Figure 3

The resolution of the enumeration problem. A) The process of enumeration: The hyperpath given by running FindAll on the hypergraph of Figure 2D. This hyperpath is not minimal as can be verified by running Minimize without constraints (i.e. R _f = {∅}). Two minimal hypergraphs connect v₈ to the source nodes v₁, v₄: the one containing hyperarcs R₄, R₇ and the one highlighted in the figure containing only the hyperarcs R₃, R₄; B) Scheme of resolution of the enumeration problem for biosynthesis of liquiritigenin whose hypergraph is represented in Figure 1. Each node in the scheme corresponds to a call to FindPath and contains the constraint sets R _f and R _n (see left bottom box), and the minimal hyperpath $\mathcal{P}$ corresponding to the given constraints. FindPath iteratively calls itself, the structure obtained is a rooted tree where each node represents a call to FindPath, and each call is characterized by the sets R _f and R _n . In particular there are no constraints on the pathway searched at the root node, corresponding to the first call to FindPath, this fact is expressed by having empty R _f and R _n . The hyperpath found is R₁, R₂, R₄, R₇. For each reaction in the hyperpath, a new call to FindPath is done, this time with the constraints induced by the hyperpath solution of the parent node. Processes that do not return hyperpaths $\left(\mathcal{P}=\left\{\varnothing \right\}\right)$ are not followed by calls to children processes, while the processes that return one hyperpath $\mathcal{P}$ have unconstrained reactions $\mathcal{P}\backslash R_f$ and are followed by as many children processes as unconstrained reactions. The children processes have as set of reactions that cannot belong to the returned pathway R _n ' the same as the father augmented by one reaction from the unconstrained set of the hyperpath $\mathcal{P}$ returned by the father process, while the set of reactions required to belong to the pathway R _f ' is given by the ones of the father augmented by all the reactions preceding the one added to R _n to get R _n '. Consider for instance the process giving as pathway $\mathcal{P}=\left\{R_1,R_2,R_5,R_9\right\}$ it contains two reactions not in R _f : R₅ and R₉ so its two children processes have R _n given by R₄ ∪ R₅ and R₄ ∪ R₉ and R _f is given by the one of the father: {R₁, R₂} union once with {∅} and once with R₅: the only unconstrained reaction preceding R₉ in the pathway $\mathcal{P}$.