitophagy leads to the desired biased degradation and that in fact there are two levels of quality control: (i) control on the level of the mitochondrial network (non-selective fission and selective fusion) that does not require mitophagy and which we call ‘blind surveillance’; and (ii) control on the level of the cell using non-selective mitophagy. In this section, our terms ‘selective’ and ‘non-selective’ refer to selection based on mitochondrial function; we assume that mitophagy always selects on mitochondrial size (i.e. only small mitochondrial fragments are targeted for mitophagy). Size-dependent mitophagy is the source of non-linearity in this model.

Blind surveillance consists of two parts. The first part involves random (blind) fission events. By random we mean that these events are not selective, and therefore a functional mitochondrion is just as likely to become fragmented as a dysfunctional one. The second part is the surveillance, which involves selective fusion events: a functional mitochondrion is more likely to fuse 24. As a consequence, a functional mitochondrion will, on average, stay fragmented for a shorter period of time than a dysfunctional mitochondrion. With blind surveillance present, mitophagy does not need to be selective because any isolated mitochondrion is more likely to be dysfunctional than healthy and, therefore, even if mitophagy is non-selective, a bias towards degrading dysfunctional mitochondria is automatically established. The strength of this effect is increased in cases of higher mitochondrial fusion (see section S1.1).

Having dysfunctional mitochondria separated from the fused network is an advantage on its own, because a dysfunctional mitochondrion fusing with an already fused and functional network may have negative effects that are not present if the mitochondrion in question remains isolated. Blind surveillance thus forms the a first level of quality control (it separates dysfunctional mitochondria from the network by using non-selective fission and selective fusion), and mitophagy forms the second level by eliminating fragmented mitochondria (which are more likely to be dysfunctional – hence even non-selective mitophagy eliminates more dysfunctional than functional mitochondria).

Experimental support

Selection on the level of fusion (mitochondria with low Δψ are less likely to fuse 28, 29, 24) and mitophagy 76-79 is supported by observation. A possible mechanism for selective fusion is the increased processing of OPA1 by OMA1 when Δψ is low 80, 81, which then results in a lower fusion probability. Selective elimination of damaged mitochondria has been observed 82, 83, although the precise mechanisms are still incompletely understood 84.

Coarse-grained quantitative model

Mathematical models of mitochondrial quality control have been constructed previously 85-87 and show that fission, selective fusion and selective autophagy together increase mitochondrial functionality. To address the specific role of network state, we construct a simple model of ordinary differential equations based on populations of healthy and dysfunctional mitochondria (for details see section S1.1). Assumptions made in this model are that only fragmented mitochondria can be removed by mitophagy (which causes the non-linearity), the total number of mitochondria is constant, and spatial distributions of mitochondria are ignored.

If the rate at which healthy mitochondria become dysfunctional is particularly slow, then the model predicts a steady state where all the dysfunctional mitochondria have been removed by non-selective mitophagy. If healthy mitochondria are allowed to become damaged, then steady states with both functional and dysfunctional mitochondria exist. When fusion rate is increased, the steady state fraction of functional mitochondria increases. We thus quantitatively illustrate that large-scale selective fusion, without need for selective mitophagy or fission, has a beneficial effect on cellular mitochondrial populations.

Limitations/Critique

Autophagy is required for this mechanism to function, although this assumption also underlies other quality control mechanisms. Our model makes several assumptions (mentioned above). A possibility for future work is to relax these assumptions (by e.g. including spatial distributions of mitochondria 86).

Summary

Large-scale fusion facilitates sharing of mitochondrial machinery, as this machinery moves through the mitochondrial network. Increased sharing may equilibrate concentrations of nuclear encoded proteins to enable better control over mitochondria 39, promote coordinated behaviour between mitochondria to synchronize gene expression 1, 88 or improve the health of the mitochondrial population by allowing mitochondria to complement each other's deficiencies 89, 90 (functional complementation).

The effect of increased fusion (or decreased fission) on diffusion rate is most important for fast-diffusing species because they will have their diffusion rate capped by fission and fusion rates. By diffusion rate we mean the effective rate of diffusion through the mitochondrial network, which will depend on fission and fusion rates. As the network becomes more interconnected, a fast-diffusing species will rapidly explore the new available spaces. Small increases in p (i.e. small increases in network connectivity may lead to large, non-linear, increases in diffusion rate, especially near the percolating value p_c. This is because when p > p_c it suddenly becomes possible to diffuse from one extreme end of the cell to the other end, whereas when p < p_c the distance that can be travelled is much more restricted. A ‘switch-like’ response is thus present for fast-diffusing species. Slow-diffusing species will locally be exposed to a relatively rapidly fluctuating network structure; thus, their diffusion rate increases linearly with fusion level, meaning that there is no ‘switch-like’ response.

Experimental support

Functional complementation through fusion has been observed in numerous experimental studies 40, 89, 90, indicating that fused mitochondria do exchange contents. As mentioned before, the effect of increased connectedness of mitochondria is likely to be largest for fast-diffusing species. We therefore compare diffusion constants of several mitochondrial proteins (and mtDNA) in Table S3.

Coarse-grained quantitative model

We can study the diffusion of species along mitochondrial networks by simulations in which the network is represented by a 2D or 3D lattice. Mitochondria are represented by the nodes in the network, and they can be connected through the edges. The edges can flicker ‘on’ and ‘off’, representing fusion and fission events respectively, with rates λ_fus and λ_fis. In this model, each mitochondrion has the potential to fuse with any of its direct neighbours independently, and does so with rate λ_fus (similarly for fission). The previously-defined value p broadly determines how likely it is that an edge is ‘on’. If the hopping rate for molecules moving between adjacent points on the network is given by λ_dif, and particles are restricted to move only along ‘on’ edges, then the average time it takes a particle to diffuse across a certain distance can be calculated in dynamic hyperfused (p_c < p < 1), mesofused (0 < p < p_c) and microfused (p ≈ 0) states (see section S1.2). Our coarse-grained model makes several assumptions, including constant mitochondrial mass (corresponding to a short timescale), an absence of mitophagy and a lattice topology (Ref. 58 provides an alternative approach).

In Fig. 2A, it is shown how the apparent diffusion coefficient varies with p for different values of τ, where τ is the relaxation time of the fluctuating bonds 91, 92 and is defined as (a lower τ means that bonds flicker more frequently between the states ‘on’ and ‘off’). This figure is based on a mean field approximation, and similar results are produced by our agent-based stochastic model. In static percolation (τ = ∞), an abrupt change in diffusion rate is present at p = 0.5 (in 2D, assuming a square lattice), but when bonds fluctuate in time, the diffusion rate changes less abruptly. For a given p, letting bonds fluctuate faster has the effect of increasing the diffusion rate, and this effect is largest around p = 0.5. The simple model shows that a microfused state results in a smaller diffusion rate than a hyperfused state, even if the microfused state has very frequent fusion and fission events. Even slowly-varying hyperfused networks thus afford more facility to spread elements throughout the cell, confirming that a hyperfused morphology confers a mixing advantage for proteins.

Limitations/Critique

Complementation may be sufficiently achieved without a need for large-scale fusion: short ‘kiss and run’ events are claimed to be responsible for most mixing because of their high frequency 43. In the spirit of ‘back-of-the-envelope’ calculations in biology 93, we use rough estimates to assess the feasibility of equilibration. The duration of association of these fusion events approximately ranges from 4 seconds to 5 minutes, with a mean of 45 seconds 43. The diffusion coefficient of Green Fluorescent Protein was measured to be 20–30 µm²/s 94, 95, meaning that in 45 seconds a distance of about can be travelled, which is more than enough to accomplish equilibration. However, even though complementation of matrix components is feasible, kiss and run events may be too short to allow for sharing of membrane proteins and nucleoids.

Summary

Mitochondrial fusion may increase ATP production, through one or more of several possible mechanisms. We suggest the following possible mechanisms of particular interest: (i) higher [ATP] is caused by fusion-induced changes in inner membrane shape; (ii) higher [ATP] is caused by fusion-induced decreases in proton leak; (iii) higher [ATP] is caused by fusion-induced decreases in mitochondrial degradation; (iv) higher [ATP] is caused by the non-linear response of ATP synthesis rate to membrane potential. (i–iii) are discussed in more detail in the SI. We note that an assumption made in all four hypotheses is that fusion causes in higher [ATP]. However, it remains to be determined whether mitochondrial fusion is the cause of an observed increase in [ATP], or whether hyperfusion and high ATP concentrations have a common cause, e.g. a recent study suggested that an increase in ATP production may precede mitochondrial fusion 96.

Experimental support

Several studies suggest that hyperfusion increases ATP levels and mitochondrial respiratory capacity 8-11. In nutrient-rich environments mitochondria tend to be fragmented, whereas under starvation they are observed to form elongated networks 9, 12, which can be interpreted as an attempt to enhance energy production in challenging environments. Hyperfusion is also observed at the G1/S transition, before energetically costly DNA replication 8. It is worth noting that in hyperfused states, concentrations of cellular ATP – not just the rate of ATP production – were measured to be higher 10. These increased concentrations suggest a physiological role for fusion beyond that of meeting extra ATP demand. This role could be to act as a cellular ‘accelerator pedal’, because an increase in ATP/ADP ratio in the cell has the consequence that many reactions will go faster 97, 98. Further support for specific hypotheses is described in sections S2.2.1, S2.2.2 and S2.2.3.

Coarse-grained quantitative model

We focus mathematically on hypothesis (iv) which we have not seen discussed elsewhere. We first consider how the membrane potentials of two individual mitochondria Δψ₁, Δψ₂ are related to the overall membrane potential of their fused product Δψ₁₊₂. A simple model is that the fused product inherits the arithmetic average of the two individuals, i.e. . However, physical calculations based on considering charged capacitors suggest a range of possibilities depending on modelled cristae structure where (see section S1.4). We do not claim that this capacitor model is a particularly realistic one (in particular, it does not necessarily capture the behaviour of mitochondria fusing in chains), but we use it to give a quantitative example of the possibility of non-averaging potentials when two charged objects are combined.

Fusion has a clear advantage in the case that . We argue that if potentials do average (i.e. ), it is still possible that total ATP synthesis rate increases (or decreases) upon fusion. This advantage is due to a sigmoidal dependence of ATP synthesis rate (r_ATP) on Δψ 99, 100 (see Fig. 3A). Referring to the illustration in Fig. 3A, fusion of mitochondria B and C causes an increase in r_ATP (i.e. r_ATP is larger post-fission than pre-fission) whereas fusing A and B causes a decrease in r_ATP and fusion of C and D leaves r_ATP approximately constant. The effect of fusion on rate of ATP synthesis therefore depends on the magnitude of the potentials of the pre-fusing mitochondria. Having mitochondria like A fusing to others causes net decreases in energy production, and this is a possible reason for why mitochondria with low Δψ have a lower fusion probability 28, 29, 24.

Limitations/Critique

The causal relationship between higher ATP levels and mitochondrial fusion is still incompletely understood, and it may be possible that fusion is the effect of high [ATP] instead of the cause. A recent study showed that the rate of inner membrane fusion was closely correlated with oxygen consumption, and that this rate increased during OXPHOS stimulation as the result of increased OPA1 cleavage by Yme1l 96. Also, in the model discussed above we have considered the role of Δψ in ATP synthesis rate, whereas actually the proton motive force (defined as ) drives ATP production 99, 100. The region of the ATP synthesis sigmoid at which mitochondria lie is not yet experimentally clear. Further critique of specific hypotheses is described in sections S2.2.1, S2.2.2 and S2.2.3.

Summary

Fusion averages out calcium concentrations across the mitochondrial network, giving rise to two advantages: (i) calcium concentrations are placed in an intermediate regime where small changes have pronounced effects on enzymatic rates, and thus tight bioenergetic control of the mitochondrial population is facilitated; and (ii) the amount of mitochondrial volume that experiences a rise in [Ca²⁺]_matrix is increased, hence boosting total energy production.

Calcium is known to stimulate certain enzymes of the TCA cycle 101-103, which in turn influences the rate of ATP production. The dependence of TCA cycle activity on calcium is sigmoidal 104, which means that the cell has maximal control over this activity when [Ca²⁺]_matrix is in the steeply rising part of the sigmoid curve, as indicated in Fig. 4A. If [Ca²⁺]_matrix is high enough to reach the plateau region of the sigmoid, the enzymes are saturated. Only mitochondria located near microdomains of high [Ca²⁺] (e.g. ER calcium channels) are thought to take up calcium efficiently 105, 106. The calcium concentration in these mitochondria rises significantly during ER depletion or calcium entry into the cell. This rise in [Ca²⁺]_matrix can be such that the saturation level is reached, and increasing [Ca²⁺]_matrix further has no effect on enzyme activity. Because fusion averages out concentrations inside the matrix, fusing a calcium-saturated mitochondrion with one with low [Ca²⁺]_matrix has the effect of shifting [Ca²⁺]_matrix towards the more controllable region. Mitochondria far from high [Ca²⁺] microdomains with lower [Ca²⁺]_matrix increase stimulation of their TCA cycles by fusing with a mitochondrion with high [Ca²⁺]_matrix. Fusion therefore increases the amount of mitochondrial volume affected by a calcium signal, which subsequently increases total energy production.

Experimental support

The idea that fusion increases the number of mitochondria that experiences a given calcium signal in the cell has been suggested before in Ref. 107.

Coarse-grained quantitative model

We focus on the uptake of calcium from the ER as the most important influence on mitochondrial calcium levels, assuming that without fusion, the mitochondria near ER calcium channels have saturated calcium-dependent TCA cycle enzymes, and the mitochondria far from the ER or plasma membrane have low [Ca²⁺]_matrix (as in Fig. 4A). The first few fusion events will increase total enzyme stimulation (and therefore ATP production) because the mitochondria fusing to the calcium saturated ones experience a significant increases in [Ca²⁺]_matrix and enzyme activation, whereas the saturated mitochondria experience little change in enzyme activation, despite their drop in [Ca²⁺]_matrix (because of the sigmoidal dependence of enzyme activity on [Ca²⁺]). Adding more and more mitochondria to the fused chain will ultimately lead to a decrease in enzyme stimulation (because [Ca²⁺]_matrix will reach the lower part of the sigmoid), meaning that there exists a certain number of fused mitochondria that leads to maximal enzyme activation. This number depends on the amount of calcium released by the ER, the number of mitochondria near the ER, the basal [Ca²⁺]_matrix present in mitochondria far from the ER, the exact shape of the sigmoid, and, furthermore, relies on the assumption that when many mitochondria are fused, [Ca²⁺]_matrix becomes so diluted as to reach the low enzyme activity regime. There will of course be many other factors that influence this ‘optimal number of fused mitochondria’ (e.g. the ratio of phosphate bound calcium to free calcium): we merely present the basic idea, which is also shown in Fig. 4B (for details see section S1.5).

Limitations/Critique

This hypothesis is based on several assumptions that are debated in the literature. While several studies show that higher calcium concentrations lead to increases in NADH production and oxygen consumption 68, 108, 109, mathematical modelling 56 and uniporter studies 110 suggest that calcium perturbations may have little effect on respiration. Other studies have observed that calcium only has an effect on NAD(P)H concentrations in glucose-stimulated conditions 111, or that only a single TCA enzyme is controlled by calcium 104. The mechanisms of calcium uptake that are mainly used, the dependence of these mechanisms on calcium concentrations, and the number of sites close to sources of calcium are also debated 105, 112 (for details, see section S1.6). Mitochondria have been observed to experience calcium transients, meaning that they extrude calcium quite rapidly after having absorbed it 53 and the simple model we discuss in section S1.5 may therefore only be valid on short timescales. Finally, calcium signalling linked to mitochondrial ultrastructure is by no means the only, or simplest, way in which the cell regulates its energy status.

Summary

By changing their morphological state, mitochondria can make themselves less susceptible to perturbations (for example, fluctuations in electrochemical membrane potential). The fused neighbours of a mitochondrial element can act as a buffer for biochemical or physical fluctuations. If failure of individual mitochondria has severe consequences for the cell, this increased robustness through fusion will be beneficial when mitochondria are subject to perturbations.

Experimental support

There are some experimental indications that fused mitochondria are better protected against stress 113. This hypothesis could account for why hyperfusion occurs during stress 10, and may also be an explanation as to why fusion protects the cell against apoptosis 10, 13-16 (or at least delays apoptosis 114), because prior to cytochrome c release (which induces apoptosis) remodelling of the cristae structure has been seen as a cause of changes in membrane potential 115. If a larger network protects against fluctuations in membrane potential, it may be able to prevent cristae remodelling, thereby potentially delaying or preventing apoptosis.

In stressful conditions, it may thus be disadvantageous for the cell to have its mitochondria fragmented because all of them are vulnerable to membrane potential fluctuations. If the cell fuses its mitochondria, they become more robust to the fluctuations, which may prevent the failure of many individual mitochondria, and subsequent cell death.

Coarse-grained quantitative model

A biophysical calculation considering the change in membrane potential upon changes in permeability of the inner mitochondrial membrane (see section S1.7.1) suggests where r is the radius of the mitochondrion and depends on the structure of the membrane (the more invaginated the membrane, the larger y). This result suggests that for large mitochondria, fluctuations of Δψ caused by transient perturbations will be of smaller magnitude. Fusing mitochondria thus protects them from perturbations, as illustrated in Fig. 3B. This model assumes spherical mitochondrial geometry. Small fragmented mitochondria are often seen to have a spherical shape and the fusion of two small spherical mitochondria may produce a mitochondrion that itself has an approximate spherical shape. The model discussed here may therefore be applied to fusion events involving small mitochondrial fragments.

We provide an alternative model that is independent of volume and surface area scaling and involves picturing mitochondria as individual agents coupled with spin-like interactions 116. This model, in which a mitochondrion prefers to be in a similar state as its neighbours, shows that groups of mitochondria are less likely to undergo a catastrophic loss of function than individuals (see section S1.7.2).

Limitations/Critique

Even though fluctuations in larger mitochondria (fused mitochondria) may be of smaller relative magnitude, they will occur more frequently because of the larger surface area which increases the probability that, for example, a pore opens or an ETC component fails. Stress does not always lead to a fused mitochondrial state, but can also lead to mitochondrial fragmentation 117, 118. It can be that the level of stress is important, and that too much stress leads to fragmentation and subsequent apoptosis.