Mitochondrial networks through the lens of mathematics

In order to understand the relationship between mitochondrial network form and function, we must have a way of quantifying each aspect. As networks, mathematics and physics provide tools and representations that can and have been used for the quantification of mitochondrial network form. These tools and representations provide more than just a means of describing the networks, however; prior work in physics and mathematics can inform our understanding of physiological steady states (via the Perron–Frobenius theorem), ergodicity (through state-space representation of networks), and variation (using global and local spatiotemporal measures) in mitochondrial networks [75, 76].

Graph theory and topology offer some of the most useful tools and representations for quantifying and comparing mitochondrial networks.

Using graph theory, we can represent mitochondrial networks at increasing levels of complexity, from unitless geometric representations at one end to spatially embedded and constrained tubules at the other. It also provides tools for deriving unique representations of individual networks, as well as local and global metrics for describing different features of those networks [71].

Tools from applied topology, while not typically used in the consideration of mitochondrial networks, provide representations for describing networks and their essential properties across multiple length scales [77]. Of these representations, we will in this work focus on the application of 'persistent homology' and its associated metrics, which provide broadly applicable and easily computable 'barcodes' that themselves can be compared with their distances measured in an informative manner [78, 79].

Additionally, following up on questions raised in the previous subsection, a better understanding of the mathematical structure underlying the space of 'mitochondrial' graphs may give additional insight into what we see under the microscope. How much do mitochondrial networks vary across a population of cells? How similarly do networks evolve in different cells? Are there steady-state distributions of mitochondrial network structures we expect to see in large populations of cells, and if so, how do they compare to networks seen over time in a single cell (i.e. is network evolution an ergodic process)?

The remainder of this work is organized as follows. Firstly, we will review some of the extant literature, give an overview of outstanding questions regarding mitochondrial network morphology and dynamics, and briefly describe relevant experimental methods. Secondly, we will explore mathematical and physical questions raised by the study of mitochondrial networks, emphasizing methods of representation and measurement, as well as provide a link to state spaces and their associated statistical mechanics. Lastly, we will describe some biological questions raised by the study of mitochondrial networks through the lenses of mathematics and physics, with an aim of describing potential links between network morphology and cellular function.

Firstly, advances in experimental techniques, in particular wider-spread adoption of microscopy with high spatiotemporal resolution (e.g. iSIM and lattice light-sheet), enable us to capture mitochondrial network dynamics on an event-by-event basis for longer periods of time than ever before [63, 80–85]. Secondly, the low cost of computational infrastructure makes storage, processing, and analysis of such data not just possible but reasonable; additionally, the same infrastructure can be used to simulate mitochondrial networks in silico, accelerating the experiment-theory feedback cycle and allowing it to be implemented within a single laboratory [86–91]. Lastly, recent widespread adoption of so-called 'black box' machine learning techniques have catalyzed a corresponding movement for understanding biological systems using interpretable theory-based approaches, and the intersection of mathematics and mitochondrial networks provides opportunities for study using both classes of techniques, along with the potential to build bridges between them.

Networks that change over time are the subject of study across a wide range of fields, from the study of social interactions networks in sociology, to neuronal rewiring within brains in neuroscience, and beyond [92–94]. However, our understanding of the 'rules' guiding the evolution of those networks is still quite narrow.

How can we develop better theory and tools for the investigation of complex, evolving networks? Ideally, we would test new hypotheses on real-world datasets of an addressable size that change in limited, understandable ways. Mitochondrial networks satisfy both of these criteria: the number of nodes and edges is computationally addressable (on the order of hundreds or fewer in S. cerevisiae) and there exist fewer than ten types of structural modifications that can occur within the network [48]. Also, the morphological changes that take place on mitochondrial networks, in combination with the rules limiting network structure, form a heretofore uncharacterized class of graphs: planar graphs exclusively of degrees 1 and 3. Having a biologically-inspired generative mechanism for these graphs complements the traditional combinatorics-based approaches to characterizing graph classes.

These considerations immediately lead to further questions, What is the space or range of network structures, i.e. graphs, that mitochondria take on inside living cells? How does the space and variety of mitochondrial network structures differ from that of mitochondrial-like networks? What is the structure of mitochondrial network space under the restriction of 'biological' graph modifications?

Although a great deal of research has been done on mitochondrial structure and function, we will limit our brief review of the literature to prior work that has included efforts at quantitatively modeling the mitochondrial network. These studies are not large in number. Susanne Rafelski and collaborators developed the segmentation and analysis software MitoGraph (figure 1), which has enabled them to show (among other results) that mitochondrial network volume scales linearly with cellular volume, that mitochondrial networks resemble 'geographical networks,' and the fission–fusion double knockouts exhibit quantitatively distinct network properties from those in wild-type yeast [47, 48, 61]. Meyer-Hermann et al developed a graph-theoretic algebraic framework for describing mitochondrial networks and their relationship with the mammalian cytoskeleton [68, 95]. Shirihai et al have conducted studies on the relationship between mitochondrial dynamics and quality control, as well as how those concepts might be related to network structure [96, 97]. Johnston et al have published many papers pertaining to mtDNA population dynamics; some of their more recent publications examine possible benefits of mitochondrial network structure from the perspectives of both cellular physiology and mitochondrial mutational burdens [38, 98–101]. Chialvo et al combine the theoretical framework from Meyer-Hermann's group and progressive coarse-graining of mitochondrial images to study mitochondrial network structures through the lenses of criticality and percolation theory [49, 50]. Quinn et al use a range of representations of mitochondrial networks, such as dynamic social networks and embeddings derived from graph convolutional neural networks, to evaluate changes in mitochondrial morphology over time [102–105]. The Mitometer algorithm from Lefebvre et al uses graph features to help track the motion of mitochondria [91].

In wild-type S. cerevisiae, mitochondria are located beneath the spheroidal cell cortex, inducing mitochondrial networks to be planar (due to the concordance between the surface of a two-dimensional sphere and the two-dimensional Euclidean plane joined with a point at infinity) [48]. We note that the planarity claim above may not hold in some cell types due to substantial differences in cytoskeletal structures, mitochondrial tethering, and other biological features, but in budding yeast it is physically enforced by the tethering of mitochondria to the cell cortex [65].

Graph representations of mitochondrial networks by both Sukhorukov and Rafelski involve assigning nodes to three-way junctions of tubules and ends of tubules, then inserting an edge between two nodes when they are physically connected by a continuous membrane [47, 48, 68, 95]. As a result, we can classify mitochondrial networks as undirected, unweighted, simple (no edges from a node to itself) planar graphs with nodes of degrees 1 or 3 that are not necessarily connected. To our knowledge, there does not exist a specific classification in mathematics corresponding to this set of graphs, and we therefore refer to it here as the set of 'mitochondria-like graphs.' However, a well-studied class of graphs forming a subset of the mitochondria-like networks are cubic graphs, which are constructed exclusively from degree-3 nodes. Although much is known about cubic graphs, the incorporation of pendant (degree-1) nodes into such graphs allows for the possibility of boron tree-like mitochondrial networks whose structures correspond to boron tree graphs (unrooted binary trees with nodes of degrees 1 and/or 3) [106]. These structures may also emanate from edges of cubic graphs or serve as bridges between them, though not all such constructions are allowed. As a result, the space of possible mitochondrial graph structures is substantially greater in size than the numbers of cubic graphs and boron trees for a given number of nodes. Another related structure is the Bethe lattice, consisting of single loop-free component with infinite nodes all of which are degree three, which can be seen as an infinite limit of the boron tree. Viana et al showed that Bethe lattices exhibit features similar to fission-fusion double-knockout mitochondrial networks, likely due to those mutants' typically tree-like mitochondrial networks [48]. At the other end of the spectrum, Brown et al modeled particle search times for intact and 'decimated' honeycomb networks, demonstrating that increasing the number of loops in such networks lead to continual improvements in efficiency [45].

Given this set, we can construct a corresponding state space ('mitochondrial network state space') by assigning different (non-isomorphic) graphs as unique states and allowing transitions between two states if a single morphological change (fission, fusion, etc) performed on the first graph induces an isomorphism between the modified and second states. Our choices of an unweighted-graph representation and isomorphism-based difference are not unique; we discuss additional representations later in the article.

Having selected a quantitative representation of mitochondrial network structures, we are now confronted with the problem of how to compare those structures. The binary evaluation of whether two mitochondrial network structures are identical is known in mathematics as the graph isomorphism problem. Although in general the time complexity of evaluating whether two graphs are isomorphic is now believed to be quasi-polynomial, planarity reduces the time to the square of the number of vertices or less [107, 108]. Planar graphs can be given unique identifiers (canonized) that have length in the order of the logarithm of the number of vertices and efficient software exists to generate these canonizations [109]. The ability to index all possible mitochondria-like graphs also provides a framework for counting the distribution of graphs in real datasets. Although great progress has been made in the development of planar graph algorithms, many basic questions are still unanswered.

Given these definitions and background, we can now ask deeper-reaching questions. Some of these questions are primarily mathematical in nature. For example, how many mitochondria-like graphs are there, for a given number of nodes? As with the general case of enumerating unlabeled planar graphs, an analytical expression to calculate this number has not yet been reported. Upper and lower bounds can be established, however.

The planarity of mitochondria-like graphs enables us to use extant results from the full family of planar graphs to set a loose upper bound of $30.061 ^ {N}$, which was derived using information-theoretic methods [110].

For a lower bound, we can consider the sum of two classes of graphs. The first is the set of unlabeled cubic outerplanar graphs, which only have nodes of degree 3 and can always be drawn in a 2D plane such that all nodes are adjacent to the outermost face of the graph. For example, a pyramid or tetrahedron with nodes assigned to its corners is not outerplanar when flattened onto the plane because it must always contain a node drawn within the boundary of that drawing. We use the size of this set as a lower bound because there does not presently exist an asymptotic scaling relation for unlabeled cubic planar graphs [111]. The second component of the lower limit is the set of 'boron trees,' which are trees with nodes of degree either one or three [112, 113].

Combining these bounds gives the following expression:

$$\begin{align*} &\frac{1.255 * (2.48)^{|V|}}{|V|^{2.5}} + \frac{0.009099 * (7.5036)^{|V|}}{|V|^{2.5}} \nonumber\\ &\quad \leqslant |M_{G}(|V|)| \lt 30.061^{|V|}, \end{align*}$$

where $|M_{G}(|V|)|$ is the number of mitochondria-like graphs with $|V|$ vertices. These bounds are loose, as they were obtained for simple connected graphs, while mitochondria-like graphs may be composed of multiple connected components that may not belong to the classes of cubic outerplanar graphs or boron trees.

It is possible to outline a recursive procedure for determining the number of mitochondria-like graphs for a given number of nodes. We reiterate that mitochondrial-like networks are not necessarily connected in a single component: at any given point in time, there is no guarantee that every node or edge is accessible to all of the others. Therefore, to calculate the number of unique (non-isomorphic) mitochondria-like graphs, we can use the following relation:

$$\begin{align} |M_{G}(|V|)| = \sum_{K \in p(|V|)} \prod_{k \in K} \left(\begin{array}{c}|M_{G}(k)| + m(k,K) - 1\\ m(k,K)\end{array}\right) , \end{align} \tag{ 3 }$$

where $p(|V|)$ is the number of partitions of $|V|$, which enumerates the unique ways to generate the positive integer $|V|$ from positive integers; K is a partition of $|V|$ unique up to permutation; k is a member of that partitioning; and $m(k, K)$ is the multiplicity of k in K (that is, how many times k appears in K).

Conceptually, we can interpret (3) as follows. A mitochondria-like graph with $|V|$ nodes may consist of a single connected component containing $|V|$ nodes, two connected components (separated networks) containing $|V| - k$ and k nodes, three connected components containing $|V| - j - k$ and j and k nodes, etc. If we add up the number of mitochondria-like graphs that contain each of the smaller number of nodes and add that to the number of mitochondria-like graphs that consist of a single connected component containing N nodes, we should end up with the total number of possible mitochondria-like graphs. However, some partitions may include multiples of the same number (e.g. $4 = 1 + 1 + 2$, with 1 showing up twice), and so we must account for identical smaller networks appearing in duplicate (requiring calculation of combinations with replacement).

Carrying out this computation is non-trivial: although the recursive definition allows previously stored results to be re-used, the process of generating all connected and non-isomorphic mitochondria-like graphs for a given number of nodes must be done exhaustively. Although we would like to take inspiration from the underlying biological processes, any strategy for doing so must take into account the constraints on morphology of mitochondria-like graphs in terms of planarity and degree. For example, it must be noted that a theoretical 'fusion' event can lead to the generation of a non-planar graph from an originally planar mitochondria-like graph (figure 3).

Zoom In Zoom Out Reset image size

This observation immediately leads to additional mathematical questions. Under what morphological operations (e.g. fission, fusion, etc) do graphs form closed sets? Can we define a fusion operation that is guaranteed to result in a mitochondria-like graph, and if so, how? One possible solution for defining a fusion process that respects biological constraints such as planarity is to represent graphs embedded in space, replacing operations on graphs with operations on projections. Some considerations related to embedding mitochondria-like graphs in space will be discussed below. Does such a fusion function, in combination with the other morphological operations, enable the generation of all mitochondrial-like networks? What features, measures, or heuristics of these graphs exhibit the most variance or are most informative?

Before moving on, we believe it worth noting that the graph-based description of mitochondria, while generally effective, is not necessarily comprehensive. For instance, donut-shaped mitochondria with toroidal skeletons have been reported in the literature as a result of cellular stress (e.g. hypoxia) and can be generated through the fusion of the two ends of an isolated rod-like mitochondrion or the fusion of two rod-like mitochondria, but our formulation does not permit such a structure [41, 114–117]. Using more generalized representations, such as pseudographs that allows loops and multiple edges between the same pair of nodes or hypergraphs, may more effectively and completely describe mitochondrial network structure [118–122]. At the time of writing, however, relatively few computational tools exist to conduct analyses using those representations.

In the preceding section we described how to enumerate the possible mitochondria-like networks that could exist subject to the constraints on graph structure. This enumeration leads to the question of which mitochondria-like graphs can actually be produced by known physical processes that operate on real mitochondria. Given the processes that sculpt the mitochondrial network, can one reach any network in the space of possible networks, starting from any other network? Is the answer to the prior question different if those processes occur randomly in space and time? In other words, are all mitochondria-like graphs reachable starting from any particular graph? Second, can we predict the distribution of mitochondria-like graphs that should be observed at steady-state? The first question falls into the realm of evolutionary graph theory while the second represents statistical mechanics.

Given the unweighted graph representation of mitochondrial networks, we can explicitly construct and define mitochondrial network state space.

We first define the states. Here, mitochondrial networks will be designated as different states if their unweighted graph representations are non-isomorphic. Two states are accessible (adjacent) if there is a single morphological change that can convert one state into the other.

Mitochondrial networks are known to undergo seven types of morphological changes, with the results of changes to minimal relevant structural elements (e.g. a single edge, component, etc) illustrated in figure 4. In depicting these local operations, we assume that in a single step only the shown nodes and edges can undergo a change. The possible operations are:

• (i)  
  Fission

  • (a)  
    Type 1: Fission maintains the number of connected components
  • (b)  
    Type 2: Fission breaks a component into two new connected components
• (ii)  
  Tip-tip Fusion (TT-fusion)

  • (a)  
    Type 1: Fusion maintains the number of connected components
  • (b)  
    Type 2: Fusion joins two connected components into one new connected component
• (iii)  
  Tip-side Fusion (TS-fusion)

  • (a)  
    Type 1: Fusion maintains the number of connected components
  • (b)  
    Type 2: Fusion joins two connected components into one new connected component
• (iv)  
  Outgrowth: a new tubule emerges from the side of an existing tubule
• (v)  
  Resorption: an extant tubule is absorbed into one of its adjacent tubules
• (vi)  
  Mitophagy: a network component is degraded
• (vii)  
  Vertex Flip: two adjacent degree-3 nodes transiently merge and re-equilibrate in a flipped orientation.

Zoom In Zoom Out Reset image size

How do these changes relate to one another? Firstly, we note that fission and TT-fusion are inverse operations, as are outgrowth and resorption. Mitophagy does not have a single-operation inverse, as there is no de novo mitochondrial biogenesis in the absence of extant mitochondria, though fission on isolated two-node components may serve as a functional inverse. Additionally, we assume that only isolated mitochondrial tubules, consisting of two degree-1 nodes connected by an edge, are subject to mitophagy and thereby destroyed. In order to account for this, we introduce the dispersion parameter α, defined as the fraction of degree-1 nodes that are adjacent to other degree-1 nodes. It is unclear whether TS-fusion has a single-operation inverse: it is possible that fission could take place at a three-way junction, but there is not yet biological evidence to support this as a sufficiently common occurrence; alternatively, in the case of pendant edges, resorption could serve as a specialized inverse operation. Vertex flips alter the local configuration of a mitochondrial network but do not change the number of each type of node, enabling more rapid exploration of graph structures.

Equipped with states and permissible transitions, we can represent mitochondrial network state space itself as a graph: the nodes are non-isomorphic mitochondrial-like networks and an edge is constructed between a pair of nodes if one of the above morphological operations converts one node into the other. A standard unweighted graph may not be the best representation of this space, however: it is reasonable to associate directionality with state transitions, as morphological operations are not their own inverses, suggesting a directed-graph state space representation. Similarly, there may be multiple morphological operations that convert one state into another, in which case a choice must be made whether to aggregate equivalent operations together, or to allow multiple (potentially directed) edges in the structure (enabled via a hypergraph, multigraph, or pseudograph structure). Lastly, assigning edge weights in state space can be useful: these could correspond to the multiplicity of morphological operations (i.e. fission on distinct edges leading to the same resultant graph), experimentally measured transition rates, and/or other relevant properties.

Construction of this state space leads to further questions, some of which are more biological in nature. What are the fundamental properties of mitochondrial network state space? Does the state space exhibit some form of detailed balance? Are the dynamics of mitochondrial network morphology ergodic? How does the set or space of mitochondrial networks that cells 'sample' compare to the space of mitochondria-like graphs? If two or more cells exhibit the same mitochondrial network 'state' at some point in time, how do the future trajectories of those cells through mitochondrial network state-space differ? If the observed distribution of graphs is substantially different from that predicted by the model, it would suggest that the model is missing one or more additional processes or else is missing regulatory linkages governing the processes: for example, a dependence of one or more of the rates on one or more structural features of the graph.

We can begin to address these questions through the development of mass-action-like equations describing the change in the number of different node types over time, with n₁ and n₃ corresponding to the number of nodes of degrees 1 and 3 in the network, respectively. Although the number of each node type is a non-negative integer, we can formulate equations in the continuum limit using a chemical-reaction-theory-like approach for the purposes of steady-state analysis.

We make two sets of assumptions. First, that TT- and TS-fusion are both second order processes, as nodes require other nodes or edges for fusion. Second, that fission, mitophagy, outgrowth, and retraction are all at most first-order, as these events do not seem to require interaction between multiple nodes but may depend on the overall size of the network. These lead to rate equations of the following form (though the exact terms may differ):

$$\begin{align} \textrm{Fission }\, \beta_{\textrm{fis,1}} = 2 k_{\textrm{fis}} |E| = k_{\textrm{fis}} \left ( n_{1} + 3 n_{3} \right ). \end{align} \tag{ 4 }$$

$$\begin{align} & \textrm{TT-Fusion }\, \beta_{\textrm{ttf,1}}\nonumber\\ & \quad = -\frac{1}{2} k_{\textrm{fus,tt}} \Bigg [ \alpha n_{1} \left ( \alpha n_{1} - 2 \right ) + 2 \alpha \left ( 1 - \alpha \right )n_{1}^{2} \phantom{\beta_{\textrm{ttf,1}} } \phantom{=} \nonumber\\ & \qquad + \frac{1}{2}\sum_{v_{i} \in L(G)} \bigg [ 2 ( 1 - \alpha ) n_{1} - 2 \nonumber\\ & \qquad - \sum_{v_{j} \in \{N_{2} ( v_{i} ) \cup N_{3} ( v_{i} )\}} {\Big ( 3 - \textrm{deg}(v_{j}) \Big )} \bigg ] \Bigg ]. \end{align} \tag{ 5 }$$

$$\begin{align} &\textrm{TS-Fusion }\, \beta_{\textrm{tsf,1}} \nonumber\\ &\;\; = -k_{\textrm{fus,ts}} \Bigg [ \alpha n_{1} \bigg ( |E| - 1 \bigg ) + \big (1 - \alpha \big ) n_{1} \bigg (|E| - 3 \bigg ) \Bigg ] \nonumber \\ \phantom{\beta_{\textrm{tsf,1}}} &\;\; =-\frac{1}{2} k_{\textrm{fus,ts}} n_{1} \left [ n_{1} + 3n_{3} + 4 \alpha - 6 \right ]. \end{align} \tag{ 6 }$$

$$\begin{align} \beta_{\textrm{tsf,3}} &= -\beta_{\textrm{tsf,1}} \nonumber \\ \textrm{Outgrowth } \beta_{\textrm{out,1}} & = \beta_{\textrm{out,3}} = k_{\textrm{out}}|E| \nonumber\\ & = \frac{1}{2} k_{\textrm{out}} \left( n_{1} + 3n_{3} \right).\qquad \end{align} \tag{ 7 }$$

$$\begin{align} \quad\!\textrm{Resorption } \beta_{\textrm{res,1}} &= \beta_{\textrm{res,3}} = -k_{\textrm{res}}(1 - \alpha) n_{1}. \end{align} \tag{ 8 }$$

$$\begin{align} \textrm{Mitophagy } \beta_{\textrm{aut,1}} & = - 2 k_{\textrm{aut}} \alpha.\qquad\qquad\quad \end{align} \tag{ 9 }$$

Here, α is the dispersion fraction defined above and $\beta_{X,Y}$ represents the rate of process X acting on nodes of degree Y. In the formula for TT-Fusion (5), L(G) refers to the leaves (degree-1 nodes) of the graph G and $N_{k} (v_{i})$ is the set of nodes in the k-neighborhood of v_i (the set of nodes accessible after k hops). The sum over neighborhoods is performed in order to avoid prohibited fusion events within each connected component, which lead to self-loops or multiple edges between the same pair of nodes. We describe mitophagy as occurring at a constant rate, though it can only affect the relevant sub-population of degree-1 nodes. The rate constants k_i are experimentally determined. We must note that these equations are unlikely to describe the full behavior of geometrically-constrained mitochondrial networks, as the breadth of fusion events permitted by this theoretical framework can allow tubules to fuse that would not be able to do so in situ, leading a prohibited network structure due to a loss of graph planarity.

Combining rates for n₁ and n₃ enables us to capture the net rate of creation and destruction of those node types, leading to rate equations of the following form:

$$\begin{align} \frac{\mathrm{d} n_{1}}{\mathrm{d} t} &= \beta_{\textrm{fis,1}} + \beta_{\textrm{ttf,1}} + \beta_{\textrm{tsf,1}} + \beta_{\textrm{out,1}} + \beta_{\textrm{res,1}} + \beta_{\textrm{aut,1}} \end{align} \tag{ 10 }$$

and

$$\begin{equation} \frac{\mathrm{d} n_{3}}{\mathrm{d} t} = \beta_{\textrm{tsf,3}} + \beta_{\textrm{out,3}} + \beta_{\textrm{res,3}}. \end{equation} \tag{ 11 }$$

Taking the steady-state limit of $\frac{\mathrm{d} n_{1}}{\mathrm{d} t} = \frac{\mathrm{d} n_{3}}{\mathrm{d} t} = 0$ for (10) and (11) leads to:

$$\begin{equation} \beta_{\textrm{fis,1}} + \beta_{\textrm{aut,1}} = 2\beta_{\textrm{tsf,3}} - \beta_{\textrm{ttf,1}}. \end{equation} \tag{ 12 }$$

By invoking our prior assumptions about the reaction orders of mitochondrial reconfiguration processes, we can rewrite (12) in terms of n₁ and n₃ to form the most general model describing those processes:

$$\begin{equation} a_{0}n_{1} + a_{1}n_{3} + a_{2} = b_{0} n_{1} ^{2} + b_{1} n_{1} n_{3} + b_{2} n_{1} \end{equation} \tag{ 13 }$$

Here, the parameters a_i and b_j are linear combinations of rate constants from (4)–(9). We exclude a first order term in n₃ on the right-hand side (RHS) of (13) as it would require tip-to-side fusion to increase in probability if the number of edges increases, regardless of the number of degree-1 nodes, while both types of nodes are required to execute the fusion operation. Similarly, we also exclude a constant term from the RHS of (13) as tip-to-side fusion is impossible for the many states lacking degree-1 nodes, prohibiting a constant baseline probability of fusion.

Rearranging (13) leads to an expression for n₃ in terms of n₁ at steady state:

$$\begin{equation} n_{3} = \frac{-b_{0} n_{1} ^{2} + \left ( a_{0} - b_{2} \right ) n_{1} + a_{2}}{b_{1} n_{1} - a_{1}}. \end{equation} \tag{ 14 }$$

Examination of n₃ vs. n₁ for mitochondrial networks measured in live yeast (figure 5) reveals generally positive scaling of n₃ as n₁ increases, implying that the leading term of a suitable model will have a positive coefficient as $n_{1} \rightarrow \infty$. The leading term of (14) implies that $n_{3} \approx \frac{-b_{0}}{b_{1}} n_{1}$ as $n_{1} \rightarrow \infty$. To satisfy the positivity requirement, either: (1) exactly one of b₀ and b₁ is negative and the other non-zero, or (2) at at least one of b₀ and b₁ is 0. If b₀ is negative, fusion must be less likely as n₁ increases. Alternatively, if b₁ is negative, then at least one of TT- and TS-fusion must decrease in probability as both n₁ and the number of edges increase. We do not believe either of these conclusions to be likely and therefore reject the model described by (14).

Zoom In Zoom Out Reset image size

Therefore, we must consider the cases in which at least one of b₀ and b₁ is 0, and in doing so obtain the following expressions:

$$\begin{align} n_{3} &= \frac{b_{0}}{a_{1}} n_{1}^{2} + \frac{b_{2} - a_{0}}{a_{1}} n_{1} - \frac{a_{2}}{a_{1}} \qquad \quad &b_{0} \neq b_{1}=0 \end{align} \tag{ 15 }$$

$$\begin{align} n_{3} &= -\frac{a_{0} - b_{2}}{a_{1}} n_{1} - \frac{a_{2}}{a_{1}} \qquad \quad &b_{0}=b_{1}=0. \end{align} \tag{ 16 }$$

We note that, as any product of n₁ and $E = n_{1} + 3 n_{3}$ must contain nonzero $n_{1}^{2}$ and $n_{1} n_{3}$ terms, $b_{0} = 0$ implies $b_{1} = 0$.

Following a similar approach as before for model (15), we find that $a_{1} \gt 0$, implying that fission and possibly mitophagy increase in probability in first-order as E increases. Conversely, in model (16) we must have either $a_{1} \lt 0$ or $a_{0} \lt b_{2}$. Both of these scenarios are permissible, so both models are potentially consistent with the data. However, fitting the data to a quadratic model with its associated constraint failed to converge, with the quadratic coefficient tending towards zero. We therefore reject the quadratic model (15) and conclude that, in this study, $b_{1} = 0$, indicating that a mass-action approach to modeling the data does not describe a mechanism of fusion that is second-order in n₁ and n₃. Biologically, this implies (through the lens of a mass-action framework) that TT-fusion should occur broadly while TS-fusion is either rare, slow, or (more likely) a predominantly local process. If TS-fusion is either rare or slow, outgrowth should be the primary generator of three-way junctions.

In contrast, a linear model fits each of the evaluated datasets with high significance but low R², implying that this model can explain only a small proportion of the data. Indeed, this model fails to capture the relatively high mean value of n₃ at the lowest value of n₁ as shown in figure 5(B). However, based on our assumptions, the linear model can be used to describe fission and mitophagy. This may be because fission and mitophagy are carried out using components that are not embedded in the mitochondrial membrane, potentially satisfying the well-mixed assumptions of mass-action kinetics, unlike a local fusion process. However,

We leave to a further study the exact determination and interpretation of the coefficients $a_{i},b_{j}$. However, using this approach, it may be possible to obtain values for k_fis and k_fis from the values of a_i and b_j based on network structures alone. If so, the results could be compared to directly measured experimental values of those rate constants obtained with microscopy.

Representing mitochondria as graphs allows the powerful apparatus of graph theory to be brought to bear on this organelle, but doing so comes at the price of ignoring some aspects of shape that are likely important for real mitochondria. Two key features ignored by the graph representation are the physical lengths of the edges and the embedding of the graph in 3D space. Integrating tubule lengths or volumes into a mitochondrial network state space is likely quite useful, as doing so should introduce a conservation law on total mitochondrial volume within a cell on short timescales (i.e. before a mitophagy event) [47]. However, because length and volume are continuously varying quantities (as opposed to the discrete number of nodes and edges), state spaces integrating these quantities should also be continuously varying. Sukhorukov et al address this through the introduction of a minimal mitochondrial unit, which has a length of approximately $0.5~\mu\mathrm{m}$ (corresponding to the diameter at which mitochondria convert from tubules to spheroids) [68]. While this approach enables effective analysis and predictions, it also restricts mitochondrial fusion to a discrete set of positions along tubules. What other approaches could be used to combine tubule lengths and network structure into a single, cohesive framework?

One appealing candidate, persistent homology, comes from the field of applied topology. Persistent homology describes the persistence of topological features of a network, such as the number of connected components or cycles, as the scale of resolution of the representation changes [78, 127, 128]. The key idea is that 'noisy' features will rapidly emerge and disappear as the length scale changes, while fundamental topological features of the data should persist for much longer. To compute persistent homology, both a topological representation of the network and an associated means of describing distances within the network are selected, then a filtration parameter is varied across the range of possible distances. The output of this analysis is a set of barcodes describing the distances scales over which n-dimensional holes (a hole in two dimensions, a void in three dimensions, etc) in the graph are present; these barcodes can be converted into a wide range of representations with well-established associated distance metrics for the study of unweighted, weighted, and directed networks [78]. We propose here the use of persistence diagrams to summarize mitochondrial network structures at a moderate level of complexity (ignoring spatial positioning but retaining length information) and then constructing a minimal state space that captures variation within and across cells over time.

We perform persistent homology analysis on mitochondrial networks as follows. As input, we take in either the unweighted or weighted graph representations of individual mitochondrial networks (figure 6(A)). From these graphs, we construct Vietoris–Rips complexes [127, 129]. Considering each node of the weighted graph as the center of a sphere, we take a parameter ε = 0 as the radius of the spheres and temporarily remove all of the edges from the graph. Next, we increase ε and reintroduce an edge to the graph once the value of ε reaches half the weight of that edge, i.e. when the circles of radius ε centered at each of the bounding nodes become tangent. Each time an edge is re-introduced, we compute the size of the homology groups of the modified graph structure. The process is equivalent for unweighted graphs, though methods for choosing how and when to reintroduce edges vary from all at once to with a delay based on some feature(s) of the nodes and edges (e.g. the presence of cycles, cliques, etc). The size of the 0th dimensional homology group counts the number of connected components of the graph. The size of the 1st homology group counts the number of cycles with at least four edges, because a three-edge loop forms the boundary of a 2-dimensional triangular simplex, which is considered to be a fundamental component for constructing simplicial complexes, rather than a void or hole. Similarly, higher dimensional homology groups have sizes corresponding to the number of higher-dimensional holes that are present [128, 130].

Zoom In Zoom Out Reset image size

With this in mind, we can consider the relationship ε and the size of the homology group. When ε reaches a value at which a new member of a homology group is introduced (e.g. a 4-cycle is formed), that construct is considered to have a 'birth' value equal to . As ε increases further, previously generated simplices may disappear, such as when two connected components re-join into a single component, or the splitting of a cycle into two via the re-introduction of an edge. The value of at which a component disappears occurs is known as the 'death' value for that component. Once ε reaches the maximum edge weight of the graph, the final edge is re-introduced, and the graphical structure no longer changes. By plotting the 'birth' and 'death' values for simplices across multiple homological dimensions, known as a persistence diagram (PD), we can get a sense of how the structure of the mitochondrial network varies over its associated length scales (figure 6(B)).

It is worth noting that this analysis is not necessarily very informative for an isolated unweighted graph, as all edges are re-introduced at a single step. However, by computing the shortest path between all pairs of nodes, then constructing a graph with the original nodes and new edges between all pairs of nodes with weights corresponding to the shortest path lengths (which may be infinite), we can perform a much more informative persistent homology analysis using this shortest-path graph (figure 6(B), bottom). Once the persistence diagram is obtained, it can be used to compare two graphs and quantify their difference. Here, we use the 'bottleneck' distance, which is the shortest euclidean distance d required for a perfect matching between points on two different diagrams such that the maximum distance between any matched pair of points is d [131]. This method of analysis can distinguish between wild-type mitochondrial networks and those in fission/fusion double knockout yeast strains, and also suggests that morphological differences within the analyzed populations are much smaller for double knockouts than for wild-type yeast (figure 6(C)).

Although the proposed state space is approximate, it captures network features across multiple scales and may provide insight into the full, latent mitochondrial state space. Persistent homology can also be extended to study the evolution of networks over time, enabling the detection of node movement along a tubule without changes to the underlying network structure [129, 132]. Although persistent homology is not commonly used in the analysis of biological datasets, there are examples of its success in the neuroscience and protein structure literatures, as well as introductions to its practical usage and implementations in multiple programming languages.

In deciding what mathematical features to use to describe graphs, it may be important to consider what aspects of structure are biologically important so as to be sure to preserve them in the representation. Although we have chosen one way to define the placement of nodes and edges in the construction of a mitochondria-like graph, other representations (such as placing nodes at the locations of mitochondrial nucleoids and edges between nucleoids in the same connected component) may prove fruitful. Also, the embedding of a graph in three dimensional space may be extremely important in terms of how the mitochondria are able to distribute biochemical products through the cell, etc. Topology provides additional tools for comparing networks embedded into surfaces. The graph-theoretic notion of isomorphism provides a necessary but insufficient condition for evaluating the equivalence of two embedded mitochondrial network structures, as a connected component may be contained with a face of a larger and different component, which would be considered identical under isomorphism to a scenario in which the smaller connected component is on the opposite side of a cell from the larger component.

To remedy this, we can use the topological concept of isotopy with fixed vertices, which (loosely) is a continuous deformation of one graph to another that preserves the relative connectivity of the edges and vertices while not introducing new loops or edge crossings [133]. Determining whether two embedded graphs are isotopic can likely be accomplished with a linear-time algorithm based on established work [108, 134–136]. Related and likely informative approaches also include homotopy groups, persistent homotopy, and analysis of polynomial invariants of mitochondrial-like networks [137–143]. Beyond topology and persistent homology, recent advances in the study of spanning trees could be applied to any graph representation of mitochondrial networks, particularly in the study of structural differences beyond isomorphism and isotopy, as well as for the deeper study of local structural features across networks [144–148].