A reconstructed network can have many functional states even in the same environment. Constraint-based optimization problems can have multiple equivalent solutions that yield the same numerical value of the objective that is being optimized. This issue is an important one in characterizing networks and is a common occurrence in large-scale networks. In this chapter, we will cover the methods that have been developed to study what are called alternative optimal solutions (AOS). We demonstrate them by applying them to the core E. coli model, and discuss illustrative studies and issues that arise with finding AOS at the genome-scale.

Equivalent Ways to Reach a Network Objective

An historical note Even for explorations of network properties, such as those illustrated in the last chapter, alternative optimal solutions with the same optimal value of the objective function are found. This issue arose in the early days of the development of FBA approaches when two solutions for the formation of E4P were discovered (see Figure 20.1). The optimization that was being performed was to compute the maximal yield of the biosynthetic precursor E4P from glucose as a substrate. Two different solutions were computed that gave a yield of 1.33 molecules E4P per molecule glucose, or an 88.7% carbon conversion. The two solutions differ in the utilization of the TCA cycle or the glyoxalate shunt to achieve the same value of the objective.

The existence of AOS is a fundamental issue in COBRA methods and understanding their biological implications and relevance is important. Thus, a fair amount of effort has been devoted towards computing and studying AOS.

Characterizing alternative optimal states As outlined in the last chapter, LP can be used to find single optimal solutions. The solution to this optimization problem may not be unique, as illustrated in Figure 20.1, and AOS may exist. Various methods have been developed to characterize AOS.