In the last chapter, we discussed the simple topological properties of the network that the stoichiometric matrix represents. In this chapter, we look deeper into the properties of the stoichiometric matrix, and how fundamental network properties can be used to obtain a more thorough understanding of the reaction network that it represents. This material is perhaps the most mathematical part of this book. It should be readily accessible to readers with formal education in the physical and engineering sciences, while readers with a life science background may find it challenging. The stoichiometric matrix is a mathematical mapping operation (recall Figure 9.7). Matrices have certain fundamental properties that describe this mapping operation. These properties are contained in the four fundamental subspaces associated with a matrix. This chapter discusses these subspaces and how we can mathematically define them and begin the process of interpreting their contents in biochemical and biological terms.

Singular Value Decomposition

Singular value decomposition (SVD) of a matrix is a well-established method used in a wide variety of applications, including signal processing, noise reduction, image processing, kinematics, and for the analysis of high-throughput biological data [15, 169]. Unlike matrices composed of experimentally determined numbers, the stoichiometric matrix is a ‘perfect’ matrix that is commonly composed of integers describing the structure of a reaction network. SVD of S can be used to analyze network properties and it is a particularly useful way to obtain the basic information about the four fundamental subspaces of S.

11.1.1 Decomposition into three matrices

Mathematical format SVD states that for a matrix S of dimension m × n and of rank r, there are orthonormal matrices U (of dimension m × m) and V (of dimension n × n), and a matrix (of dimension m ×n) with diagonal elements Σ = diag(σ1, σ2, …, σr) with σ1 ≥ σ2 ≥ … ≥ σr > 0 such that:

S = UΣVT

where the superscript T denotes the transpose.