INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

A constraints-based approach to the mathematical modeling and in silico study of reconstructed metabolic networks has been developed (Palsson, 2000). This approach successively imposes governing constraints on a biochemical reaction network including connectivity, thermodynamic irreversibility, and maximum flux capacities to limit the steady-state flux solutions to a closed solution space (Schilling et al., 1999, 2000). Markedly, no kinetic parameters are used in defining this space. However, if the kinetic parameters are known the precise location of the steady-state flux distribution in the solution space can be found using a model that involves simultaneously solving a system of differential equations. Such comprehensive kinetic models are available for the human red blood cell (Mulquiney and Kuchel, 1999; Lee and Palsson, 1991; Joshi and Palsson, 1989, 1990), and such a precise solution can be calculated.

Steady-state analysis of stoichiometric networks has been reviewed recently (Schilling et al., 1999) and the steady-state solution space is a convex hull, or cone, where the edges are so-called "extreme pathways." An algorithm to compute the extreme pathways has been described (Schilling et al., 2000) and applied to the genome-scale metabolic network of Hemophilus influenzae (Schilling and Palsson, 2000). The extreme pathways for a genome-scale network are large and challenging to both compute and interpret (Papin et al., 2002). However, the subsequent imposition of gene expression regulation significantly reduces the number of allowable extreme pathways under a given condition (Covert et al., 2001b).

The difficulties associated with the computation and interpretation of large numbers of extreme pathways for real metabolic networks have hampered their detailed biochemical and physiological study. These computational issues arise from both the size and complexity of the metabolic networks. Although these cone-generating vectors correspond to biochemical pathways that represent steady-state flux maps, they have yet to be examined in detail for a biologically realistic metabolic system to determine their characteristics and usefulness in analyzing and interpreting integrated metabolic functions. The human red blood cell provides an attractive case to study the extreme pathways. Its metabolism contains four basic classical pathways: glycolysis, the pentose pathway, adenosine nucleotide metabolism, and the Rapoport-Leubering shunt. Unlike most metabolic networks, the red cell does not need to generate biomass; its main task is to produce the necessary cofactors (ATP, NADPH, and NADH) for maintaining its osmotic balance and electroneutrality and fighting oxidative stresses. The relatively simplistic demands on the red cell network serve to reduce the system's complexity and help make the computation of its extreme pathways manageable. The human red blood cell model accounts for 39 metabolites and 32 internal metabolic reactions, as well as 12 primary exchange and 7 currency exchange fluxes. The extreme pathways of this simple metabolic network can be readily calculated from the stoichiometric matrix. Herein we study these extreme pathways.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The metabolic network and its stoichiometric matrix

Thirty-nine metabolites are included in the red blood cell network (Table 1). To calculate the extreme pathways, those metabolites that can be exchanged with the system boundary (both as primary exchanges and currency exchanges) are identified (Schilling et al., 2000). Primary exchange metabolites are those that can be thought of as being transported across the cell membrane and exchanged with the environment (Fig. 1). Currency exchanges are exchanges of cofactors and metabolites that are produced and used by the metabolic network for such tasks as running the Na/K Pump (ATP), reducing met-Hemoglobin (NADH), fighting oxidative stresses to the cell via glutathione reduction (NADPH), and regulating hemoglobin oxygen affinity (2,3-DPG). The elemental composition of each metabolite in the network is given in Table 2 and it is used to ensure that every reaction in the network is elementally balanced. The metabolic reactions, excluding transporters, are shown in Table 3. The full red blood cell stoichiometric matrix is derived from these reactions and the corresponding metabolic map is shown in Fig. 1.

View larger version (48K): [in this window] [in a new window]   FIGURE 1   The red blood cell metabolic map depicting the classical pathways of glycolysis, the Rapoport-Leubering shunt, the pentose phosphate pathway, and adenosine metabolism (demarcated by the dashed lines). In addition, exchange fluxes with the system boundary are defined that include substrates/by-products, small molecules, and cofactors. Internal usage reactions for NADPH, NADH, ATP, and 2,3-DPG are also depicted. Note that the map does not include the F26BP bypass, which does not significantly change the extreme pathway analysis of the system; it simply adds another type II futile cycle that dissipates ATP.

Extreme pathway calculation and classification

The extreme pathways are calculated based on:
where S is the red blood cell stoichiometric matrix. We constrain all fluxes (v_i) to be positive so that no reaction can be run "negatively," violating the laws of thermodynamics. Thus, reversible reactions are broken down into their forward and backward components (Schilling and Palsson, 1998). Any steady-state flux distribution (v) within the cone can be described as a nonnegative linear combination of the extreme edges of the cone:
where the extreme pathways (p_i) are a set of generating vectors that describe a conical solution space (Schilling et al., 1999).

Extreme pathways can be divided into three main categories: type I, which are through pathways that utilize primary exchange fluxes as defined in Table 2; type II, which are futile cycles that only utilize currency exchange fluxes and degrade charged cofactors such as ATP; and type III, which are simply reversible reactions with no exchange fluxes involved (Schilling et al., 1999). Note that the type I pathways include traditional pathways with a single substrate in and a single product out, and the simultaneous production of cofactors.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Extreme pathway structure of the red blood cell

The computation of the extreme pathways for the red blood cell metabolic network resulted in 36 type I, 3 type II, and 16 type III extreme pathways. The type I and II extreme pathways are of most interest and will be focused on herein. The net reactions (exchanges only) are contained in Table 1 for both type I and II extreme pathways.

The type I and II extreme pathways can be described in detail based on their function and corresponding steady-state flux map. The complete collection of all 39 steady-state flux maps referred to below can be found in Fig. 5.

Glucose to pyruvate (GP)

Pyruvate/lactate conversion (PL)

2,3-DPG production (DPG)

Inosine to pyruvate (IP)

Salvage pathways using adenine (SP)

Nucleotide incorporation/removal via glucose and adenine (GA)

Nucleotide incorporation/removal via inosine and adenosine (IA)

Adenosine to inosine (AI)

Dissipation of ATP (type II pathways, DIS)

Maximal fluxes through the extreme pathways

Maximum flux capacities of the reactions serve to "cap off" the steady-state solution cone forming a closed polytope (Fig. 2 A). Every reaction has an estimated absolute V_max value of 1 × 10⁶ molecules/s/µm³ set by physico-chemical limitations. However, a few enzymes in the metabolic network will have a lower maximum flux capacity, i.e., low V_max due to kinetic limitations of the enzyme (Table 3). The enzyme with the lowest V_max in an extreme pathway serves as the "bottleneck" and determines the maximum possible flux through that extreme pathway. For instance, the low uptake rate of inosine limits the fluxes through all the extreme pathways in which it is utilized.

View larger version (26K): [in this window] [in a new window]   FIGURE 2   The three extreme pathways of this sample system form a conical solution space. (A) Full capabilities of the system where the cone is capped off at the minimum Vmax for each pathway; (B) A simulation of an enzymopathy in which the maximum flux capacity of v3 is greatly reduced. This reduction in the maximum capacity results in a shift down the p3 axis, thus significantly shrinking the size and volume of the steady-state solution cone, and hence the metabolic capabilities of the system.

Response to metabolic loads in red blood cell metabolism

There are four main physiologic loads experienced by the red blood cell, and the extreme pathways can be interpreted in terms of these metabolic demands.

1.	ATP loads are a combination of the Na/K ATPase-driven pump used in osmotic and ion balance as well as general ATP-related cell maintenance. ATP loads are experienced at the normal physiologic steady state as the pump must constantly be run and cellular volume maintained. Such loads are modeled by the conversion of ATP to ADP and Pi. Hence, some combination of extreme pathways that convert ADP to ATP must be utilized, which includes the GPs and the IPs, which is consistent with the nominal steady state in which ~80% of the flux in the system is through glycolysis (GP1 and GP5), ~15% is through the PPP (GP2-4, GP6-8), and the rest is through the adenosine reactions (IP1-4). If ATP is produced by the system in excess, it can be dissipated via one of the type II futile cycle pathways (DIS1-3);
2.	Oxidative loads in the red blood cell are combated via glutathione reduction, which must then be reoxidized by NADPH. There is a basal level of oxidative load on the cell at the physiologic steady state. Oxidative loads are modeled via the oxidation of NADPH to NADP. Because NAPDH can only be produced in the oxidative branch of the PPP, pathways GP2-4 and/or GP6-8 must be utilized;
3.	2,3-DPG loads are experienced in conditions such as high altitude, where the oxygen affinity of hemoglobin must be altered. However, at the physiologic steady state there is no drain on 2,3-DPG. The only pathways that produce 2,3-DPG which can be drained from the system are DPG1-6 which, while not "turned on" in nominal steady state, will become active under load;
4.	NADH loads are used to convert the unusable form of methylated hemoglobin (met-Hb) into the oxygen-carrying form of hemoglobin. Under normal, steady-state conditions, however, there is no drain on NADH. Hence, all the extreme pathways that result in the production of pyruvate are not technically utilized (GP1-8 and IP1-4). Rather, these pathways function in tandem with PL2 to balance NADH and produce lactate (Fig. 3). Note that such combinations of extreme pathways lie on a "face" of the conical solution space and represent an elementary flux mode (Schuster et al., 2000).

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Extreme pathways GP1 and PL2 can be "added" to produce a pathway on the face of the solution cone (an elementary mode) that balances NADH, as is the case in the physiologic steady state of the red blood cell.

Extreme pathways and the interpretation of physiological responses

The stoichiometry of a metabolic network, and hence the extreme pathway structure of that network, is relatively easy to obtain as compared with acquiring detailed kinetic knowledge about each enzyme in the network. A number of valuable physiological insights can be obtained from network structure and basic reaction capacity limitations, as the following examples demonstrate.

Projection of pathways based on production of key cofactors

View larger version (42K): [in this window] [in a new window]   FIGURE 4   Inset: A schematic of how a high-dimensional cone can be projected into a 2-D plane. (A) Projection of the red blood cell high-dimensional flux cone as defined by the extreme pathways into a 2-D cofactor space. The nominal steady-state value is shown by the dashed arrow. The red blood cell's capacity to respond to loads (hatched region) is defined as the difference between the steady-state operating point (black diamond) and the edge of the solution space representing the maximum capabilities of the cell (dotted line). Any loads outside the solution space are not attainable. The results from repeated dynamic simulation of stepwise increasing energy and oxidative loads on the red blood cell are plotted on the graph with open black circles using the Jamshidi model (Jamshidi et al., 2001). Note that the kinetic model is slightly more restrictive than the stoichiometric one. (B) 2-D projection into the cofactor space in which an enzymopathy has shortened GP1 to GP1' (decreased the minimum Vmax). The cell's ability to respond to energy loads (black hatched region) is decreased as compared with the normal cell (gray hatched region).

Change in global adenosine inventory in the red blood cell

Adjustment in 2,3-DPG concentration

SNPs and maximal capacities

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Extreme pathway analysis has been applied to the human red blood cell metabolic network. The resulting extreme pathways were analyzed and classified based on their structure and functional capabilities. The results of this study are 1) the establishment of a complete set of extreme pathways for a biologically meaningful system, 2) the finding that some of the extreme pathways correspond to "classical" biochemical pathways but most do not, and 3) a demonstration that extreme pathways can be used to interpret the steady-state solution space with respect to network capabilities.

Previously, extreme pathway analysis was applied to sample systems without real biological meaning (Schilling et al., 2000). Such systems helped in establishing the algorithm and interpreting the results, but provided no real biological insight. At the other extreme, the analysis has been applied to genome-scale metabolic network, resulting in an immense number of extreme pathways for which a detailed interpretation is not possible. Only statistical properties of these large sets of data could be obtained yielding limited insight into cellular physiology (Papin et al., 2002). This red blood cell study represents a situation where the full set of extreme pathways was calculated, detailed, and used for physiological interpretation. Scaling such detailed analysis to a genome-scale represents an unmet challenge in this field.

The systemic extreme pathways of the red blood cell metabolic network were fully enumerated and described (Fig. 5). In contrast to traditional experimental discovery and heuristic definitions of metabolic pathways, extreme pathway analysis provides a unique, mathematically defined way to identify systemically meaningful metabolic pathways that may be unintuitive, but no less informative. Interestingly, "historical" pathways such as glycolysis (GP1 in Fig. 5) and nucleotide salvage pathways (SP1-3 in Fig. 5) are extreme pathways. However, a majority of the extreme pathways are nontraditional multiple input-multiple output pathways such as those in which glucose and inosine are used in tandem to produce pyruvate and hypoxanthine, as well as the cofactors ATP and NADH (GP4 in Fig. 5). Such nontraditional pathways are an example of how extreme pathway analysis can elucidate systemic properties resulting from network interconnectedness and complexity, an essential feature of emerging systems biology.

View larger version (115K): [in this window] [in a new window]   FIGURE 5   Steady-state flux maps for all type I and type II red blood cell extreme pathways.

Extreme pathway analysis can also be used to interpret and predict the systemic consequences of maximum capabilities of individual reactions in the network (Fig. 4 A); a priori, a seemingly impossible task for a model based solely on stoichiometric information. In this case, the maximum cofactor production capacity of the red blood cell was conservatively predicted by the extreme pathway structure and the tolerance to elevated metabolic demands defined. The incorporation of basic transport and reaction V_max values and approximate metabolite concentration data expands the utility of extreme pathway analysis to include estimation of time constants with respect to pathway usage and prediction of the effect of enzyme defects on systemic function (Fig. 4 B).

With the emergence of systems biology comes a need for new methods for defining and understanding metabolic pathways as they pertain to network-scale functions. A recent article by Marcotte begs the question: "Why not abandon the old representation of pathways and instead work directly with the networks?" (Marcotte, 2001). Many different methods for such pathway definitions have been proposed to understand whole-cell metabolism, including Ouzounis's database definitions (Ouzounis and Karp, 2000), Schuster's elementary flux modes (Schuster et al., 1999, 2000), and Schilling's extreme pathways (Schilling et al., 2000). Such computational methods are essential for providing the link between mathematics and biology. Extreme pathways provide a unique way to define metabolic pathways in the era of systems biology. The main challenges that currently face extreme pathway analysis are their computation for genome-scale networks and the physiological interpretation of the results.

The present study represents the first complete analysis of the extreme pathway structure of a real metabolic system. Previously, the extreme pathway algorithm has either been applied to simple example systems for which the pathways could be interpreted but were not physiologically relevant, or to a genome-scale model for which the pathways were most certainly meaningful but were simply too numerous to individually interpret. The human red blood cell provides an ideal test bed that combines simplicity with physiologic significance. The results from the extreme pathway analysis show that network structure and capacity constraints provide a strong basis for analysis and interpretation of the physiology of the red blood cell metabolic network. Genome-scale metabolic networks can now be reconstructed from genomic and other data sources (Covert et al., 2001a). The use of extreme pathways for the analysis of such reconstructed networks and their relation to whole-cell functions will thus become critical in the advancement of systems biology.