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Figure 3 | BMC Systems Biology

Figure 3

From: Enumerating metabolic pathways for the production of heterologous target chemicals in chassis organisms

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 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 ( P = { } ) are not followed by calls to children processes, while the processes that return one hyperpath P have unconstrained reactions P\ 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 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 P= { R 1 , R 2 , R 5 , R 9 } 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 P.

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