Exploring the gap between dynamic and constraint-based models of metabolism
Highlights
► Constraint-based and full kinetic models share the same set of steady states. ► Partial rate-constant knowledge can further constrain kinetic models. ► Such reductions can be transferred from kinetic to constraint-based formulations.
Introduction
The prevalence of systems approaches to biological problems has renewed interest in mathematical models as fundamental research tools for performing in silico experiments of biological systems (Kitano, 2002). In the context of metabolic engineering, models of metabolism play an important role in the simulation of cellular behavior under different genetic and environmental conditions (Stephanopoulos, 1998). Typical experiments include knockout simulations to study how metabolic flux distributions readjust throughout a given network. With the selection of an optimal set of knockouts or changes in enzyme expression levels, it is desirable to optimize the production of compounds of industrial interest (Burgard et al., 2003, Patil et al., 2005).
Systems of ordinary differential equations (ODEs) have been applied in different areas to model dynamical systems. In the context of metabolic networks, they describe the rate of change of metabolite concentrations. These dynamic models contain rate law equations for the reactions as well as their kinetic parameters and initial metabolite concentrations. Building this type of model requires insight into enzyme mechanism to select appropriate rate laws, as well as experimental data for parameter estimation. Therefore, their application has been more limited, but areas of application include central metabolic pathways of well-studied organisms such as Escherichia coli (Chassagnole et al., 2002) and Saccharomyces cerevisiae (Rizzi et al., 1997). There are, however, some recent efforts to overcome these limitations in the reconstruction of large-scale dynamic models, such as through the hybrid dynamic/static approach (Yugi et al., 2005), the ensemble modeling approach (Tran et al., 2008), and the application of approximate kinetic formats using stoichiometric models as a scaffold (Smallbone et al., 2010, Jamshidi and Palsson, 2010). Nevertheless, these techniques have so far been applied to very few organisms.
On the other hand, advances in genome sequencing have facilitated the reconstruction of genome-scale metabolic networks for several organisms, with over 50 reconstructions available to date (Oberhardt et al., 2011). Due to the lack of kinetic data at the genome scale, this type of model only accounts for reaction stoichiometry and reversibility. Analysis is performed under the assumption of steady state using a constraint-based formulation that is underdetermined, resulting in a continuous space of solutions for the reaction flux distributions. This uncertainty of the flux distributions requires additional conditions to determine unique solutions and predictions. Often this takes the form of an optimization based on a particular assumption, such as optimal biomass growth for wild-type (Edwards and Palsson, 2000) and minimization of cellular adjustments for knock-out strains (Segrè et al., 2002, Shlomi et al., 2005). The inclusion of regulatory constraints, introduced by Covert and Palsson (2003), is a current approach to reduce the size of the solution space and eliminate infeasible solutions. One limitation of the constraint-based approach is the inability to express transient behavior. In order to simulate fermentation profiles, a few methods have been developed to integrate the variation of external concentrations while assuming an internal pseudo-steady state (Oddone et al., 2009, Leighty and Antoniewicz, 2011).
The two most common model types in use, therefore, represent two extremes. The dynamic ODE formulation contains detailed mechanistic information that gives solutions of the transient dynamic approach to equilibrium from any given set of initial conditions (generally concentrations of enzymes and metabolites), as well as the steady state specified by metabolite concentrations that depend on total enzyme concentrations (for the usual case where they are treated as fixed) but often do not depend on the initial metabolite concentrations. Steady-state fluxes are readily computed from the steady-state concentrations and the rate laws. The constraint-based formulation seems minimalist by comparison: it has no mechanistic knowledge of any of the chemical reactions beyond their stoichiometry, its solutions have fluxes at steady state but no information regarding concentrations or dynamics, and rather than giving a unique solution, it produces a high-dimensional continuum of steady-state solutions (referred to as a flux cone). The dynamic formulation needs significant information (parameters in term of rate constants and total enzyme concentrations, as well as reaction mechanisms to give rate laws), but generally rewards that effort with unique and detailed solutions. The constraint-based formulation requires less (no parameters except maximum fluxes) but delivers less.
Because of these significant differences between dynamic and constraint-based formulations, they treat the effects of network perturbations, which might be undertaken as part of a metabolic engineering study, very differently. A dynamic formulation will make very specific predictions about the response to a gene knockout, for example, but generally such models lack information about gene regulatory changes that accompany metabolic changes, and so without foreknowledge to adjust relative enzyme concentrations, such predictions can be significantly in error. Constraint-based formulations can access all possible steady-state solutions but can only rely on relatively simple heuristics to select among them, and are uncertain how to include specific information on gene regulatory changes.
Here we explore further the relationship between these formulations by essentially considering the continuous ensemble of dynamic formulations obtained by varying parameters (principally rate constants and enzyme concentrations) and compare the steady-state solutions to those from the corresponding constraint-based formulation. We find an equivalence between the sets of steady states when only maximum flux constraints are present, but that more specific constraints and enzyme concentrations can be directly incorporated to define a reduced dynamic ensemble that is significantly more informative regarding possible steady-state solutions than the constraint-based formulation.
Section snippets
Models
We have used a dynamic model of the central carbon metabolism of E. coli (Chassagnole et al., 2002) available at the Biomodels database (Le Novere et al., 2006). The model was converted from its original SBML format into a MATLAB (The Mathworks; Natick, MA, USA) file that was used for all computations in this work. The model consists of a total of 18 metabolites and 31 reactions, including several enzymatic reactions, one exchange reaction, and a few lumped versions of biosynthetic pathways.
Results
In order to explore the gap between both types of formulations, we analyzed and compared the dynamic and constraint-based formulations of the same model of the central carbon metabolism of E. coli (Chassagnole et al., 2002) (see Methods for model formulation).
Our goal is to compare the steady states achievable by the two model types. Intuitively the dynamic formulation has more constraints than the constraint-based one because the later only enforces the steady-state condition and maximum flux
Discussion
We have analyzed and compared dynamic and constraint-based formulations of the same model for the central carbon metabolism of E. coli (Chassagnole et al., 2002). The constraint-based version does not account for metabolite concentrations, and it does not express transient behavior. Therefore, the formulations can only be compared in their common domain, which is the steady-state flux distribution.
The constraint-based model defines a solution space for the steady-state flux distribution (called
Conclusions
In this work we have explored the solution spaces of both dynamic and constraint-based models in order to bring together top-down and bottom-up approaches, and we have proposed methods of treating each as well as their interrelation.
Dynamic model reconstruction is a bottom-up approach for iteratively building large-scale metabolic pathways with kinetic detail. Due to a lack of experimental data, differences in experimental conditions, and measurement uncertainty, the kinetic parameters are
Acknowledgments
This research was supported by PhD Grants SFRH/BD/35215/2007 and SFRH/BD/25506/2005 from the Fundação para a Ciência e a Tecnologia (FCT) and the MIT–Portugal Program through the project “Bridging Systems and Synthetic Biology for the Development of Improved Microbial Cell Factories” (MIT-Pt/BS-BB/0082/2008).
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