Large-scale computation of elementary flux modes with bit pattern trees

Motivation: Elementary flux modes (EFMs)-non-decomposable minimal pathways-are commonly accepted tools for metabolic network analysis under steady state conditions. Valid states of the network are linear superpositions of elementary modes shaping a polyhedral cone (the flux cone), which is a well-studied convex set in computational geometry. Computing EFMs is thus basically equivalent to extreme ray enumeration of polyhedral cones. This is a combinatorial problem with poorly scaling algorithms, preventing the large-scale analysis of metabolic networks so far. Results: Here, we introduce new algorithmic concepts that enable large-scale computation of EFMs. Distinguishing extreme rays from normal (composite) vectors is one critical aspect of the algorithm. We present a new recursive enumeration strategy with bit pattern trees for adjacent rays-the ancestors of extreme rays-that is roughly one order of magnitude faster than previous methods. Additionally, we introduce a rank updating method that is particularly well suited for parallel computation and a residue arithmetic method for matrix rank computations, which circumvents potential numerical instability problems. Multi-core architectures of modern CPUs can be exploited for further performance improvements. The methods are applied to a central metabolism network of Escherichia coli, resulting in approximate to 26 Mio. EFMs. Within the top 2% modes considering biomass production, most of the gain in flux variability is achieved. In addition, we compute approximate to 5 Mio. EFMs for the production of non-essential amino acids for a genome-scale metabolic network of Helicobacter pylori. Only large-scale EFM analysis reveals the > 85% of modes that generate several amino acids simultaneously.