Studying organisational closure in biological systems with process-enablement graphs

At the heart of many contemporary theories of life is the concept of biological self-organisation: organisms have to continuously produce and maintain the conditions of their own existence in order to stay alive. The way in which these varying accounts articulate this concept, however, differs quite significantly. As a result, it can be difficult to identify self-organising features within biological systems, and to compare different descriptions of such features. In this article, we develop a graph-theoretic formalism – process-enablement graphs – to study the organisational structure of living systems. A process-enablement graph is a directed graph where the vertices represent processes, the edges represent direct enablements, and a cycle within the graph captures a self-organising component of a physical system in a general and abstract way. We use our notion of a process-enablement graph to provide a concise definition of organisational closure in the language of graph theory. Further, we define a class of graph homomorphism which allows us to compare biological models as process-enablement graphs. These homomorphisms facilitate a comparison of descriptions of self-organisation in a consistent and precise manner. We apply our formalism to a range of classical theories of life including autopoiesis, $(F,A)$-systems, and autocatalytic sets. We demonstrate exactly how these models are similar, and where they differ, with respect to their organisational structure. While our current framework does not demarcate living systems from non-living ones, it does allow us to better study systems that lie in the grey area between life and non-life.