Pathways as basis vectors for the null space are useful for studying the capabilities of a network and to determine network properties. An alternative approach to characterizing the contents of solution spaces is uniform random sampling. This approach involves obtaining a statistically meaningful number of solutions uniformly distributed throughout the entire solution space and then studying their properties. Randomized sampling of candidate states throughout an entire solution space gives an unbiased assessment of its properties and can result in significant biological insights and understanding.

The Basics

A simple flux split Uniform random sampling can be illustrated by looking at a simple flux split (see Figure 14.1). The null space can be shown in two dimensions using its two conical basis vectors:

b1 = (1, 0, 1) and b2 = (0, 1, 1)

Note that the maximum weight that can be placed on these vectors is α1,max = 6 and α2,max = 8, whereas the minimum values in both cases is zero. There is an additional constraint association with reaction 3 that states that

α1,max + α2,max≤ α3,max = 14

thus both α1,max and α2,max cannot be attained simultaneously.

The null space can be enclosed with a parallelepiped by ignoring the constraint in Equation (14.2), which is a parallelogram in two-dimensions.