Assembly in Directed Hypergraphs
Christoph Flamm, Daniel Merkle, Peter F. Stadler
TL;DR
The paper reframes assembly theory as a hypergraph problem, showing that assembly pathways correspond to shortest $B$-hyperpaths and embedding assembly constructs within directed hypergraphs. By establishing a 1-1 correspondence with acyclic $B$-hypergraphs and extending to general hypergraphs, it connects assembly to grammar-based compression, retrosynthesis, and rule-based graph rewriting via $DPO$. It further develops computational approaches, including an ILP formulation and DP methods, and analyzes cyclization effects and alternative cost measures to assess synthetic and evolutionary complexity. The work unifies assembly with known hyperpath and synthesis-planning frameworks, enabling principled computation of complexity measures and opening avenues for applying assembly concepts to broader chemical reaction networks.
Abstract
Assembly theory has received considerable attention in the recent past. Here we analyze the formal framework of this model and show that assembly pathways coincide with certain minimal hyperpaths in B-hypergraphs. This makes it possible to generalize the notion of assembly to general chemical reaction systems and to make explicit the connection to rule based models of chemistry, in particular DPO graph rewriting. We observe, furthermore, that assembly theory is closely related to retrosynthetic analysis in chemistry. The assembly index fits seamlessly into a large family of cost measures for directed hyperpath problems that also encompasses cost functions used in computational synthesis planning. This allows to devise a generic approach to compute complexity measures derived from minimal hyperpaths in rule-derived directed hypergraphs using integer linear programming.