On dynamically generating relevant elementary flux modes in a metabolic network using optimization

On the geometry of elementary flux modes

Elementary flux modes (EFMs) play a prominent role in the constraint-based analysis of metabolic networks. They correspond to minimal functional units of the metabolic network at steady-state and as such have been studied for almost 30 years. The set of all EFMs in a metabolic network tends to be very large and may have exponential size in the number of reactions. Hence, there is a need to elucidate the structure of this set. Here we focus on geometric properties of EFMs. We analyze the distribution of EFMs in the face lattice of the steady-state flux cone of the metabolic network and show that EFMs in the relative interior of the cone occur only in very special cases. We introduce the concept of degree of an EFM as a measure how elementary it is and study the decomposition of flux vectors and EFMs depending on their degree. Geometric analysis can help to better understand the structure of the set of EFMs, which is important from both the mathematical and the biological viewpoint.

How to Tackle Underdeterminacy in Metabolic Flux Analysis? A Tutorial and Critical Review

Metabolic flux analysis is often (not to say almost always) faced with system underdeterminacy. Indeed, the linear algebraic system formed by the steady-state mass balance equations around the intracellular metabolites and the equality constraints related to the measurements of extracellular fluxes do not define a unique solution for the distribution of intracellular fluxes, but instead a set of solutions belonging to a convex polytope. Various methods have been proposed to tackle this underdeterminacy, including flux pathway analysis, flux balance analysis, flux variability analysis and sampling. These approaches are reviewed in this article and a toy example supports the discussion with illustrative numerical results.

Novel column generation-based optimization approach for poly-pathway kinetic model applied to CHO cell culture

Mathematical modelling can provide precious tools for bioprocess simulation, prediction, control and optimization of mammalian cell-based cultures. In this paper we present a novel method to generate kinetic models of such cultures, rendering complex metabolic networks in a poly-pathway kinetic model. The model is based on subsets of elementary flux modes (EFMs) to generate macro-reactions. Thanks to our column generation-based optimization algorithm, the experimental data are used to identify the EFMs, which are relevant to the data. Here the systematic enumeration of all the EFMs is eliminated and a network including a large number of reactions can be considered. In particular, the poly-pathway model can simulate multiple metabolic behaviors in response to changes in the culture conditions.

We apply the method to a network of 126 metabolic reactions describing cultures of antibody-producing Chinese hamster ovary cells, and generate a poly-pathway model that simulates multiple experimental conditions obtained in response to variations in amino acid availability. A good fit between simulated and experimental data is obtained, rendering the variations in the growth, product, and metabolite uptake/secretion rates. The intracellular reaction fluxes simulated by the model are explored, linking variations in metabolic behavior to adaptations of the intracellular metabolism.

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Novel method to model multiple states by a poly-pathway kinetic model.

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EFMs relevant to data identified by column generation (CG)-based optimization.

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CG optimization enables use of networks much larger than systematic enumeration.

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A kinetic model simulates changes in metabolic rates linked to available amino acids.

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The flux distribution of each metabolic state is visualized in the original network.