A Semigroup Formalism for Biochemical Reaction Networks

Biological systems comprise processes on time and length scales ranging over many orders of magnitude; from chemical reactions with characteristic time scales in the order of nanoseconds to changes in gene expression requiring several hours. Modern experimental techniques enable high temporal and spatial resolutions in the measurements of single cells and suggest that both spatial and temporal domains in between the two mentioned extremes are densely filled with processes and structures. Such experiments are discussed as a motivation for the theoretical work. In particular, they demand possibilities of coarse-graining over several scales.

We develop a formalism for biochemical reaction networks using finite semigroups. It emphasizes the catalytic function of reactants within the networks thus can be used to decide whether the network is self-sustaining. Then, a correspondence between coarse-graining procedures and semigroup congruences respecting the functional structure is established. A family of congruences that leads to a rather unusual coarse-graining procedure is discussed: Thereby, the network is covered with local patches in a way that the local information on the network is fully retained, but the environment of each patch is no longer resolved. Whereas classical coarse-graining procedures would fix a particular local patch and delete detailed information about the environment, the algebraic approach keeps the structure of all local patches and allows the interaction of functions within distinct patches. Possible extensions of this work are sketched.