Characterization and Design of Phase Spaces and Yield Spaces in Genome-Scale Metabolic Models

Production rates and yields are key parameters of biochemical transformation processes. While optimal rates and phase spaces are readily studied with flux-balance analysis (FBA) approaches, optimal yields and yield spaces are rarely systematically analyzed. Often elementary flux modes (EFMs) are used to characterize yields and yield-optimal pathways. However, EFMs characterize the unbounded flux cone and are incompatible with non-zero flux bounds and allocation constraints often used in FBA.

By resorting to the concept of elementary flux vectors (EFVs), it is possible to generalize the idea of unique metabolic pathways to also account for inhomogeneous linear constraints. We show that any rate-optimal FBA solution sits in an optimal polyhedron spanned by (certain) EFVs. This holds true not only for rate-optimal but for yield-optimal solutions too, which cannot be found by standard FBA approaches.

Next, we demonstrate that (optimal) yield spaces can be readily calculated even in genome-scale metabolic models by linear-fractional programing without explicitly enumerating EFVs. Although phase spaces and yield spaces often are of similar shapes (and therefore sometimes confused), they carry very different information. In a realistic analysis based on E. coli, we show how these complementary pieces of information can be used to understand and optimally shape the metabolic capabilities of cell factories with any desired yield and/or rate requirements.

We conclude that EFVs provide an unifying framework for the theoretical description and analysis of any constraint-based model under arbitrary linear constraints. More specifically, EFVs close the gap between biased FBA approaches and unbiased EFM approaches and allow one to fully characterize and shape metabolic phase spaces and yield spaces. This reinforces the fundamental importance of EFVs (or EFMs) as the ``coordinates of metabolism´´. However, an explicit enumeration of EFVs is not required as phase spaces and yield spaces can be efficiently computed even in genome-scale metabolic networks.