Flux balance analysis accounting for metabolite dilution
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Flux balance analysis is a common method for predicting steady-state flux distributions within metabolic networks, accounting for the growth demand for the synthesis of a predefined set of essential biomass precursors. Ignoring the growth demand for the synthesis of intermediate metabolites required for balancing their dilution leads flux balance analysis to false predictions in some cases. Here, we present metabolite dilution flux balance analysis, which addresses this problem, resulting in improved metabolic phenotype predictions.
KeywordsFlux Distribution Flux Balance Analysis Intermediate Metabolite Efficient Pathway Extreme Pathway
area under curve
flux balance analysis
metabolite dilution flux balance analysis
mixed integer linear programming
receiver operating characteristic.
A practical approach to gaining biological understanding of complex metabolic networks requires the development of mathematical modeling, simulation, and analysis techniques. Traditional modeling techniques are based on mathematical approaches that require detailed and accurate information regarding reaction kinetics as well as enzyme and metabolite concentrations [1, 2]. The lack of sufficient data limits the current applicability of such methods to small-scale systems. This hurdle is surpassed through the use of constraint-based modeling (CBM), which serves to analyze the functionality of genome-scale metabolic networks by relying solely on simple physical-chemical constraints [3, 4]. Genome-scale CBM models have already been constructed for more than 50 organisms , including common model microorganisms [6, 7], industrially relevant microbes [8, 9, 10, 11], various pathogens [12, 13, 14, 15], and recently for human cellular metabolism . Flux balance analysis (FBA) is a key computational approach within the CBM modeling framework [17, 18, 19] and is frequently used to successfully predict various phenotypes of microorganisms, such as their growth rates, uptake rates, by-product secretion, and knockout lethality (see [3, 5, 20] for reviews).
The uptake and secretion of a pre-defined set of metabolites from and to the environment is facilitated via the definition of exchange reactions in the stoichiometric matrix S . A pseudo growth reaction is defined to simulate the utilization of metabolites during growth, consuming the most abundant biomass constituents based on experimentally determined concentrations (that is, the j-th component in Open image in new window denotes the steady-state concentration of metabolite j). The objective of FBA is to find a steady-state flux distribution, Open image in new window , satisfying Equation 2 alongside additional enzymatic directionality and capacity constraints , together permitting a maximal growth rate μ. Accounting only for linear constraints, the resulting space of feasible flux distribution described by FBA is convex (forming a high-dimensional polytope), in which optimal biomass producing solutions can be efficiently searched for via linear programming (LP).
The employment of a pseudo growth reaction in FBA to represent the utilization of metabolites as part of growth poses two fundamental problems. First, the metabolite composition of cellular biomass significantly varies across different growth media, genetic backgrounds and growth rates [22, 23, 24]. Indeed, previous work by Pramanik and Keasling [22, 23] has shown that using the correct experimentally measured biomass composition of Escherichia coli under different growth media and growth rates significantly improves FBA flux predictions. However, as FBA is commonly applied to probe metabolic behavior under diverse genetic and environmental conditions for which no metabolite concentration data are available, it has become common practice to employ a constant biomass composition across all conditions . Second, the growth reaction in various CBM models commonly accounts for no more than a few dozen metabolites, for which measured concentrations are available under a specific condition . Ignoring the growth-associated dilution of the remaining metabolites (those not included in the biomass composition in use; required by Equation 1) may result in the prediction of biologically implausible flux distributions, leading to false predictions of gene essentiality and growth rates, as shown in the Results. This problem has been recently addressed by Kruse and Ebenhöh , who suggested a method that is based on network expansion to compute the set of producible metabolites under a given growth medium. This method, however, does not enable the prediction of feasible flux distributions that account for the growth-associated dilution of all intermediate metabolites. Another approach, recently suggested by Martelli et al. , predicts metabolic fluxes based on Von Neumann's model, which maximizes the growth rate in a metabolic network without assuming mass-balance nor utilizing prior knowledge of a biomass composition. However, similarly to FBA, flux distributions predicted by this method do not fully account for the growth-associated dilution of all intermediate metabolites.
In this paper we describe a variant of FBA, metabolite dilution flux balance analysis (MD-FBA), which aims to predict metabolic flux distributions by accounting for the dilution of all intermediate metabolites that are synthesized under a given condition. As shown below, accounting for growth dilution of intermediate metabolites is especially important for metabolites that participate in catalytic cycles, many of them being metabolic co-factors. Since CBM assumes a steady-state flux distribution and does not predict the actual concentration of the intermediate metabolites, we consider a uniform minimal dilution rate for all intermediate metabolites produced via a non-zero flux through some reaction (assuming a uniform concentration for all intermediate metabolites, following ).
Next, we describe the implementation of MD-FBA as a mixed-integer linear programming (MILP) optimization problem and demonstrate its applicability in predicting metabolic phenotypes, outperforming the commonly used FBA method.
MD-FBA - accounting for growth-associated dilution of all intermediate metabolites
Our method, MD-FBA, aims to predict a feasible flux distribution through a metabolic network under a given environmental and genetic condition, by maximizing the production rate of the biomass (that is, the flux through the biomass reaction) while satisfying a stoichiometric mass-balance constraint, accounting for the growth-associated dilution of all produced intermediate metabolites, and satisfying enzymatic directionality and capacity constraints embedded in the model (similarly to FBA). MD-FBA is formulated as a MILP problem as defined in the Materials and methods.
Applying MD-FBA to predict metabolic phenotypes in Escherichia coli
As a benchmark for the prediction performance of MD-FBA, we applied it to the genome-scale metabolic network model of E. coli  to predict growth rates and gene essentiality under a diverse set of growth media and gene knockouts. The model of Feist et al.  accounts for 1,260 metabolic genes, 2,382 reactions and 1,668 metabolites.
This study presents MD-FBA, a variant of FBA for predicting metabolic flux distributions by accounting for growth-associated dilution of all metabolites in a context-dependent manner. The method predicts feasible flux distributions maximizing the production rate of a predefined biomass while accounting for the dilution of all intermediate metabolites, and most importantly, for all metabolic co-factors involved in the process. MD-FBA was shown to successfully predict E. coli's gene essentiality under a variety of growth media and knockout strains, displaying a significant improvement upon the prediction performance of the commonly used FBA method.
MD-FBA has two notable limitations, which may contribute to the relatively low improvement in growth rate prediction accuracy (compared to the marked advantage in predicting gene knockout lethality). First, MD-FBA employs a uniform lower bound on the dilution rate of intermediate metabolites which, along with the absence of reactions outside the scope of the network model that degrade intermediate metabolites, implicitly reflects the assumption of a uniform concentration of all intermediate metabolites. A natural extension of MD-FBA would be to consider different lower and upper bounds on concentrations of different metabolites, based on concentration statistics gathered via metabolomic measurements across a variety of conditions (for example, ). Notably though, changing the lower bound employed here to a range of possible values and incorporating an upper bound on dilution rates across all metabolites did not improve the prediction performance (data not shown). Second, MD-FBA, similarly to FBA, is based on the assumption that microbial species aim to maximize their growth rate and hence search for feasible flux distributions that maximize biomass synthesis rate. However, previous studies have questioned this hypothesis, suggesting alternative possible optimization criteria. Future studies should investigate the potential usage of such optimization criteria with MD-FBA . More generally, CBM methods that do not rely on optimization may also benefit from variants that account for metabolite dilution during growth.
A marked disadvantage of MD-FBA is its dependence on MILP, which is computationally more demanding than LP, utilized by FBA. To improve the run-time of MD-FBA, the amount of integer variables in the MD-FBA formulation may be reduced by employing a previous method to identify the metabolic 'scope' of the medium nutrients. Specifically, Handorf et al.  investigated the capacity to produce metabolites from available medium nutrients by applying FBA and a network expansion algorithm, resulting in a production scope for each set of medium metabolites. A potential improvement in run-time may be achieved by calculating the scope of the input growth medium and assigning integer variables only for metabolites in that derived scope, as all the other metabolites will never be able to satisfy their dilution demand. Speeding up the run-time may be of importance when applying MD-FBA to larger networks, such as the recently published human model , or when probing the network under multiple knockout configurations [37, 38].
An interesting comparison can be made between MD-FBA and a method developed by Price et al.  for eliminating futile cycles via the identification of type III extreme pathways (that is, a unique set of convex basis vectors of the flux distribution solution space that do not include exchange reactions). While the extreme pathways method enables the elimination of thermodynamically impossible loops, MD-FBA removes infeasible solutions due to dilution demands. Notably, the latter method also implicitly eliminates type III extreme pathways since these pathways do not satisfy dilution demands of the participating metabolites. Additionally, MD-FBA eliminates solutions that do not involve type III extreme pathways as demonstrated in Figure 1b: when metabolite X is absent from the growth medium, the cycle involving reactions v4 and v8 cannot be activated based on MD-FBA, since the dilution of co-factor C cannot be satisfied, although this cycle is not part of a type III extreme pathway.
Another appealing application of MD-FBA could be the identification of missing reactions in the model by comparing predicted phenotypes with measured ones, in line with previous works using FBA for this purpose . For example, suppose that in Figure 1a the biosynthetic pathway for metabolite C, through reactions v6 and v7, was not included in the model. In this case, MD-FBA would predict metabolic flow through reactions v2 and v3, such that the enzymes catalyzing these reactions are essential, contrary to experimental essentiality data. Utilizing a method similar to that used by Reed et al. , using MD-FBA can infer the missing reactions, v6 and v7. Employing FBA for this purpose would not work since FBA predicts v2 and v3 to be non-essential, as the activity of reactions v4 and v8 do not depend on the presence of reactions v6 and v7.
While this work applied MD-FBA to predict metabolic phenotypes in E. coli, for which a comprehensive and accurate metabolic network model exists, the method can also be applied to any one of a growing number of reconstructed network models . Importantly, the application of MD-FBA to other network models is straightforward and requires no model-specific data curation. To facilitate simple usage of MD-FBA, we provide an implementation of the method in the supplemental website . A particularly interesting potential application of MD-FBA would be for modeling malignant proliferating cells in human cancer, potentially revealing the activity of biosynthetic pathways for various co-factors required to balance their growth-associated dilution. The latter may utilize the recently published model of human cellular metabolism by  or . Overall, we expect that future use of MD-FBA will promote improved metabolic phenotypic predictions across a variety of organisms, growth conditions and genetic alterations.
Materials and methods
Metabolite dilution flux balance analysis
where a mass-balance constraint, accounting for the dilution of all active metabolites, is formulated in Equation 3. Equation 4 assigns a positive dilution rate above a pre-defined threshold (denoted by ε) for active metabolites, produced in some non-zero rate in the flux distribution Open image in new window . In our application of the method for E. coli we set ε = 10-4 μmol/mg, which represents a common concentration of intermediate metabolites . Notably, the model's predictions were robust to different choices of ε values (data not shown). Enzyme directionality and capacity constraints are formulated in Equation 5 by imposing Open image in new window and Open image in new window as lower and upper bounds on flux values.
A simplified formulation assuming a constant growth rate of μ = 1 in Equation 8 (for calculating the dilution rate of intermediate metabolites) gave qualitatively similar results to the above linear formulation (data not shown). The commercial solver CPLEX running on 64-bit Linux machines was used for solving LP and MILP problems within a few dozens of seconds per problem.
We are grateful to Hadas Zur, Naama Tepper and Yoav Teboulle for their fruitful comments. This study was supported in part by a fellowship from the Edmond J Safra Bioinformatics program at Tel-Aviv University. ER's and TS's research is supported by grants from the Israel Science Foundation.
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