Optimization-based models have been used to predict cellular behavior for over 25 years. The constraints in these models are derived from genome annotations, measured macro-molecular composition of cells, and by measuring the cell's growth rate and metabolism in different conditions. The cellular goal (the optimization problem that the cell is trying to solve) can be challenging to derive experimentally for many organisms, including human or mammalian cells, which have complex metabolic capabilities and are not well understood. Existing approaches to learning goals from data include (a) estimating a linear objective function, or (b) estimating linear constraints that model complex biochemical reactions and constrain the cell's operation. The latter approach is important because often the known observed biochemical reactions are not enough to explain observations, and hence there is a need to extend automatically the model complexity by learning new chemical reactions. However, this leads to nonconvex optimization problems, and existing tools cannot scale to realistically large metabolic models. Hence, constraint estimation is still used sparingly despite its benefits for modeling cell metabolism, which is important for developing novel antimicrobials against pathogens, discovering cancer drug targets, and producing value-added chemicals. Here, we develop the first approach to estimating constraint reactions from data that can scale to realistically large metabolic models. Previous tools have been used on problems having less than 75 biochemical reactions and 60 metabolites, which limits real-life-size applications. We perform extensive experiments using 75 large-scale metabolic network models for different organisms (including bacteria, yeasts, and mammals) and show that our algorithm can recover cellular constraint reactions, even when some measurements are missing.
Similarly, the metabolic fluxes in a cell can be predicted by assuming that the cell is solving an optimization problem shaped by its evolutionary history @cite_48 . The problem includes constraints that model biochemical reactions'' consuming and producing metabolites at specific stoichiometric ratios, and an objective that depends on the fluxes through the reactions. For example, the metabolism of fast-growing microbes (mid-log phase) is predicted accurately by a linear problem such as . As another example, the metabolism of mammalian cells (e.g., hybridoma), is explained well by minimization of the flux through the reaction that makes reactive oxygen species @cite_48 . Studies have identified alternative objective functions that best predict microbial metabolism under different growth conditions @cite_38 . These objectives include maximizing ATP yield per flux unit and maximizing ATP or biomass yield.
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To which extent can optimality principles describe the operation of metabolic networks? By explicitly considering experimental errors and in silico alternate optima in flux balance analysis, we systematically evaluate the capacity of 11 objective functions combined with eight adjustable constraints to predict 13C-determined in vivo fluxes in Escherichia coli under six environmental conditions. While no single objective describes the flux states under all conditions, we identified two sets of objectives for biologically meaningful predictions without the need for further, potentially artificial constraints. Unlimited growth on glucose in oxygen or nitrate respiring batch cultures is best described by nonlinear maximization of the ATP yield per flux unit. Under nutrient scarcity in continuous cultures, in contrast, linear maximization of the overall ATP or biomass yields achieved the highest predictive accuracy. Since these particular objectives predict the system behavior without preconditioning of the network structure, the identified optimality principles reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states. Genome-scale flux balance models of metabolism provide testable predictions of all metabolic rates in an organism, by assuming that the cell is optimizing a metabolic goal known as the objective function. We introduce an efficient inverse flux balance analysis (invFBA) approach, based on linear programming duality, to characterize the space of possible objective functions compatible with measured fluxes. After testing our algorithm on simulated E. coli data and time-dependent S. oneidensis fluxes inferred from gene expression, we apply our inverse approach to flux measurements in long-term evolved E. coli strains, revealing objective functions that provide insight into metabolic adaptation trajectories. An optimization-based framework is introduced for testing whether experimental flux data are consistent with different hypothesized objective functions. Specifically, we examine whether the maximization of a weighted combination of fluxes can explain a set of observed experimental data. Coefficients of importance (CoIs) are identified that quantify the fraction of the additive contribution of a given flux to a fitness (objective) function with an optimization that can explain the experimental flux data. A high CoI value implies that the experimental flux data are consistent with the hypothesis that the corresponding flux is maximized by the network, whereas a low value implies the converse. This framework (i.e., ObjFind) is applied to both an aerobic and anaerobic set of Escherichia coli flux data derived from isotopomer analysis. Results reveal that the CoIs for both growth conditions are strikingly similar, even though the flux distributions for the two cases are quite different, which is consistent with the presence of a single metabolic objective driving the flux distributions in both cases. Interestingly, the CoI associated with a biomass production flux, complete with energy and reducing power requirements, assumes a value 9 and 15 times higher than the next largest coefficient for the aerobic and anaerobic cases, respectively. © 2003 Wiley Periodicals, Inc. Biotechnol Bioeng 82: 670–677, 2003. Background Optimization theory has been applied to complex biological systems to interrogate network properties and develop and refine metabolic engineering strategies. For example, methods are emerging to engineer cells to optimally produce byproducts of commercial value, such as bioethanol, as well as molecular compounds for disease therapy. Flux balance analysis (FBA) is an optimization framework that aids in this interrogation by generating predictions of optimal flux distributions in cellular networks. Critical features of FBA are the definition of a biologically relevant objective function (e.g., maximizing the rate of synthesis of biomass, a unit of measurement of cellular growth) and the subsequent application of linear programming (LP) to identify fluxes through a reaction network. Despite the success of FBA, a central remaining challenge is the definition of a network objective with biological meaning.