This monograph describes a mathematical model for the metabolic networks that are present in every living organism. The authors pursue a surprising connection between metabolic networks and the geometry of polyhedral cones. The notion of a connection like that is relatively new, but it is proving to be of real value.

Metabolism, the collection of chemical reactions inside living cells, extracts energy from
nutrients and uses it to build cell materials. The goal of this book is to present a structure of metabolic pathways that can be described quantitatively. This involves attention to elementary flux modes, which are fundamental metabolic pathways. Elementary flux mode analysis enables decomposition of a metabolic network into minimal functional units.

These are found to be closely related to extreme rays, which are geometrical descriptors of polyhedral cones. The cone defines all possible steady-state distributions for a metabolic network based on stoichiometric (and thermodynamic) constraints. Elementary flux modes are extreme rays of the cone. Every valid flux vector can be expressed as a non-negative linear combination of elementary flux modes.

A metabolic network can be encoded by a matrix – called a stoichiometric matrix. The objective of this monograph is to characterize all the flux vectors through this network. The admissible flux vectors for the network are describable as a subset of the null space of this matrix. As it turns out, this subset is a polyhedral cone. Finding the elementary flux modes is equivalent to finding the generators of the polyhedral cone – vectors for which all the elements of the cone can be written as linear combinations of them using only non-negative coefficients.

Polyhedral cones were first explored in the context of computational geometry, and they are a central concept here. Computing the generators is a combinatorial problem, and the usual method of solving it is the Double Description method (an algorithm first appearing in game theory), described in some detail by the authors.

The book focuses on mathematical and computational issues. No background in biology is expected. The authors aimed to make the text accessible to both mathematicians interested in biological systems, and biologists wanting to understand what the mathematical interpretation with polyhedral cones means and how it might be used. Linear algebra is the primary tool here. Basic knowledge of linear algebra – especially matrix theory – is required, and this includes the concepts of bases, rank, dimension, and null spaces. (These are reviewed in a brief appendix.) Proofs and some additional details are relegated to chapter notes.

Both mathematicians and biologists may find they have issues with this book. For mathematicians, it provides too little of the biology to be satisfying, while biologists – especially those without strong mathematical backgrounds - might be overwhelmed.

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry in diverse applications. He did his PhD work in dynamical systems and celestial mechanics.