Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
Review
. 2023 Jul 14;20(5):051001.
doi: 10.1088/1478-3975/acdcdb.

Mitochondrial networks through the lens of mathematics

Greyson R Lewis  1   2   3 , Wallace F Marshall  2   3
Affiliations
Review

Mitochondrial networks through the lens of mathematics

Greyson R Lewis et al. Phys Biol. .

Abstract

Mitochondria serve a wide range of functions within cells, most notably via their production of ATP. Although their morphology is commonly described as bean-like, mitochondria often form interconnected networks within cells that exhibit dynamic restructuring through a variety of physical changes. Further, though relationships between form and function in biology are well established, the extant toolkit for understanding mitochondrial morphology is limited. Here, we emphasize new and established methods for quantitatively describing mitochondrial networks, ranging from unweighted graph-theoretic representations to multi-scale approaches from applied topology, in particular persistent homology. We also show fundamental relationships between mitochondrial networks, mathematics, and physics, using ideas of graph planarity and statistical mechanics to better understand the full possible morphological space of mitochondrial network structures. Lastly, we provide suggestions for how examination of mitochondrial network form through the language of mathematics can inform biological understanding, and vice versa.

Keywords: applied topology; graph theory; mathematical biology; mitochondria; persistent homology; statistical mechanics.

PubMed Disclaimer

Figures

Figure 1.
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.
Figure 2.
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.
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.
Figure 4.
Figure 4.
Morphological operations in mitochondrial networks.
Figure 5.
Figure 5.
Mitochondrial networks of budding yeast (strain SRY123, mitochondria labeled with mRuby2, generous gift of S. Rafelski lab) grown in synthetic complete media with a carbon source of either glucose (A) or glycerol (B) were captured at 20 time points spaced 30 seconds apart using spinning disk confocal microscopy. Networks were converted into graphs using MitoGraph, processed using pynauty, analyzed via linear regression using statsmodels, and plotted using matplotlib and seaborn [, , –126]. Data (magenta X’s) from a previous study [48] is shown for data subject to the mitochondria-like network (‘MLN’) definitions (section 3.1) from this work (C) as well as in its totality (D). Black boxes and error bars designate means and one standard deviation of n 3 for individual values of n 1. Curves of best fit are shown by thicker black dashed lines, with 95% confidence intervals in thin black lines.
Figure 6.
Figure 6.
Processing the graphical structure of mitochondrial networks through a persistent homology pipeline reveals differences between the mitochondrial networks of wild-type and fission-fusion double knockout yeast. A graph extracted from a mitochondrial network (A) can be processed into PDs using measured tubule lengths (B, top, ‘weighted’) or assigning each edge a unit length (B, middle and bottom). If length is ignored, the graph can be considered either in its native unweighted form (B, middle) or converted into a fully-connected weighted graph using the graph geodesic distance (B, bottom), whose value is the shortest path between a specified pair of nodes. Single points on a persistence diagram may have multiplicity greater than 1. Once all graphs are processed, pairwise distances between PDs of different mitochondrial networks can be computed via the Bottleneck metric, leading to the generation of heatmaps (C). Data in (C) are separated into wild-type (top half of each heatmap) and double-knockout (bottom half), showing increased similarity of pairwise distances within a single strain as compared to different strains. H 0, H 1, and H 2 refer to homology groups in 0, 1, and 2 dimensions, respectively.
Figure 7.
Figure 7.
Linear scaling of edges with nodes in mitochondrial networks. The dashed lines in each panel demarcate the theoretical edge-node scaling limits of slope 12 (lower) and 32 (upper).
See this image and copyright information in PMC

Similar articles

See all similar articles

References

    1. Thompson D W. On Growth and Form Canto. Cambridge: Cambridge University Press; 1992.
    1. Karnkowska A, et al. A eukaryote without a mitochondrial organelle. Curr. Biol. 2016;26:1274–84. doi: 10.1016/j.cub.2016.03.053. - DOI - PubMed
    1. Zhang Z-W, Cheng J, Xu F, Chen Y-E, Du J-B, Yuan M, Zhu F, Xu X-C, Yuan S. Red blood cell extrudes nucleus and mitochondria against oxidative stress. IUBMB Life. 2011;63:560–5. doi: 10.1002/iub.490. - DOI - PubMed
    1. Green D R, Reed J C. Mitochondria and apoptosis. Science. 1998;281:1309–12. doi: 10.1126/science.281.5381.1309. - DOI - PubMed
    1. Wang C, Youle R J. The role of mitochondria in apoptosis. Annu. Rev. Genet. 2009;43:95–118. doi: 10.1146/annurev-genet-102108-134850. - DOI - PMC - PubMed

LinkOut - more resources