Constraint-based modeling has proven to be a useful tool in the analysis of biochemical networks. To date, most studies in this field have focused on the use of linear constraints, resulting from mass balance and capacity constraints, which lead to the definition of convex solution spaces. One additional constraint arising out of thermodynamics is known as the "loop law" for reaction fluxes, which states that the net flux around a closed biochemical loop must be zero because no net thermodynamic driving force exists. The imposition of the loop-law can lead to nonconvex solution spaces making the analysis of the consequences of its imposition challenging. A four-step approach is developed here to apply the loop-law to study metabolic network properties: 1), determine linear equality constraints that are necessary (but not necessarily sufficient) for thermodynamic feasibility; 2), tighten V^sub max^ and V^sub min^ constraints to enclose the remaining nonconvex space; 3), uniformly sample the convex space that encloses the nonconvex space using standard Monte Carlo techniques; and 4), eliminate from the resulting set all solutions that violate the loop-law, leaving a subset of steady-state solutions. This subset of solutions represents a uniform random sample of the space that is defined by the additional imposition of the loop-law. This approach is used to evaluate the effect of imposing the loop-law on predicted candidate states of the genome-scale metabolic network of Helicobacter pylori.

INTRODUCTION

Constraint-based modeling has emerged as an important tool for assessing the properties of reconstructed genome-scale biochemical networks (1). Central to this modeling philosophy is the imposition of fundamental physico-chemical constraints on reconstructed networks to define a solution space that contains all the allowable functional states of networks that do not violate the constraints. To date, the constraint-based modeling approach has primarily used linear constraints to define convex solution spaces, which can be characterized by constraint-based optimization methods. Linear programming has been used for such purposes as predicting the lethality of knockouts (2), optimal growth rates (3), ranges of achievable fluxes (4), and even the endpoint of in vitro evolution (5). Bilevel optimization methods have been used to generate strain designs (6,7) and to calculate the minimum gene expression changes after gene knockout (8). Unbiased methods not requiring the statement of an objective function have also been used to characterize these solution spaces. For example, the edges of this space form a set of convex basis vectors that have led to the development of network-based pathways (9,10). The nonnegative combination of these edge vectors can be used to span the solution space. More recently, uniform random sampling of these convex solution spaces has been utilized to find a high-flux backbone in Escherichia coli metabolism (11), the effect of enzymopathies in human red blood cells (12), and the effects on mitochondrial metabolism of diabetes, ischemia, and diet (13).

Some physico-chemical constraints of importance in analyzing biochemical network functions, such as thermodynamic constraints, may be stated in the form of bilinear or nonlinear constraints that led to nonconvex spaces. This effort has been pioneered recently by the work of D. Beard, H. Qian, and colleagues (14-18). Systemic thermodynamic constraints are analogous to Kirchhoff's second law for electrical circuits (19). It simply states that the net flux around a closed biochemical loop must be zero because there is no net thermodynamic driving force (14). It is thus known as the "loop law" for reaction fluxes. Flux balance solutions violating the loop-law should be eliminated from the set of allowable network states. The present study will 1), develop methods to incorporate the loop-law in the procedure for uniformly sampling metabolic network states; and 2), demonstrate the effect of including these constraints on sampling candidate states of Helicobacterpylori's genome-scale metabolic network.

MATERIALS AND METHODS

H. pylori metabolic network and associated constraints

The metabolic network for H. pylori used herein was recently reconstructed (13). Briefly, H. pylori iIT341 GSM/GPR is a genome-scale metabolic model (GSM). where reactions are associated to the protein(s) that catalyze them, and to the associated gene(s). In total, iIT341 GSM/GPR accounts for 341 metabolic genes, 476 internal reactions, 411 internal metabolites, and 74 external metabolites. iIT341 GSM/GPR is connected with its in silico environment by 74 exchange reactions. The corresponding S-matrix has 485 metabolites and 558 reactions, including a biomass function, and demand functions for thiamin, menaquinone 6, biotin, and heme (protoheme).

These two equations can only be satisfied simultaneously if the net flux around a biochemical loop is equal to zero, resulting in the loop-law for reaction fluxes. Throughout this article, the term "loop law" will always refer to the constraint on the reaction fluxes.