You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: 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.

[2]: 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.
While the aforementioned studies have focused on using pre-defined optimization problems, a number of studies have investigated data-driven estimation of cellular goals. Two types of method exist. Methods of the first type estimate how important the different chemical reactions are for the cell's operation, i.e., estimate @math in . For example, ObjFind [2] does this through a nonconvex optimization formulation, and the more recent invFBA [1] solves a linear program. The second type, and the focus of our work, estimate the stoichiometric coefficients of a new chemical reaction, i.e., estimate a new column for matrix @math in . This approach is important because often the known biochemical reactions are not enough to explain observed data. We want to extend the model complexity automatically by learning new chemical reactions, i.e., new columns for @math .