Systems biology paradigm has provided insights in the maintenance of robustness for biological processes involving a multitude of interconnected elements (e.g., genes, proteins, metabolites) [1
]. In addition, recent advances in metabolomics have provided a large amount of highly reproducible data [2
], allowing reconstruction and analysis of genome-scale metabolic networks [5
]. These developments in metabolomics technologies have challenged computational systems biology with the need to accurately describe the dynamics of metabolic networks in order not only to glean the flux rates at different time points, representing the temporal flux (re)distribution, and the interdependent metabolic profiles, but also to identify key elements for metabolic engineering [6
Metabolic flux analysis (MFA) has propelled the development of computational methods for analysis of metabolic networks [11
]. Flux balance analysis (FBA), as one of the most prominent of the MFA methods, is based on linear programming (LP) whereby a given objective function (e.g., biomass yield) is optimized under the assumption that the system operates at steady state under the constraints given by the stoichiometric matrix [11
]. By optimizing an objective function, the linear program identifies one feasible flux distribution from the set of fluxes satisfying the constraints imposed by the mass-balance equations and reaction bounds [15
]. Consequently, the biological implications of the optimal flux distribution depend on the choice of the objective function [17
]. Maximization of biomass is one of these functions, which is assumed particularly suitable for microbial models [18
]. For eukaryotic cells (e.g., in plants) where biomass or yield may not be the primary goal, a different objective function has to be determined. For instance, cellular maintenance at minimal efforts has been proposed to be one of the alternatives [19
]. Nevertheless, finding an adequate objective of a metabolic network, and especially a sub network related to particular metabolic processes, remains a problem of ongoing interest [18
]. However, the steady-state assumption on which FBA is based precludes the analysis of the dynamics of metabolite concentrations and flux (re)distribution. Furthermore, the classical FBA ignores the possibility that perturbed metabolic networks may not immediately regulate towards the (assumed) optimal objective.
Based on the hypothesis that fluxes in metabolic networks, altered by removal of a reaction, undergo a minimal redistribution compared to those of the wild type, minimization of metabolic adjustment (MOMA) [21
] and regulatory on/off minimization (ROOM) [22
] have been devised as two contending alternatives for analysis of perturbed metabolic network models. MOMA predicts the flux redistribution which has the smallest Euclidean distance to the wild type flux distribution obtained by FBA, while ROOM minimizes the number of (significant) flux changes from the wild type flux distribution. As a consequence, large modifications in single fluxes are prevented in MOMA; however, such large modifications may be required for rerouting metabolic flux through alternative pathways [22
], which has been observed in experiments [23
]. Existing studies have demonstrated that ROOM outperforms MOMA and FBA in the flux prediction of the final metabolic steady state, albeit, in the particular case of pyruvate kinase knockout in E. coli
The analysis of metabolic network dynamics has traditionally been facilitated by models based on ordinary differential equations (ODEs) which require a large amount of information for simulating the temporal metabolic changes [9
]. To this end, the phenomenological parameters of specific enzyme kinetics (e.g., Michaelis-Menten or Hill) have to be determined by accurate measurements of enzyme activities and data-fitting to experimentally obtained (time-course) data. In turn, the obtained fits are often used to make predictions and draw conclusions based on the postulated kinetic model.
On the other hand, dynamic FBA (DFBA) offers the alternative to predict time-resolved metabolic profiles with limited knowledge of enzyme kinetics [6
]. Unlike, the analyses based on FBA, which focus on the steady-state behavior, DFBA offers the means to analyze transient (non-steady) states. In addition, DFBA has been combined with MOMA, resulting in the M-DFBA approach which has been employed for predicting the dynamics of photosynthetic metabolism and positing hypotheses about its robustness under different CO2
and water conditions [7
]. M-DFBA extends the MOMA hypothesis from minimal redistribution of metabolic fluxes in perturbed metabolic networks to a minimal fluctuation of the profile of metabolite concentrations over time.
We point out that the mechanisms describing the temporal
changes in the metabolic state, characterized by the metabolite concentrations and flux rates, are principal to the notion of robustness, as used with DFBA-based approaches. For instance, the posited mechanism in M-DFBA is that metabolic networks operate to minimize the fluctuations in metabolic concentrations over time. This suggests alterations to the definition of robustness as a property that maintains system function in the case of external
] to include the internal
perturbations due to changes in the dynamic state of the system's elements. External perturbations arise due to changing environmental conditions, such as stress conditions (environmental perturbations), but also by changes in the structure of a metabolic networks (e.g., caused by gene knock-outs, also known as genetic perturbations). In contrast, internal perturbations are due to temporal changes of the metabolic state characterized by the coupling of metabolite concentrations and flux distributions. Considering mass actions kinetics, which states that the flux rate of a reaction is proportional to the product of the concentrations of the participating reactants [26
]. As a result, even in this simplest kinetic law, a change in metabolite concentrations may affect flux rates, which, in turn, as a result of the mass balances have an effect on the concentrations. Here, we investigate a set of biochemically plausible hypotheses which can be used to characterize the mechanisms responsible for maintaining the metabolic network robustness due to internal perturbations. These mechanisms can in turn be used to simulate the dynamics of a given metabolic network relying purely on the stoichiometry and a limited amount of information regarding the phenomenological constants.
Driven by the idea of mutually influencing system elements, central to the systems biology paradigm, we argue that minimal fluctuations of metabolic profiles may only represent one possible explanation of the temporal changes in metabolic state. In this study, we design the ROOM-based DFBA approach (abbreviated to R-DFBA) by coupling the principle of ROOM with DFBA. R-DFBA extends the premise of the ROOM approach, which relies on significant flux changes, by considering the minimization of the total number of significant changes of metabolite concentrations. Furthermore, the coupling of ROOM and DFBA renders it possible to use the advantages of ROOM compared to MOMA in a dynamic setting.
In addition, we modify and extend the proposed R-DFBA and the existing M-DFBA approaches to consider minimizing fluctuations not only in concentrations but also in flux levels, which result in seven
proposed methods based on different optimization functions and programming formulations. Including classical DFBA and the two known variants of M-DFBA [8
], a total of ten different approaches are presented to predict and analyze the dynamics of metabolite concentrations and flux levels over time. Finally, the accuracy of a dynamic FBA-based approach in simulating the dynamics of metabolic networks must be established based on the comparison of the obtained results and the outcome of a well-defined kinetic model for a given metabolic network. Here, by comparing the results of applying DFBA, M-DFBA, R-DFBA and their extensions to those of the kinetic models for the Calvin-Benson cycle and plant carbohydrate metabolism, our analysis discriminates between the different mechanisms which result in the apparent metabolic network dynamics and robustness. Our quantitative and qualitative comparative analyses demonstrate that the extensions based on R-DFBA outperform the existing DFBA-based approaches.