Network-based analysis of metabolic regulation in the human red blood cell
Introduction
It is now possible to construct cell-scale networks describing various cellular functions (Covert et al., 2001). A major challenge we are now faced with is to develop in silico modeling strategies that generate meaningful network-level analyses. Various modeling philosophies and approaches have been pursued, including deterministic kinetic models (Joshi and Palsson, 1989; Tomita et al., 1999; Werner and Heinrich, 1985), stochastic models (Arkin et al., 1998; Elowitz et al., 2002; McAdams and Arkin, 1997), and constraint-based models (Edwards et al., 2002; Palsson, 2000; Price et al., 2003a; Schuster et al., 2000). Constraint-based models use governing constraints, such as mass balance and maximum reaction rates, to restrict potential cellular behavior. This modeling approach has been applied to metabolism and results in a solution space in which all feasible steady state flux distributions reside (Covert et al., 2001; Palsson, 2000; Papin et al., 2003). The edges of this space, which form a convex cone, can be calculated and are called the extreme pathways (Fig. 1a). All valid steady state flux distributions through the metabolic network are nonnegative linear combinations of these extreme pathways. If all kinetic parameters are known, then kinetic models can be used to calculate a single solution that lies within the convex cone formed by the extreme pathways (Fig. 1a). If the kinetic constants are not fully known, then partial kinetic and flux information can be used to approximate the location of physiologically meaningful solutions. Thus, to characterize which regions of the steady state solution space are of interest, there is a need to develop methods that bridge the kinetic and constraint-based approaches. Furthermore, we need to find candidate regulatory strategies that direct solutions to these regions.
The human red blood cell provides an excellent model system for the development of biologically relevant computational methodologies. A kinetic whole-cell simulator based on a previously formulated model (Joshi and Palsson, 1989) has been developed for human red blood cell metabolism (Jamshidi et al., 2001). The extreme pathways for the red blood cell metabolic network are known, along with the maximum allowable flux through many of the reactions (Wiback and Palsson, 2002). Thus, both the constraint-based and kinetic models are available to describe the functionalities of the complete human red blood cell metabolic network. The kinetic regulation of metabolism in the red blood cell is also well known. For these reasons, the red blood cell network provides a good test system for evaluating analytical methods.
Regulation can be thought of as the process by which a cell “chooses” its state within the constrained solution space in which it must operate. Study of metabolic regulation has focused on the identification of large numbers of individual regulatory events, their characteristics, and the molecules that participate in them (Davidson, 2001; Ptashne and Gann, 2002). Now with reconstructed metabolic networks available we can approach the study of regulatory issues from a network-based perspective, focusing on the regulatory needs of the network as a whole. Understanding these systemic regulatory needs can provide a holistic view of the organization of individual regulatory components and how they are used to satisfy cellular objectives.
Mathematical approaches have been used to study cellular regulation. Elementary modes (Schuster et al., 2000), closely related to the concept of extreme pathways, have recently been used to make predictions regarding gene expression levels under changing environmental conditions (Stelling et al., 2002). Here we apply singular value decomposition of extreme pathway matrices to study metabolic regulation from a network-based perspective.
Section snippets
Extreme pathways in human red blood cell metabolism
Extreme pathways define the edges of a high dimensional cone that circumscribes all possible flux distributions in a metabolic network (Fig. 1a) (Schilling et al., 2000). They are calculated from a stoichiometric matrix, S, where the columns contain the stoichiometric coefficients of the metabolic reactions and rows correspond to the metabolites. Reversible internal reactions are decoupled into two separate reactions for the forward and reverse directions. The application of mass balance (Eq.
Applying SVD to extreme pathway matrices
The SVD of the extreme pathway matrix, P, has been used to characterize the steady state solution space circumscribed by the extreme pathways (Price et al., 2003b). Since the first eigenpathway corresponds to the greatest contribution to the reconstruction of P, it is the single vector that best characterizes the solution space (see Fig. 1b). By decoupling all reversible exchange fluxes in the extreme pathways, all elements of P are non-negative, ensuring that the first eigenpathway is a valid
SVD of the human red blood cell extreme pathway matrix
The rank of the Vmax-limited red blood cell extreme pathway matrix, P, was 23. The first eigenpathway has a fractional contribution of 47% (Fig. 2). Combined, the first five eigenpathways have a cumulative fractional contribution of 86%, while the cumulative fractional contribution of first nine eigenpathways is 96%. When the extreme pathways were normalized to a unit length rather than the limiting Vmax, more eigenpathways were needed to describe the same cumulative fractional contribution in
Discussion
The study of metabolic regulation has historically focused on the identification and characterization of individual regulatory events. Now that we can reconstruct full metabolic reaction networks we can address the need for regulation from a network-based perspective. This study focused on interpreting regulation from a network-based perspective using singular value decomposition of the extreme pathway matrix for human red blood cell metabolism. Each of the first five eigenpathways obtained by
Acknowledgements
The authors would like to thank Iman Famili and Dr. Radhakrishnan Mahadevan for insightful discussions, as well as Natalie Duarte for helpful feedback on this manuscript. We would like to acknowledge the support of the NIH (GM 57089) and the Whitaker Foundation (Graduate Research Fellowship to JP).
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These authors contributed equally to this work.