Cells are highly complex structures that exhibit a wide range of geometrical features on scales ranging from symmetric protein complexes to entire organelles. Although relationships between form and function in biology have been appreciated at a conceptual level for centuries, it was only with the advent of modern mathematics and systematic measurement methods that cell biological systems could begin to be considered as quantitatively-describable entities within a framework that might mechanistically relate their forms and functions. While biology in the twenty-first century has advanced by leaps and bounds in its ability to identify and describe the components that comprise cell biological systems across multiple scales, our understanding of the origins of structures comprising these systems is still in its infancy.

During the scant hundred years since the publication of D'Arcy Thompson's seminal On Growth and Form, humankind's ability to observe, capture, and measure biological form has undergone a series of improvements spanning multiple orders of magnitude in both spatial and temporal resolution [1]. From the continued development of every facet of light microscopy, to our rapidly improving capacity to engineer fluorescent labels into living systems, to the creation of computational systems and structures enabling collection of enormous quantities of data, we are presently bombarded on a near-daily basis with new observations of living phenomena. In parallel, mathematics and physics have grown to such a point that the simulation of fluids in three dimensions is routinely possibly using a reasonably inexpensive desktop computer. Recent trends in artificial intelligence, particularly within the development of neural networks, have demonstrated the creation of new tools to measure, evaluate, and compare images at a heretofore-unseen pace in an automated manner.

Yet, for all of the discovery and innovation that has occurred over the past century, there seems to exist a gap between the studies of mathematics and biological form. This is due in large part to a focus of effort on discovering the molecules responsible for biological phenomena, which has proven to be highly productive. We have reached the point now that we are drowning in data, and it becomes more important than ever to start to reach a conceptual understanding of how biological form arises, a question that is inherently mathematical in nature. In line with this reflection should be an awareness that there exists an enormous variety of biological phenomena whose associated forms present rich opportunities for discovery. Here, we focus on one such form and opportunity: the study of mitochondrial networks.

The presence of mitochondria, the 'powerhouses of our cells,' is a near-unifying hallmark of eukaryotes—only in 2016 was the first eukaryotic organism (Monocercomonoides) discovered that lacked mitochondria [2]. Although there exist subsets and subtypes of eukaryotic cells that lack mitochondria, such as red blood cells, these are by and large exceptions [3]. Indeed, mitochondria are critical for a wider range of cellular functions than just generating ATP as fuel: they enable the critical self-destructive cell apoptosis response, help regulate calcium ion levels in conjunction with the endoplasmic reticulum (ER), generate energy from lipids through fatty-acid metabolism, as well as modulate the reactive oxygen species required by a range of physiological processes including pancreatic function and aging [4–11]. Given the pervasiveness of mitochondrial function in biological regulatory processes, it should not be surprising that mitochondrial dysfunction is associated with a wide range of human diseases, including Alzheimer's and Parkinson's disease [12].

Unlike most other organelles, mitochondria cannot be created de novo, but instead are derived from growing extant mitochondria. As a result, proper cycles of cellular division typically require the replication of mitochondrial DNA (mtDNA), which independently encodes a small number of proteins critical for mitochondrial function and may also serve other roles, such as innate immune signaling [13–15]. Copies of mtDNA are distributed across the set of all mitochondria in the cell and range in per-cell copy number from in human sperm cells to in human oocytes, though they are typically present on the order of 102–103 copies in many human cell types [16–18]. Although not yet fully understood, mtDNA seem to be associated with sites and specific types of mitochondrial fission in combination with ER contacts [19–23].

Although mitochondria are often depicted as bean-shaped objects, in fact in most cells they form ramifying networks of tubules (figure 1(A)). These tubular networks undergo constant rearrangement, both in structure and position, sometimes in concert with or as a result of interactions with other organelles [19–21, 24–26]. Much is known about the molecular and cellular processes that sculpt these networks [27–37]. However, the biological impacts of mitochondrial network form are not well understood, nor is it clear what determines the specific form of the network in any given cell type.

Figure 1. After 3D imaging (A), data from the mitochondrial marker channel is run through MitoGraph, which segments the mitochondria and establishes the graphical skeleton (B), leading to the weighted graphical structure (C). In (C), edges are not drawn to scale—instead, edge thickness denotes the relative mitochondrial tubule length.

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Why should we care about the structure of mitochondrial networks? A prominent hypothesis for the function of mitochondrial networks is that the combination of membrane potential-dependent fusion and nonspecific fission enables the cell to retain productive and degrade dysfunctional mitochondria [38], suggesting that network structure might reflect overall mitochondrial function. A variety of work has shown that a cell's mitochondrial network changes structure and dynamics in response to cell cycle progression, environmental perturbations, and cell state [39–41]. The extremes of eliminating mitochondrial fission and fusion lead to substantial changes in cellular physiology and fetal development [42]. Apoptosis is associated with mitochondrial fragmentation and swelling, while increased mitochondrial fusion in budding yeast may increase their replicative lifespan [43, 44]. Diffusive transport within networks is more efficient when loops are present, as is typically the case for wild-type mitochondria [45, 46]. Broadly speaking, it seems that major changes to mitochondrial morphology are associated with an enormous variety of altered cellular states.

To date, it remains unknown how mitochondrial dynamics induce or reflect changes in cellular state; by and large, only extreme and irreversible endpoints have been studied [47–50]. As a result, it seems that a better understanding of how mitochondrial dynamics are associated with transitions through cellular state space would at least provide a non-terminal imaging-based means of tracking such transitions and may even inform their nature. In order to describe the mitochondrial dynamics, we must have a way to quantitatively describe mitochondrial structure and morphological changes; additionally, such a quantitative description may provide a useful state space in which to track changes in cellular state. Using graph theory, topology, and mass-action-like kinetics, we provide a range of approaches to quantitatively describing mitochondrial structure and morphological changes.

In this article, we will focus on mitochondria of the budding yeast, Saccharomyces cerevisiae (S. cerevisiae). We focus on this organism for several reasons. First, yeast mitochondria are best understood in terms of genetics and molecular biology, with a host of known mutations that alter mitochondria morphology and function [51–60]. Second, the small size of yeast cells has enabled the complete mitochondrial network structure to be determined in individual cells, something that is far more difficult for larger mammalian cells [47, 48, 61–63]. Finally, and most importantly for our present purposes, the mitochondria of budding yeast are confined to the surface of the cell, which allows important constraints on the mathematical representation of their structure [64, 65].

Generally, the shape of an isolated mitochondrion exists on a spectrum ranging from nearly-spheres to tubules. These forms reflect the physical structures comprising and associated with mitochondria. As organelles bound by a lipid bilayer and containing fluid, small and isolated mitochondria typically revert to a minimal-energy spherical shape [42]. However, when bound to the ER, the plasma membrane, and/or the rigid rod-like structural proteins within a cell, individual mitochondria often take on a tubular form. The movement of bound molecular motors or other bound proteins can induce structural and corresponding morphological changes in mitochondria: this can take the form of either a change in tubule length or the drawing-out of a branching tubule from an existing mitochondrion (termed here as 'outgrowth').

Although mitochondria can be quantitatively characterized by a number of possible morphological descriptors, the fact that mitochondria are so often present in the form of interconnected tubules suggests that networks are natural descriptors of their overall shape and structure [66, 67]. In fact, the structure of such a network is mandated by a combination of physics (via membrane strain minimization) and mathematics (via graph theory).

If we represent the backbone of the mitochondrial network as a mathematical graph (figure 2), where we assign nodes (points, vertices) at the centers of Y-junctions and the ends of tubules, then join those nodes by edges (lines) if they are connected by a continuous tubule, we can immediately learn some network constraints.

Figure 2. (A) Graphs are mathematical objects made up of points (nodes, vertices) and lines (edges) connecting none, some, or all possible pairs of nodes. (B) The degree of a node in an unweighted graph is the number of edges emanating from that node: teal nodes are degree-1 (nodes A, D, and F) while magenta nodes are degree-3 (nodes B, C, and E). (C) Nodes and edges can have weights that correspond to useful quantities, such as edge weights representing the cost to travel between two city-nodes and node weights representing the cost of staying in that city. Edges can be undirected (no arrow) or directed (arrow), with travel from one node to another allowed if and only if there is an edge beginning at the first node and ending (arrowhead) at the second node. Cycles are paths through a graph that start and end at the same node (e.g. the loop formed by the magenta nodes and their edges to one another). The number of connected components of a graph is the number of disconnected islands of the entire graph: a graph consisting of two copies of the graph in (B) would have two connected components.

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Nodes of degree 1, corresponding to ends of tubules, are permitted. Two-way tubule junctions do not have a clear biological interpretation; Sukhorukov et al describe them abstractly as regularly-spaced locations along mitochondria where fusion can occur [68]. We see three-way Y-junctions in nearly every experiment, corresponding to nodes of degree 3. Four-way X-junctions, however, are rarely encountered in budding yeast mitochondrial networks (as opposed to e.g. 3D networks in cardiomyocytes). This may be due to the configuration being topologically unstable, such that a small perturbation at the four-way X junction can quickly resolve to two stable Y-junctions (a 'T1 elementary topological transition'), or due to differences in the Helfrich energies of the configures favoring two Y-junctions [69, 70]. Following the same reasoning, we suggest that junctions defined where mitochondrial tubules are fused will only transiently have a degree greater than 3.

Mathematics provides additional constraints. The handshaking lemma states that, if we add up the number of edges emanating from each vertex (the degree of each vertex), it equals twice the total number of edges in the graph,

where V(G) is the set of nodes of our graph G and is the total number of edges in G [71]. Since is an even number, and deg is always odd (the end of a tubule has a single edge emanating from it, and a three-way junction has three), there must necessarily be an even number of nodes in the graph.

We can go a step further: the Euler characteristic of a surface χ, a topological property of surfaces that depends on the number of holes or handles on the surface, is equal to the number of vertices (), minus the number of edges (), plus the number of faces (), minus the number of connected components () for a graph embedded on that surface [71, 72]. Because the network incompletely covers the sphere, the space it occupies is equivalent to a standard 2D plane, inducing a planar graph restriction on mitochondrial networks. In budding yeast, the mitochondrial network sits underneath the surface of a spheroid, and for such a closed surface, χ = 2. This leads to a relationship between , , and for the mitochondrial networks. Each face of our graph consists of at least three edges, and each edge can be shared by at most two faces. Therefore,

Combining these inequalities leads to (2), providing an upper bound for the number of edges in a mitochondrial network in terms of the number of vertices and connected components:

This result, which follows from Euler's formula for planar graphs, gives us yet another constraint on the structure of the mitochondrial network. By representing the network with a mathematical structure, we can rapidly limit the space of possible mitochondrial network structures that we have to consider inside budding yeast. The Universe of networks is further constrained via Kuratowski's theorem, which describes particular structures prohibited in the planar networks we consider here [71].

In many organisms, proteins exist and function to induce physical alteration of mitochondrial structure, both within a single tubule and across multiple tubules. The most well-known of these processes are fission, fusion, and mitophagy. Fission describes the division of a mitochondrial tubule, though not necessarily into two isolated mitochondria. Fusion describes the opposite process: the joining of two mitochondrial tubules to become a single unit. Mitophagy, or the autophagy of mitochondria, is a cellular process for degrading mitochondria, likely to remove dysfunctional entities from the cell. In addition to the processes described above, mitochondrial morphology may also be altered through 'resorption' (the absorption of a terminal or pendant mitochondrial branch into its adjoining tubule), though this process is poorly characterized; its inverse process is 'outgrowth,' described above [64].

Through a combination of the mitochondrial structural processes described so far, an enormous range of mitochondrial superstructures can be constructed, and indeed found, within cells. Such mitochondrial networks are readily visible through a high-power light microscope; under observation, these networks can be seen to undergo both large-scale motion and reconfiguration, demonstrating that the mitochondrial network of a cell is a dynamic system. This knowledge is not new; indeed, there are reports of mitochondrial networks published in the late 1800s [73, 74]. What remains unclear is what determines network morphology for mitochondria, in particular, are local processes like fission and fusion, taking place at random, sufficient to explain the networks that we see, or does the cell need to actively monitor and adjust its mitochondrial networks? Answering this type of question requires a way to predict the types of networks that a given set of morphological processes should produce, their statistical distribution, and the effect of mutations on the network organization. This, in turn, requires a conceptual framework for thinking about mitochondrial structures and dynamics that can bridge the gap between small scale local processes like fission/fusion and large scale properties like network connectivity. Graph theory provides such a framework, and our goal here is to explore the implications of graph theory results for mitochondrial structure, and vice versa.

2.1. How can studying mathematics and physics inform our understanding of mitochondrial networks?

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.

2.2. Why is it now an opportune time to heavily study the intersection between mathematics and mitochondrial networks?

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.

2.3. How can studying mitochondrial networks benefit mathematics and physics?

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?

2.4. What is the current state of the field?

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].

We begin with some definitions.

Networks are often represented as graphs (figure 2): mathematical structures consisting of points (nodes, vertices) that may be connected to one another by lines (edges) (figure 2(A)). The degree of a node counts the number of edges emanating from that node (figure 2(B)). Edges may be assigned weights (numerical values) describing some characteristic of that edge (e.g. tubule length, distance between rail hubs, pipe diameters, etc) (figure 2(C)). Graphs with weighted edges are known as 'weighted graphs' and those lacking weights as 'unweighted graphs.' Weighted graphs may be embedded into spaces whose axes have the same units as the edge weights; we call these 'spatially embedded graphs.' Increasing the information associated with the representation of a network (e.g. weights, locations) correspondingly increases the complexity of analysis. For that reason, we focus here mainly on unweighted graphs, though we do not mean to imply that weighted or spatially-embedded graph representations have no role in representing further aspects of mitochondrial networks [71].

3.1. Mitochondrial networks as unweighted graphs

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 , 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:

where is the number of mitochondria-like graphs with 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:

where is the number of partitions of , which enumerates the unique ways to generate the positive integer from positive integers; K is a partition of unique up to permutation; k is a member of that partitioning; and 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 nodes may consist of a single connected component containing nodes, two connected components (separated networks) containing and k nodes, three connected components containing 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. , 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).

Figure 3. Theoretical fusion event leading to non-planar graph. Mitochondrial-like networks must be planar, so the depicted fusion event is not biologically permitted, but is still mathematically possible, requiring additional constraints on fusion processes. Nodes represent physical three-way junctions and ends of mitochondrial tubules; edges signify a mitochondrial tubule connecting the two nodes. Yellow nodes from the planar graph on the left undergo fusion to form the non-planar graph on the right due to an unresolvable edge crossing. It is not possible to draw the resulting graph in a plane such that each red node has edges to each blue node.

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