Large scale methods to enumerate extreme rays and elementary modes

Approaching biological systems quantitatively using mathematical modeling techniques has gained increasing popularity in recent years. In systems biology, the biological complex is considered as a whole, as opposed to studying individual components and interactions. However, this can be very challenging even for simple bacteria, and various modeling techniques have been proposed approaching the enormous complexity at different levels. Simple methods at a high level of abstraction many times lack predictability and significance; detailed models capturing thermodynamic particulars are often limited due to gappy mechanistic knowledge and unknown kinetic parameters. Structural analysis as an intermediate attempt is based on usually wellknown network stoichiometries. Constraint-based methods like linear or convex optimization are able to predict reaction fluxes, growth rates or viability of knockout mutants with high confidence. However, their significance depends heavily on the underlying objectives, and alternative solutions in the flux space are mostly ignored. Comprehensive approaches exist, aiming at analyzing the flux space as a whole. The flux space is a high dimensional solution space for reaction fluxes, bounded by linear constraints on the flux values. Pathway analysis methods define minimal functional modes in the network that are able to comply with the constraints. All possible operation modes are superpositions of such elementary modes. Methods performing the analysis are transforming the descriptive constraints into generative basis vectors. Mathematically, the flux cone shapes a polyhedral cone, and algorithms arise from computational geometry, performing a representation conversion for the cone. Extreme ray enumeration, facet enumeration, vertex enumeration and convex hull are representatives of this family, and they are all related, if not equivalent. Unfortunately, the computation struggles with combinatorial explosion and is computationally intensive. At the beginning of this work, no implementation was applicable to genome scale metabolic networks. Improving the algorithms towards genome scale application is the declared goal of this thesis. The double description method is an algorithm to enumerate extreme rays of a polyhedral cone—or of elementary modes in biological terminology. It has proven efficient especially for degenerate problems, where points and constraints are not in general (e.g. random) position. Most biochemical networks lead to degenerate problems, hence the double description method is usually chosen for pathway computations. Our own implementation is also based on this method, and the present work describes the most important aspects to attain an efficient implementation. Various parts of the algorithm are performance critical and must be considered: it starts with input data, and we review and propose methods to sort and compress the data structures. We also show how to deal with different number types, since we need exact arithmetic for certain ill conditioned problem cases. A central part of the algorithm deals with elementarity of the