ABSTRACT The human red blood cell (hRBC) metabolic network is relatively simple compared with other whole cell metabolic networks, yet too complicated to study without the aid of a computer model. Systems science techniques can be used to uncover the key dynamic features of hRBC metabolism. Herein, we have studied a full dynamic hRBC metabolic model and developed several approaches to identify metabolic pools of metabolites. In particular, we have used phase planes, temporal decomposition, and statistical analysis to show hRBC metabolism is characterized by the formation of pseudoequilibrium concentration states. Such equilibria identify metabolic "pools" or aggregates of concentration variables. We proceed to define physiologically meaningful pools, characterize them within the hRBC, and compare them with those derived from systems engineering techniques. In conclusion, systems science methods can decipher detailed information about individual enzymes and metabolites within metabolic networks and provide further understanding of complex biological networks.

INTRODUCTION

Biology has progressed with the detailed study of various "model" experimental systems, such as Drosophilia melanogaster, Saccharomyces cerevisiae, and Escherichia coli. It is likely that the same approach will be used for developing methodologies for the computational analysis of complex biological processes. Metabolism represents a logical starting point because it has been studied through mathematical modeling and computer simulation for a number of years (Bailey, 1998; Fell, 1996; Heinrich and Schuster, 1996; Reich and Sel'kov, 1981). The red cell has played a special role in the development of mathematical models of metabolism given its relative simplicity and the detailed knowledge about its molecular components.

Mathematical models of red blood cell (RBC) metabolism have been studied since the early 1970s (Brumen and Heinrich, 1984; Heinrich et al., 1977; Heinrich and Rapoport, 1973; Holzhutter et al., 1985; Rapoport et al., 1981; Schauer et al., 1981; Schuster et al., 1988; Werner and Heinrich, 1985). The models have steadily grown in scope leading to comprehensive RBC metabolic models in the late 1980s and 1990s (Holzhutter et al., 1990; Jacobasch and Rapoport, 1996; Joshi and Palsson, 1989a; Mulquiney and Kuchel, 1999; Rae et al., 1990; Schuster et al., 1988, 1989, 1990).

DISCUSSION

High-throughput experimental technologies and bioinformatics are placing an increased emphasis on mathematical modeling and computer simulation of cellular processes. The metabolic network in the hRBC will provide a useful "model" system for computational studies. Herein, we used a mathematical model of hRBC metabolism to study the use of computer simulation, modal analysis, phase plane analysis, and statistical analysis with the goal to unravel the complexities of this relatively simple metabolic network. The study showed 1) that these methods are well suited to study the relationship between all the detailed individual biochemical events and kinetic constants and the overall physiological function of red cell metabolism, 2) that network-wide definitions of metabolic pools and their charges with various metabolic currencies result in a few integrated measures that can be used to interpret overall behavior, 3) the metabolic charges are kept within a narrow range by the regulatory structure of the cell, indicating that the network strives to maintain homeostasis as well as it can given the constraints placed on it, and 4) the integrated function of a metabolic network can be described by a reduced set of variables.

Integrated models of red cell metabolism have been constructed for some 30 years (Ataullakhanov et al., 1981; Brumen and Heinrich, 1984; Heinrich and Rapoport, 1975; Jacobasch et al., 1983a,b; Lew and Bookchin, 1986; McIntyre et al., 1989; Rapoport and Heinrich, 1975; Schuster et al., 1988; Thorburn and Kuchel, 1987; Yoshida and Dembo, 1990). The result is a comprehensive, but not quite fully complete, model of all the major metabolic events in this simple cell. A full model contains not only the kinetic equations that describe enzyme kinetics, but also various constraints such as osmotic pressure and electroneutrality (Joshi and Palsson, 1989b; Werner and Heinrich, 1985). Given the numerical challenges associated with the physicochemical constraints, many studies, such as the present one, are focused on the kinetics alone.

Using phase planes, statistical time series, and modal analysis we can readily interpret fast motions in the network and pool metabolite concentrations into aggregate dynamic variables. This process is aided by representing the data as multiple matrices of pair-wise phase plane traces of metabolites representing dynamics on different time scales. Then using generalizations of Atkinson's energy charge (Atkinson, 1977) as a guide, further pooling leads to the definition of aggregates of concentrations that represent the network wide inventory of key metabolic currencies, such as highenergy phosphate bonds and two different types of redox potentials. Using these systemic variables, we can think of red cell metabolism as a battery that has a charge and capacity for these different metabolic currencies (e.g., see Reich and Sel'kov, 1981).

Dynamic simulation of the responses of the metabolic charges to various external loads is illuminating. The simulations show that, due to the interconnectivities of these pools, the cell responds in an analogous fashion to redox and energy loads by shifting its resources from one form to the other. The simulations also show that the network wide charges change numerically within narrow ranges in response to these loads. Thus, the overall regulatory scheme is coordinated to achieve homeostasis in terms of these pooled variables, but not necessarily in terms of the individual concentrations. If the loads placed on the cell are too severe, it cannot achieve this homeostasis, and its entire metabolism collapses. Therefore, the in silico analysis leads to the definition of what may be thought of as "emergent properties" of the network.

These in silico results call for further work on the hRBC. First, an experimental program should be designed to address the interpretations and predictions made herein. Further, a similar in silico study should be performed with the full model that includes all the physicochemical constraints on the cell. Second, an in silico analysis of the regulation that achieves the global properties described herein is needed. Deciphering the relative roles of network structure, mass action kinetics, and deliberate metabolic regulation of individual enzymes is needed. Finally, the lessons learned from the hRBC may be generalized to metabolic networks in cells that have less kinetic detail available by using the type of aggregate variables defined herein and the determination of the kinetic constants needed to construct such a description.

Financial support was provided in part by a National Science Foundation Graduate Research Fellowship DGE-9616051 (to K.J.K.), a NASA Graduate Student Researchers Fellowship NGTS-50367 (to K.J.K.), the National Institutes of Health I P20 RR15598-01, the Department of Energy DE-FG02-01ER63200, and the University of Delaware Research Foundation LTR20010426.

Submitted September 21, 2001, and accepted for publication April 30, 2002.

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Kenneth J. Kauffman,* John David Pajerowski,* Neema Jamshidi,^ Bernhard O. Palsson,^ and Jeremy S. Edwards*

*Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 USA; and ^Department of Bioengineering, University of California-San Diego, La Jolla, California 92093 USA

J. D. Pajerowski's current address is Physiome Sciences, Inc., 150 College Road West, Suite 300, Princeton, NJ 08540.

Address reprint requests to Jeremy Edwards, Department of Chemical Engineering, University of Delaware, Newark, DE 19716; Tel.: 302-8318072; Fax: 302-831-1048; E-mail: edwards@che.udel.edu.

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