An Algebraic Approach to Representing Mitochondrial Dynamics

Abstract This paper addresses the increasing need for comprehensive mathematical descriptions of cell organization by examining the algebraic structure of mitochondrial network dynamics. Mitochondria are cellular structures involved in metabolism that take the form of a network of membrane-based tubes that undergo continuous re-arrangement by a set of morphological processes, including fission and fusion, carried out by protein-based machinery. Because of their network structure, mitochondria can be represented as graphs, and the morphological operations that take place in the cell, referred to as mitochondrial dynamics, can be represented by changes to the graphs. Prior studies have classified mitochondrial graphs based on graph-theoretic features, but an alternative approach is to focus not on the graphs themselves but on the set of morphological operations inducing mitochondrial dynamics, since this may provide a simpler representation. Moreover, the operations are what determine the graphs that will be generated in a biological system. Here we show that mitochondrial dynamics constitute a groupoid that includes the automorphism group of each mitochondria graph. For multi-component mitochondria we define a graph structure that encapsulates the structure of mitochondrial dynamics. Using one of the morphological operations we can define an equivalence relation among mitochondrial graphs that allows us to replace the complex structure of the full groupoid with a vastly simpler groupoid representation based on equivalence classes. Using these formalisms we define a distance metric for similarity between mitochondrial structures based on an edit distance. In the course of defining these structures we provide a mathematical motivation for new experimental questions regarding mitochondrial fusion, the impacts of cell division on mitochondrial morphology, and the presence of a single giant component in some cell types. This work points to a general strategy for formulating a cell structure state-space, based not on the shapes of cellular structures, but on relations between the dynamic operations that produce them. Competing Interest Statement The authors have declared no competing interest. Footnotes improved bounds on distance metric; added a vector groupoid representation; added new figures to explain representations