Exploring the gap between dynamic and constraint-based models of metabolism

Abstract

Systems biology provides new approaches for metabolic engineering through the development of models and methods for simulation and optimization of microbial metabolism. Here we explore the relationship between two modeling frameworks in common use namely, dynamic models with kinetic rate laws and constraint-based flux models. We compare and analyze dynamic and constraint-based formulations of the same model of the central carbon metabolism of Escherichia coli. Our results show that, if unconstrained, the space of steady states described by both formulations is the same. However, the imposition of parameter-range constraints can be mapped into kinetically feasible regions of the solution space for the dynamic formulation that is not readily transferable to the constraint-based formulation. Therefore, with partial kinetic parameter knowledge, dynamic models can be used to generate constraints that reduce the solution space below that identified by constraint-based models, eliminating infeasible solutions and increasing the accuracy of simulation and optimization methods.

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.