Mixed-Integer Optimization for Loopless Flux Distributions in Metabolic Networks

TL;DR

Overall, the combinatorial Benders' decomposition is the most promising of the tested approaches with which it could solve most instances, however, the model size and numerical instability pose a challenge to the combinatorial Benders' method.

Abstract

Constraint-based metabolic models can be used to investigate the intracellular physiology of microorganisms. These models couple genes to reactions, and typically seek to predict metabolite fluxes that optimize some biologically important metric. Classical techniques, like Flux Balance Analysis (FBA), formulate the metabolism of a microbe as an optimization problem where growth rate is maximized. While FBA has found widespread use, it often leads to thermodynamically infeasible solutions that contain internal cycles (loops). To address this shortcoming, Loopless-Flux Balance Analysis (ll-FBA) seeks to predict flux distributions that do not contain these loops. ll-FBA is a disjunctive program, usually reformulated as a mixed-integer program, and is challenging to solve for biological models that often contain thousands of reactions and metabolites. In this paper, we compare various reformulations of ll-FBA and different solution approaches. Overall, the combinatorial Benders' decomposition is the most promising of the tested approaches with which we could solve most instances. However, the model size and numerical instability pose a challenge to the combinatorial Benders' method.

Figures (7)

• Figure 1: Simple model with internal loop
• Figure 2: Used reactions in (a) the FBA solution which contains an internal cycle, and (b) the ll-FBA solution.
• Figure 3: Comparing the number of BiGG instances optimally solved by the big-M and convex hull formulations, by solving \ref{['ll-FBA (big-M)']} directly, and by solving the problem with CB (CB (big-M MIS 0.5 %)). Note that some instances remain unsolved by the combinatorial Benders' approach, as shown in \ref{['Tab:termination_CB']} and \ref{['Tab:termination_mis']}.
• Figure 4: Performance of CB master problem variants to directly solving ll-FBA
• Figure 5: Performance of CB with multiple cuts, comparing the number of optimally solved BiGG instances of solving ll-FBA (indicator/big-M) directly and running CB. We experiment with the number of cuts added per iteration of CB depending on the instance size. MIS 0.5% means that at most $k$ cuts are added per iteration, where $k$ is 0.5% of the number of reactions of the model.