Predicting internal cell fluxes at sub-optimal growth

Schultz and Qutub BMC Systems Biology (2015) 9:18 DOI 10.1186/s12918-015-0153-3 METHODOLOGY ARTICLE Open Access Predicting internal cell fluxes at sub-optimal growth André Schultz* and Amina A Qutub Abstract Background: Flux Balance Analysis (FBA) is a widely used tool to model metabolic behavior and cellular function. Applications of FBA span a breadth of research from synthetic engineering of biofuels to understanding evolutionary adaptations. FBA predicts metabolic reaction fluxes that optimize a given objective. This objective is generally defined for unicellular organisms by a theoretical reaction which simulates biomass production. FBA has been extremely successful at predicting in E. coli growth rates under different media and gene essentiality, amongst other things. In order to improve predictions, additional constraints are coupled with optimization of the biomass function. Studies have suggested, however, that unicellular organisms - like multicellular organisms - do not grow at optimal rates. To further improve FBA predictions, particularly of internal cell fluxes, new techniques to explore the sub-optimal solution space need to be developed. Results: We present an innovative FBA method called corsoFBA based on the optimization of protein cost at sub-optimal objective levels. Our method shows good agreement with experimental data of E. coli grown at different dilution rates. Maintaining the objective function close to its maximum value predicts metabolic states that closely resemble low dilution rates; while higher dilution rates can be mirrored by lowering the biomass production value. By using a modified version of Extreme Pathways, we are also able to quantify the energy production and overall protein cost for all possible pathways in the central carbon metabolism. Conclusion: Metabolic flux distributions at the optimal objective can be substantially different from the near-optimal distributions. Importantly, the behavior of E. coli central carbon metabolism can be better predicted by exploring the sub-optimal FBA solution space. The corsoFBA method presented here is able to predict the behavior of PEP Carboxylase, the glyoxylate shunt and the Entner-Doudoroff pathway at different glucose levels, a behavior not predicted by the minimization of metabolic steps and FBA alone. This technique can be used to better predict internal cell fluxes under different conditions, and corsoFBA will be of great help for the study of cells from multicellular organisms using Flux Balance Analysis. Keywords: Flux balance analysis, COBRA, BiGG, Genome-wide metabolic reconstructions, Constraint based models, Extreme pathways, Sub-optimal growth, Protein cost Background Genome-wide metabolic reconstructions provide a powerful platform for the analysis of metabolic pathways. Reconstructions for at least sixty-five different species spanning fifty-one different genera, including humans [1,2], are available today [3]. Such reconstructions have been used successfully in several fields of research, including metabolic engineering, evolutionary analysis and metabolic network properties [4]. The mathematical *Correspondence: Department of Bioengineering, Rice University, 6500 Main Street, Houston, USA formulation of these reconstructions are called Constraint Based Models. These models are defined at the core by a sparse stoichiometric matrix, where each column represents a reaction, each row a metabolite, and each entry the corresponding stoichiometric coefficient. A vast array of computational methods for the analysis of Constraint Based Models is available, and the number continues to grow [5,6]. Perhaps the most widely used of these methods is Flux Balance Analysis (FBA). FBA returns a flux distribution through the entire metabolism which optimizes (minimizes or maximizes) a given objective function or reaction, such as ATP or biomass © 2015 Schultz and Qutub; licensee BioMed Central. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Schultz and Qutub BMC Systems Biology (2015) 9:18 production. This flux distribution satisfies a steady-state assumption, such that there is no net production or consumption of any metabolite. A pre-defined set of constraints, such as upper and lower bounds for each reaction and substrate availability, is also defined [7]. This technique is most commonly applied to metabolic reconstructions of unicellular organisms with the optimization of a theoretical Biomass reaction. This reaction consumes resources such as amino acids, nucleotides and ATP at the rate the given organism would need in order to grow and multiply [8,9]. FBA alone has predicted in E. coli the uptake and release rates of certain metabolites [10-12], cell growth rate under different environmental conditions [10,11] and gene essentiality [10] with great success. However, the prediction of internal cell fluxes remains a challenge [13], mainly due to four reasons: 1. The FBA solution is not unique. There are several fluxes within the model that are not well defined, but can exist within certain bounds while the objective function is being optimized [7]. These fluctuations then define an FBA solution space, rather than a single, unique output. 2. Organisms might not be operating at maximum capacity [14-18]. In this case, the objective function might not be fully optimized, but instead be in a near-optimal or sub-optimal state. Furthermore, Flux Variability Analysis shows that the FBA solution space increases drastically when considering a near-optimal to optimal state [18], exacerbating the possibility of multiple FBA solutions. 3. The observed metabolic state is also not unique. That is, a single flux distribution cannot explain all the flux states observed in the experiments, and variation occurs within the bacterial population. Wintermute et al. [19] proposes a cloud theory for metabolic regulation, where bacteria are allowed to fluctuate within a near-optimal solution space. The study also demonstrates that the variability of fluxes within this region matches the observed variability within the data. 4. Thermodynamically infeasible loops can appear in the FBA output. These are sets of pathways termed “type III pathways ” [20], which are composed of internal reactions that can be combined to yield a steady-state with no net input or output. These cycles can clutter the FBA output, hindering any subsequent analysis [5]. To better predict internal fluxes, some studies have relied on additional thermodynamic constraints [21]. Several of these studies successfully reduce the FBA (...truncated)