Reconstructing biochemical cluster networks

Motivated by fundamental problems in chemistry and biology we study cluster graphs arising from a set of initial states S ⊆ ℤⁿ₊ and a set of transitions/reactions M ⊆ ℤⁿ₊ × ℤⁿ₊. The clusters are formed out of states that can be mutually transformed into each other by a sequence of reversible transitions. We provide a solution method from computational commutative algebra that allows for deciding whether two given states belong to the same cluster as well as for the reconstruction of the full cluster graph. Using the cluster graph approach we provide solutions to two fundamental questions: (1) Deciding whether two states are connected, e. g., if the initial state can be turned into the final state by a sequence of transition and (2) listing concisely all reactions processes that can accomplish that. As a computational example, we apply the framework to the permanganate/oxalic acid reaction.