A key feature in the application of FBA is the optimization of an objective function subject to fundamental constraints on cellular function 91125. FBA requires a stoichiometric network reconstruction, usually represented as a stoichiometric matrix, S, in which components are delineated as rows, component interactions as columns, and stoichiometric relationships as coefficients (see Figure 1 for examples of biochemical networks and the corresponding S matrices). This reconstruction, coupled with reaction fluxes, comprises the principal constraint in FBA, i.e., at steady-state, for each component, the sum of the stoichiometric coefficients multiplied by the corresponding reaction fluxes must equal zero to ensure that mass is balanced within the system.

Commonly used objective functions (i.e., stoichiometric objectives or "objective reactions") in FBA of metabolic networks include biomass production 2627, energy production or consumption 13, and byproduct production 14. The objective function used in FBA is assumed to be linear by a first-order approximation, since this form simplifies computation and reduces the number of parameters to be defined in the objective experimentally. Changes in a linear objective under varying growth conditions have been observed 12.

FBA-predicted flux distributions have displayed agreement with experimentally-measured flux data in some cases 12. Furthermore, FBA has yielded insights into optimal targets for metabolic engineering 15 and adaptive evolution of E. coli strains 28, among other applications 29. Additionally, FBA has successfully predicted metabolic phenotypes of cellular systems by hypothesizing biomass production as the objective function 791012192730313233343536. However, other objectives may be equally successful at predicting phenotypes for particular conditions 20. Furthermore, a metabolic objective may change at different stages of an organism's life cycle 37. Some studies suggest a greater role for environmental factors in determining cellular objectives than is assumed when FBA is employed with a biomass production objective 38. Ultimately, as additional network reconstructions are completed and experimental flux data are obtained, there is increasing interest in determining realistic objective functions to explain systemic behavior 18192039.

The BOSS framework initially takes the form of a bi-level optimization problem that minimizes the sum-squared error between experimentally-measured (in vivo) fluxes and framework-computed (in silico) fluxes ("outer problem") (see Figure 2(a), line 1), subject to the condition that a cellular objective is simultaneously maximized ("inner problem") (lines 2–5). The inner problem takes the canonical form of a FBA problem, wherein an objective reaction introduced to the network is maximized (line 2) subject to thermodynamic (line 3), mass balance (line 4), and uptake (line 5) constraints. The coefficients of the objective reaction are unknown and part of the solution space.

In order to make this bi-level optimization problem computationally tractable, we reformulated it as a single-level optimization problem via the duality theorem of LP, as previously described 340 (see Figure 2(b)). Specifically, the revised form is comprised of an objective that aims to minimize the sum-squared error between experimentally-measured and framework-computed fluxes (line 1) subject to a set of primal (lines 3 through 5) and dual constraints (lines 7 through 9). The objective in line 1 is equivalent to the outer objective of the bi-level optimization problem (see Figure 2(a), line 1), and the primal constraints in lines 3 through 5 are equivalent to the constraints in the original bi-level optimization problem (see Figure 2(a), lines 3 through 5). Two additional constraints are included in the single-level optimization framework. Specifically, the value of the primal and dual problems are set equivalent to one another (line 2), and the flux distribution corresponding to the new objective reaction is normalized to a specific value (line 6). As described in the caption for Figure 2, the decision variables in this optimization are: (1) the stoichiometric coefficients of the objective reaction, S_{i, objective} where i ∈ N, or the set of metabolites, and "objective" denotes the new objective reaction; (2) the framework-computed fluxes v^experimental; (3) the dual variable g associated with the uptake constraint; and (4) the dual variables u indicating shadow prices on the mass balance constraints for each metabolite in the system.

Due to the large number of decision variables, large solution space, existence of multiple local optima, and inherent non-convexity of the problem, we instituted a multiple restart approach, wherein the optimization is run multiple times, each time with different randomly-selected starting values for the decision variables (within a specified range) (see Figure 3(b), item 3). The output of this approach is a series of putative objective reactions, one for each restart. Objective reactions that yield a poor value in the outer optimization (i.e., for which the sum-squared error between experimental and calculated fluxes is relatively high) are removed from the solution set. The remaining putative objective reactions are then clustered into groups based on similarity, and the most populous cluster is chosen as the consensus objective reaction (see Figure 3(b), item 4, as well as Additional file 2). This last step seeks to eliminate solutions to the outer problem that constitute suboptimal local minima. It is assumed that the global minimum to the outer problem (i.e., the best match between BOSS-derived and experimental fluxes, thereby leading to the likely objective reaction) will draw from a larger portion of the initial state space than any local minimum, and thus will gather the greatest proportion of the restarts when solutions are clustered.

This single-level optimization with multiple restarts, followed by clustering and subsequent averaging of the most populous cluster, ultimately yields a stoichiometrically-weighted reaction for which the metabolic network is optimized. This reaction represents a best hypothesis for the network objective function based on available data about the system. Determination of the objective reaction in this manner offers a means to gain insight about network and sub-network objectives and behavior.

The previously reconstructed S. cerevisiae central metabolic network 2223 was used to assess the BOSS framework (see Additional file 2). This network is a subset of the genome-scale S. cerevisiae metabolic reconstruction 4142 and comprises the major carbohydrate metabolism pathways of glycolysis, pentose phosphate, and the citrate cycle, the principal energy metabolism pathway of oxidative phosphorylation, and a precursor biomass synthesis reaction. The network is comprised of 60 metabolites participating in 62 reactions, including six exchange reactions, 55 intracellular reactions, and one precursor biomass synthesis reaction.

A flux distribution for the central metabolic network was previously characterized experimentally 2223. This distribution was obtained by GC-MS tracing of ¹³C-labeled isotopomers, and corresponds to yeast cells cultivated in batch culture with glucose as the limiting substrate and at a maximum specific growth rate, μ_max, of 0.37 h^-1. (See 23 for complete experimental details.) The BOSS framework was applied to this data set.

In addition, the precursor biomass synthesis reaction, which balances the major metabolites from central metabolism that contribute to biomass, was hypothesized to be the "true" objective function of the system for testing the BOSS framework. This hypothesis was based on the underlying premise described previously that organisms have maximized their growth performance through natural selection 12. The precursor biomass synthesis reaction was previously characterized experimentally 2223. It is important to note that BOSS does not require an assumption of an objective function, and we demonstrate below the success of BOSS at generating a hypothesized objective function de novo.

The general implementation scheme for BOSS is illustrated in Figure 3. Specifically, we inputted the stoichiometric network reconstruction and isotopomer flux data for the S. cerevisiae central metabolic network into BOSS (see Figure 3(a)) and assessed the objective reaction that it derived (see Figure 3(b)). We evaluated two conditions, one in which the hypothesized objective function of precursor biomass synthesis was included in the network reconstruction inputted into BOSS and another in which it was excluded. In the second case, BOSS yielded a zero-vector for the objective reaction, suggesting that the true objective was already a part of the network reconstruction. Therefore, we altered BOSS slightly to identify which of the reactions was the likely objective: we removed, one by one, each reaction flux from the set of experimental flux data (v^experimental in Figure 2), and identified in each case the sum-squared error between the BOSS-derived objective reaction and the reaction corresponding to the removed flux. The reaction that exhibited the smallest sum-squared error through this method was identified as the objective function. This approach was utilized to ensure that, if the actual system objective is already in the stoichiometric network reconstruction, the outer optimization problem of BOSS does not force flux to go through that reaction and rather allows the flux corresponding to the new objective function column to be maximized as part of BOSS.

We considered one additional in silico experiment to evaluate how well our framework handles the kind of variability that is characteristic of actual flux measurements. Specifically, we assessed how a varying amount of noise introduced to the set of experimental flux data affects the objective function derived by BOSS. (See Additional file 3 as well for a discussion of how BOSS handles a paucity of flux data, as applied to a prototypic system.) To perform this test, additional steps were performed during data preparation (see Figure 3(a), item 2). First, each experimental flux for the S. cerevisiae metabolic network was varied randomly via a normal distribution, with standard deviation equal to a set percent of the initial value of the flux. Different percent variances, ranging from five percent to 25 percent in increments of five percent, were considered. Fluxes with an initial value of zero were varied with a standard deviation equal to that of the smallest nonzero flux in the system. BOSS was fed these varied fluxes as "experimental" fluxes. To avoid biasing results toward one particular variant of the "experimental" fluxes, we repeated multiple times this process of randomly varying the experimental flux distribution. In other words, for any given percent variance, we tested multiple noisy flux data sets (n = 20). Consequently, when considering the results of any BOSS run on noisy flux data, we evaluated the coefficients of the objective reaction for each noisy flux data set independently.

In inferring the objective function for the S. cerevisiae metabolic network with BOSS, two conditions were evaluated. In the first one, the hypothesized objective reaction (i.e., precursor biomass synthesis) was excluded from the stoichiometric network reconstruction. BOSS identified coefficients approximately equal to that of the precursor biomass synthesis reaction (see Figure 4(a)). The sum-squared error between the BOSS-computed objective reaction and the experimentally-derived precursor biomass synthesis reaction, normalized to the magnitude of the precursor biomass synthesis reaction (SSE_S), was 8.242 × 10^-29. The magnitude of this SSE_S suggests that BOSS derives an objective reaction equivalent to the precursor biomass synthesis reaction, well within the limits of numerical tolerance.

We also implemented BOSS on the complete S. cerevisiae central metabolic network, i.e., without removing the precursor biomass synthesis reaction. When the precursor biomass synthesis reaction was thus included in the set of stoichiometric reactions, BOSS yielded a zero-vector for the objective reaction (SSE_S = 655.0), suggesting that the actual system objective was already a part of the stoichiometric network reconstruction (see Figure 4(b)). Subsequently, we individually removed each of the reaction fluxes from the pool of experimental fluxes defined in the outer optimization problem of BOSS (i.e., v^experimental in Line 1 of Figure 2(a)) and observed the resultant stoichiometric coefficients that BOSS derived for the objective function. SSE_S values between the reaction whose flux was removed from v^experimental and the BOSS-derived objective reaction given removal of that flux were compared for each reaction in the system (see Figure 4(c)). The reaction with the smallest SSE_S (between the BOSS-derived objective and the experimentally-characterized reaction whose flux was excluded) was the hypothesized system objective of precursor biomass synthesis (SSE_S = 4.210 × 10^-4). This result further validated the ability of BOSS to identify objective reactions with previously known as well as unknown stoichiometry (see Figures 4(c) and 4(d)). Interestingly, the reaction with the next smallest SSE_S (= 42.20) was the ATP maintenance reaction, a reaction that has also been evaluated as an objective of metabolic networks under some conditions 32143. All other reactions in the network had SSE_S values that were at least an order of magnitude higher. This key result is also a direct validation of the hypothesis that biomass production is a reasonable objective function for many metabolic networks; with the experimentally-generated flux data, BOSS derived an objective function highly correlated with biomass production although there was no such assumption a priori.

It is important to note that application of FBA to the S. cerevisiae metabolic network does not yield the experimental flux distribution measured by 23. Because FBA problems have large convex solution spaces, many different flux distributions yield equally optimal results. Therefore, it is significant that BOSS is able to infer the correct coefficients for the precursor biomass synthesis reaction in spite of this difference. Furthermore, although maximization of biomass synthesis is hypothesized as the objective of a metabolic network, the experimental flux data are calculated based on isotopomer measurements and the underlying system stoichiometry but is not biased toward this objective. Thus, the characterization of biomass as the system objective by BOSS is a novel validation of this theoretical assumption.

To account for current limitations in experimental (isotopomer) technologies for measuring reaction fluxes, the S. cerevisiae central metabolic network was evaluated by BOSS when different levels of noise were introduced into the experimental flux data. Different percentages of deviation for the flux data were evaluated to determine how quickly the accuracy of the BOSS-computed objective reaction degrades with noisy data.

The results for the S. cerevisiae network when varying levels of noise are introduced into the individual flux values are illustrated in Figure 5. For the purposes of this evaluation, based on the results described above, the hypothesized objective reaction of precursor biomass synthesis was included as part of the underlying network stoichiometry inputted into BOSS, and the set of experimental fluxes consisted of all fluxes except for the flux corresponding to this reaction. Figure 5(a) illustrates the normalized sum-squared error between the BOSS-derived objective reaction and the hypothesized objective function of precursor biomass synthesis (SSE_S) for different percent variances in experimental flux data, ranging from five percent to 25 percent in increments of five percent. (As described above, for each increment, 20 unique "initial" flux distributions, each with the same level of "noise," were computed, and the results for each of these 20 flux distributions were treated independently of the others. Therefore, the plot in Figure 5(a) illustrates the mean SSE_S across these 20 distributions, as well as the associated standard deviation within SSE_S across these 20 distributions.) Up to about 15 percent variance in the flux data, SSE_S < 1000, i.e., BOSS is able to generate reconstructions of the hypothesized objective reaction orders of magnitude more accurately than when random flux data are inputted into BOSS (see "Control Case: Random Flux Data" below). As expected, with increasing noise, the results of BOSS exhibit increasing mean SSE_S values and the variability of SSE_S across different initial flux sets is higher. These data support the utility of our framework in identifying objective functions for biological systems given the current state of experimental flux measurements.

To further highlight BOSS's performance on noisy flux data, snapshots of the objective reactions that it generated at different flux variances are illustrated. Specifically, Figures 5(b), 5(c), and 5(d) correspond to the coefficients identified by BOSS (blue error bars) overlaid on the precursor biomass synthesis reaction (gray bars) for flux variances of 5, 15, and 25 percent, respectively. (Note that, for each flux variance, 20 unique "initial" flux distributions were computed. Consequently, the values for the coefficients derived by BOSS correspond to a representative result for one of these 20 initial flux sets.) For smaller flux variances, the pattern of the coefficients identified by BOSS follows that of the hypothesized network objective of precursor biomass synthesis that was used to generate the noisy flux data (see Figure 5(b)). As expected, the more noisy the flux data, the less precise the objective reaction identified by the framework (see Figure 5(c)).

It is important to note that this analysis involves artificially adding noise to an experimental flux distribution. Although the original experimental fluxes are based on a steady-state metabolic model of yeast central metabolism, it is likely that there is some degree of noise already within the data set as part of the experimental protocol. Consequently, the studies with noise presented here include even higher degrees of noise than the five to 25 percent spectrum evaluated. Nevertheless, the results suggest that BOSS performs well, to an extent, with noisy experimental data.