[Paper Review] Removing both Internal and Unrealistic Energy-Generating Cycles in Flux Balance Analysis

Εlad Noor|arXiv (Cornell University)|Mar 13, 2018

Microbial Metabolic Engineering and Bioproduction30 references17 citations

TL;DR

This paper introduces semi-thermodynamic FBA (st-FBA), a novel constraint-based modeling approach that eliminates thermodynamically infeasible energy-generating cycles (EGCs) by imposing selective thermodynamic constraints only on energy currency metabolites—such as ATP, pyrophosphate, and proton gradients—without requiring extensive thermodynamic parameters. The method effectively removes internal and futile cycles while preserving biologically relevant flux distributions, offering a computationally efficient and biologically plausible alternative to both loopless FBA and full thermodynamic FBA.

ABSTRACT

Constraint-based stoichiometric models are ubiquitous in metabolic research, with Flux Balance Analysis (FBA) being the most widely used method to describe metabolic phenotypes of cells growing in steady-state. Of the many variants of constrain-based modelling methods published throughout the years, only few have focused on thermodynamic issues, in particular the elimination of non-physical and non-physiological cyclic fluxes. In this work, we revisit two of these methods, namely thermodynamic FBA and loopless FBA, and analyze the strengths and weaknesses of each one. Finally, we suggest a compromise denoted semi-thermodynamic FBA (st-FBA) which imposes stronger thermodynamic constrains on the flux polytope compared to loopless FBA, without requiring a large set of thermodynamic parameters as in the case of thermodynamic FBA. We show that st-FBA is a useful and simple way to eliminate thermodynamically infeasible cycles that generate ATP.

Motivation & Objective

• To address the persistent issue of thermodynamically infeasible flux cycles in Flux Balance Analysis (FBA), particularly those that generate ATP without substrate input.
• To overcome the limitations of existing methods: thermodynamic FBA (TFBA), which requires extensive thermodynamic parameters, and loopless FBA (ll-FBA), which fails to eliminate all energy-generating cycles.
• To develop a compromise method that enforces stronger thermodynamic constraints than ll-FBA but avoids the data-intensive parameterization of TFBA.
• To ensure that only biologically plausible flux distributions are retained, especially in genome-scale metabolic models.

Proposed method

• st-FBA extends standard FBA by introducing bounds on the standard transformed Gibbs free energy of formation ($\Delta_{f}G'^{\prime}$) exclusively for energy currency metabolites, such as ATP, pyrophosphate, and protons.
• It uses a mixed-integer linear programming (MILP) formulation with binary variables to enforce reaction directionality based on thermodynamic feasibility, similar to loopless FBA.
• The method fixes the $\Delta_{f}G'^{\prime}$ values of key energy carriers to physiological concentration ranges (e.g., 1 mM for ATP), while leaving other metabolites unconstrained.
• It applies constraints such that $\Delta_{r}G'^{\prime} \leq -\varepsilon$ for forward reactions and $\geq \varepsilon$ for reverse reactions, ensuring thermodynamically feasible flux directions.
• The approach leverages the kernel of the internal stoichiometric matrix ($\ker(\mathbf{S}_{\text{int}})^\perp$) to compute reaction Gibbs energies from metabolite formation energies.
• A network-embedded semi-thermodynamic analysis (NEST) is used to evaluate the thermodynamic feasibility of flux distributions across the network.

Experimental results

Research questions

• RQ1 Can a method be developed that removes energy-generating cycles (EGCs) more effectively than loopless FBA without requiring full thermodynamic parameterization?
• RQ2 To what extent do selective thermodynamic constraints on energy currency metabolites improve the biological realism of FBA solutions?
• RQ3 How does st-FBA compare to thermodynamic FBA and loopless FBA in eliminating internal and futile cycles while preserving physiologically relevant flux distributions?
• RQ4 What is the impact of constraining only a small subset of metabolites (e.g., ATP, H+) on the overall solution space of genome-scale models?

Key findings

• st-FBA successfully eliminates all thermodynamically infeasible energy-generating cycles (EGCs), including internal cycles that produce ATP without substrate input.
• The method achieves this by constraining only the formation energies of key energy currency metabolites (e.g., ATP, pyrophosphate, H+), avoiding the need for extensive thermodynamic data.
• Compared to loopless FBA, st-FBA provides stronger thermodynamic enforcement, removing cycles that ll-FBA fails to eliminate.
• Compared to thermodynamic FBA, st-FBA requires far fewer parameters—only bounds on a small set of metabolites—making it more practical for large-scale models.
• The implementation in COBRApy confirms that st-FBA is computationally efficient and readily integrable into existing FBA workflows.
• NEST analysis confirms that st-FBA solutions are thermodynamically feasible across all reactions, with no reaction violating the $\Delta_{r}G'^{\prime} \leq -\varepsilon$ or $\geq \varepsilon$ constraints.