Trans-dimensional Bayesian model averaging for $^{13}$C-based metabolic flux analysis: Evidence-based flux inference under structural model uncertainty
Accurate quantification of intracellular metabolic fluxes is central to systems biology and biotechnology. Flux estimation relies on biochemical network models, with $^{13}$C metabolic flux analysis (MFA) being the state-of-the-art approach. However, isotope labeling data are often insufficient to uniquely support a single network formulation. In such cases, flux estimates become model-dependent, highlighting the need for methods that explicitly account for structural uncertainty. Bayesian model averaging (BMA) provides a principled framework for this purpose, but its application to $^{13}$C-MFA has so far been restricted to uncertainty in reaction bidirectionality within fixed network topologies. We introduce a scalable Bayesian inference framework for $^{13}$C-MFA, Bayesian model set averaging, that applies BMA to encompass uncertainty in reactions and pathways. Our approach combines reversible jump Markov chain Monte Carlo for trans-dimensional exploration of model spaces with diffusive nested sampling for robust estimation of model evidences, enabling averaging over large families of metabolic network models. Using illustrative and application-scale synthetic case studies, we demonstrate that the method yields robust flux estimates, reveals when multiple network configurations are statistically indistinguishable, and recovers data-supported model structures. Importantly, rather than committing to a single model, the framework manages structural uncertainty: under limited data, competing models are retained, whereas increasing data informativeness improved model and flux recovery. The approach scales to billions of model variants, providing a practical foundation for uncertainty- and misspecification-aware quantitative flux inference in $^{13}$C-MFA.