The FBA solution space kernel: introduction and illustrative examples

The solution space of an FBA-based model of cellular metabolism, can be characterised by extraction of a bounded, low dimensional kernel (the SSK) that facilitates perceiving it as a geometric object in multidimensional flux space. The aim is to produce an amenable description, intermediate between the single feasible extreme flux of FBA, and the intractable proliferation of extreme modes in conventional solution space descriptions. Fluxes that remain fixed are separated off while the focus of interest is put on the subset of variable fluxes that have a nonzero but finite range of values. For unbounded fluxes, a finite subrange that geometrically corresponds to the variable flux range is determined and is supplemented by a limited set of rays that encapsulates their unbounded aspects. In this way the kernel emphasises the realistic range of flux variation allowed in the interconnected biochemical network by e.g. limited nutrient uptake, an optimised objective and other model constraints. This work builds on the full presentation of the kernel approach in a research monograph.

Calculations are performed with the publicly available software package SSKernel, the source code and user manual of which is included as a supplementary file.

It is demonstrated how knowledge of the SSK and accompanying rays can be exploited to explore representative flux states of the metabolic network. Noting that bioengineering interventions such as gene knockouts modify the solution space, new tools based on the kernel analysis are presented here that predict the effects of such interventions on a target flux constructed to represent a desired metabolic output. A simple metabolic model is used first to demonstrate the special concepts and constructions needed to define and compute the SSK. The demonstration model is tweaked to produce typical behaviours of larger models, but with kernels in 1, 2 or 3 dimensions that are explicitly displayed to visualise the concepts. General applicability to models where visualisation is inaccessible, is illustrated by showing evaluation of potential bioengineering strategies for a genome scale model.

SSKernel is a flexible interactive tool that facilitates an overview of the FBA solution space as a multidimensional geometric object, in terms of a manageable number of parameters. It allows exploration of effects on this solution space from metabolic interventions and can be used to investigate bioengineering strategies to manipulate cellular metabolism.