Networklevel analysis of metabolic regulation in the human red blood cell using random sampling and singular value decomposition
 Christian L Barrett^{1},
 Nathan D Price^{2} and
 Bernhard O Palsson^{1}Email author
DOI: 10.1186/147121057132
© Barrett et al; licensee BioMed Central Ltd. 2006
Received: 13 August 2005
Accepted: 13 March 2006
Published: 13 March 2006
Abstract
Background
Extreme pathways (ExPas) have been shown to be valuable for studying the functions and capabilities of metabolic networks through characterization of the null space of the stoichiometric matrix (S). Singular value decomposition (SVD) of the ExPa matrix P has previously been used to characterize the metabolic regulatory problem in the human red blood cell (hRBC) from a network perspective. The calculation of ExPas is NPhard, and for genomescale networks the computation of ExPas has proven to be infeasible. Therefore an alternative approach is needed to reveal regulatory properties of steady state solution spaces of genomescale stoichiometric matrices.
Results
We show that the SVD of a matrix (W) formed of random samples from the steadystate solution space of the hRBC metabolic network gives similar insights into the regulatory properties of the network as was obtained with SVD of P. This new approach has two main advantages. First, it works with a direct representation of the shape of the metabolic solution space without the confounding factor of a nonuniform distribution of the extreme pathways and second, the SVD procedure can be applied to a very large number of samples, such as will be produced from genomescale networks.
Conclusion
These results show that we are now in a position to study the network aspects of the regulatory problem in genomescale metabolic networks through the use of random sampling.
Contact: palsson@ucsd.edu
Background
In recent years, chemicallydetailed metabolic networks capable of simulating growth have been reconstructed at the genomescale for nearly twenty organisms [1]. Any metabolic network can be represented as a stoichiometric matrix, and by assuming steadystate conditions one can compute the "space" in which all allowable steadystate flux distributions reside. This space can be spanned by a set of extreme pathways (ExPas) [2, 3], and for a given metabolic network this set is unique, invariant and irreducible [4]. ExPas are so named because they form the edges of the steadystate solution space of the network and thus characterize its "extreme" functions. Mathematically, the ExPas are a set of conical basis vectors that forms a convex cone in highdimensional space. In this manner, the extreme pathways circumscribe all of the possible steadystate flux distributions possible in the metabolic network. Biochemically, each ExPa corresponds to a steadystate flux distribution through the network, and is more than just a linear set of reactions linking substrate to product. Specifically, each ExPa describes the relative fluxes among a set of reactions needed for the conversion of substrate(s) into product(s), and does so by creating all byproducts needed to maintain systemic balances and by simultaneously maintaining all cofactor pools at steady state. The development of such networkbased pathway definitions, including the elementary flux modes [5–7], has led to a systems view of network properties [8]. Genomescale metabolic pathway analysis [5, 9, 10] holds much promise for understanding fundamental aspects of network functionality, robustness, and regulation [11, 12], for metabolic engineering [13, 14], and for elucidating many other systems properties [12, 15–23].
The number of ExPas for a reaction network grows exponentially with network size, limiting their use for the analysis of large networks. Because of this limitation, ExPas have mainly been used to analyze realistic, but small, networks such as the hRBC [22]. In further work [4, 24], singular value decomposition (SVD) was applied to the ExPa matrices (P), resulting in two key insights. First, it was shown that SVD of P can compute the effective dimensionality of the regulatory problem from a network perspective. Second, it was shown that the 'eigenpathways' that correspond to the modes of the SVD identified key network branch points that can represent key control points (reactions) in the network. These points constitute a minimal set of reactions whose control sets the steady state flux state for the entire network, represented as a specific location in the steadystate solution space. SVD rank orders these key branch points in accordance with the shape of the solution space and thus rank orders the effectiveness by which the branch points can be used to move the network's flux distribution around in the solution space. These features were demonstrated by showing that the dominant features of the first five eigenpathways correspond to actual major points of control in the hRBC and can effectively locate the solution in a 5dimensional space [24]. SVD of P can thus define the magnitude and nature of the regulatory problem associated with controlling a metabolic network.
Given the fundamental understanding of network properties that can be derived from the SVD of P, it would be natural to perform the same analysis on the much larger genomescale metabolic networks that have already been reconstructed. Unfortunately, this is not currently possible due to the NPhard nature of calculating ExPas. Even if computing ExPas were possible on a genomescale, the likely result would be matrices containing extremely large numbers of ExPas, many more than would be needed using uniform random sampling. This number of ExPas presents an additional formidable hurdle for the SVD calculation, which is an O(n^{2}) algorithm [25].
In this work, we demonstrate an approach that addresses this problem. As mentioned above, the set of ExPas for a metabolic network defines the shape of the steadystate solution space of the network. The information that SVD of the set of ExPas provides is a description of the space. The eigenvectors that are produced by the SVD indicate, in rank order, the directions corresponding to the greatest variance in the highdimensional space, and the singular values give the relative magnitude of this variation. A potentially better approach is to characterize the shape of the solution spaces by generating uniform random samples within it. Each sample solution is used as a column in a matrix (W). With enough samples, a "picture" of the solution space emerges, which can then be analyzed using SVD in the same manner as was done previously for the ExPas. In this work, we performed uniform random sampling of the steadystate solution space of the hRBC metabolic network and applied SVD to the matrix, W, of these samples. We show that similar, though not identical, results are obtained through SVD of W as were obtained through SVD of P. Thus, this study provides a new method for the calculation of Principle Regulatory Modes (PRMs) that improves upon the previous method by being more tractable for genomescale networks due to its independence from ExPas. Additionally, the interpretation of the results of the new method are more straightforward because it characterizes the shape of the solution space uniformly, without being affected by the nonuniformity of the distribution of ExPas on the edges of the solution space.
Results
Ascertaining sampling sufficiency
Comparing the modes derived from P and W
Metabolite abbreviations used in Figures 2–6.
Abbr.  Metabolite  Abbr.  Metabolite 

13DPG  1,3diphosphoglycerate  GLU  glucose 
23DPG  2,3diphosphoglycerate  HX  hypoxanthine 
2PG  2phosphoglycerate  IMP  inosine monophosphate 
3PG  3phosphoglycerate  INO  inosine 
6PGC  phosphogluconate  LAC  lactate 
6PGL  phosphoglucolactone  PEP  phosphenolpyruvate 
ADE  Adenine  Pi  inorganic phosphate 
ADO  Adenosine  PRPP  5phosphoribosyl1pyrophosphate 
DHAP  dihydroxyacetone phosphate  PYR  pyruvate 
E4P  erythrose4phosphate  R1P  ribose1phosphate 
F6P  fructose6phosphate  R5P  ribose5phosphate 
FDP  fructose1,6bisphosphate  RL5P  ribulose5phosphate 
G6P  glucose6phosphate  S7P  sedoheptulose7phosphate 
GA3P  glyceraldehyde3phosphate  X5P  xylulose5phosphate 
Reaction abbreviations used in Figures 2–6. The corresponding reaction stoichiometries are also included.
Abbr.  Enzyme Name  Metabolic Reaction 

ADA  Adenosine deaminase  ADO + H2O → INO + NH3 
AdPRT  Adenine phosphoribosyl transferase  PRPP + ADE + H2O → AMP + 2 Pi 
AK  Adenosine kinase  ADO + ATP → ADP + AMP 
ALD  Aldolase  FDP ↔ GA3P + DHAP 
AMPase  Adenosine monophosphate phosphohydrolase  AMP + H2O → ADO + Pi + H 
AMPDA  Adenosine monophosphate deaminase  AMP + H2O → IMP + NH3 
ApK  Adenylate Kinase  2 ADP ↔ ATP + AMP 
DPGase  Diphosphoglycerate phosphatase  23DPG + H2O → 3PG + Pi 
DPGM  Diphosphoglyceromutase  13DPG → 23DPG + H 
EN  Enolase  2PG ↔ PEP + H2O 
G6PDH  Glucose6phosphate dehydrogenase  G6P + NADP → 6PGL + NADPH + H 
GAPDH  Glyceraldehyde phosphate dehydrogenase  GA3P + NAD + Pi ↔ 13DPG + NADH + H 
HGPRT  Hypoxanthine guanine phosphoryl transferase  PRPP + HX + H2O → IMP + 2 Pi 
HK  Hexokinase  GLU + ATP → G6P + ADP + H 
IMPase  Inosine monophosphatase  IMP + H2O → INO + Pi + H 
LD  Lactate dehydrogenase  PYR + NADH + H ↔ LAC + NAD 
PDGH  6phosphoglycononate dehydrogenase  6PGC + NADP → RL5P + NADPH + CO2 
PFK  Phosphofructokinase  F6P + ATP → FDP + ADP + H 
PGI  Phosphoglucoisomerase  G6P ↔ F6P 
PGK  Phosphoglycerate kinase  13DPG + ADP ↔ 3PG + ATP 
PGL  6phosphoglyconolactonase  6PGL + H2O ↔ 6PGC + H 
PGM  Phosphoglyceromutase  3PG ↔ 2PG 
PK  Pyruvate kinase  PEP + ADP + H → PYR + ATP 
PNPase  Purine nucleoside phosphorylase  INO + Pi ↔ HX + R1P 
PRM  Phosphoribomutase  R1P ↔ R5P 
PRPPsyn  Phosphoribosyl pyrophosphate synthetase  R5P + ATP → PRPP + AMP 
RPI  Ribose5phosphate isomerase  RL5P ↔ R5P 
TA  Transaldolase  GA3P + S7P ↔ E4P + F6P 
TK1  Transketolase  X5P + R5P ↔ S7P + E4P 
TK2  Transketolase  X5P + E4P ↔ F6P + GA3P 
TPI  Triose phosphate isomerase  DHAP ↔ GA3P 
XPI  Xylulose5phosphate epimerase  RL5P ↔ X5P 
Assessing the correspondence of inferred reaction regulatory importance
One of the primary insights resulting from the application of SVD to P was an understanding of the relative importance amongst all of the reactions in terms of regulatory control. The aim of this work was to show that similar insight can be obtained through uniform random sampling. To do this, we followed a previously developed procedure [26] and scored each reaction by the l^{2}norm (length) of its loading vector, which is a vector of the reaction's contribution to each of the eigenvectors resulting from the SVD procedure (see Methods section). The rationale for this scoring method is based on our goal of identifying those reactions most critical for controlling flux through the metabolic network. These control reactions are expected to have larger contributions to the eigenvectors resulting from the SVD procedure. The impact of these larger contributions is reflected in a larger absolute value of the loading vector, signifying a larger distance of the reaction from the origin. So in this context the l^{2}norm quantifying the length of the loading vector is appropriate for our purposes.
Scores allow us to impart a rank ordering on the reactions. The degree to which the SVD of W and the SVD of P provide similar information about the regulatory importance of the reactions, then, is measured by the similarity in which they rank order the reactions by their scores. We measured this similarity by computing the rank correlation coefficient between the two rank orderings of reactions. Just like Pearson's correlation coefficient, the rank correlation has a range [1, 1]. We used two related rank correlation measures [27]: Kendall's tau and Spearman's rho. We found the reaction rankings to be highly correlated, resulting in a tau value of 0.88 with Pvalue 1.1*10^{15} and a rho value of 0.97 with Pvalue 4.0*10^{12}. Based on these results, we claim that the sampling methodology produces similar insight into identifying the most important reactions for flux control as did the ExPabased approach.
Reconstruction of physiological flux distributions
Genomescale sampling
Discussion
In previous work [24] it was demonstrated that SVD of the extreme pathway matrix for the hRBC metabolic network revealed the essential dimensionality of the regulatory problem. Furthermore, it was shown that the dominant features of the eigenpathways corresponding to the five largest singular values corresponded to control points in the hRBC metabolic network. These results are significant because they demonstrate a potentially powerful method for developing networkbased understanding of regulation in genomescale metabolic networks. Unfortunately, the method relies on the computation of the extreme pathways for these networks – a computational exercise that is currently infeasible. In an effort to circumvent this obstacle, we investigated the utility of randomly sampling the steadystate solution space of the hRBC metabolic network, and through SVD of the matrix of sampled solution points arrived at comparable results to those obtained via SVD of P. We present three main results. First, we found that the sampling procedure did lead to the same estimates of the dimensionality of the regulatory problem for the metabolic network studied. Secondly, we demonstrated that the resulting PRMs reconstruct physiological flux distributions as efficiently as did the eigenpathways. Thirdly, we showed that the flux adjustments represented by the PRMs were similar, and similarly interpretable, as those represented by the eigenpathways. In particular, we found that the ranking of reactions by their regulatory "importance" in the two approaches was highly correlated with high statistical significance.
It is important to note that there is no a priori reason for believing that SVD of P would be superior to SVD of W. Thus, while the focus of this work was to explain the results of SVD of W in the context of what had already been shown for SVD of P, it is not necessary that an effort be made to force SVD of W to reproduce SVD of P. Instead, what was shown herein is that the approaches can be thought of as attacking the same problem and that SVD of W provides a way forward into genomescale matrices to the line of inquiry that was begun with SVD of P, since sampling of genomescale metabolic states has been accomplished at the genomescale for E. coli and H. pylori [29, 30] – whereas the computation of P for these networks has proven infeasible. The results of the two methods will not generally be identical. In the case of SVD of P not only the shape of the space, but, more precisely, the distribution of the pathways along the edges of the space is what is being characterized. Because the samples are generated uniformly in the solution space, it is the shape of the solution space alone that is being characterized in this approach. Thus, the interpretation of SVD of W is more straightforward than is the interpretation of SVD of P.
We used physiological flux distributions computed from a kinetic model and compared how well PRMs and eigenpathways reconstructed them. As can be seen in Figure 7, the PRMs performed just as well, if not better, than the eigenpathways. Indeed, in three instances, the PRMs arrived at essentially complete reconstructions utilizing fewer PRMs. Based on these results, we estimate the regulatory problem for the hRBC to be roughly fivedimensional – in agreement with the estimate based on the eigenpathways.
The fractional projection of eigenpathways and PRMs onto one another.
from\onto  eigpath1  eigpath2  eigpath3  eigpath4  eigpath5  PRM1  PRM2  PRM3  PRM4  PRM5 

eigpath1  0  0.021  0.021  0.126  0.384  0.950  0.107  0.077  0.247  0.054 
eigpath2  0.022  0  0.039  0.187  0.300  0.028  0.983  0.016  0.195  0.168 
eigpath3  0.025  0.045  0  0.011  0.252  0.212  0.189  0.671  0.899  0.231 
eigpath4  0.114  0.165  0.009  0  0.108  0.049  0.029  0.280  0.378  0.763 
eigpath5  0.387  0.295  0.214  0.120  0  0.282  0.284  0.430  0.584  0.560 
PRM1  0.957  0.027  0.180  0.055  0.282  0  0  0  0  0 
PRM2  0.108  0.966  0.160  0.033  0.284  0  0  0  0  0 
PRM3  0.078  0.016  0.569  0.311  0.430  0  0  0  0  0 
PRM4  0.248  0.192  0.761  0.421  0.584  0  0  0  0  0 
PRM5  0.055  0.165  0.195  0.848  0.560  0  0  0  0  0 
The flux maps in Figures 2, 3, 4, 5, 6 demonstrate additional satisfying correspondence between PRMs and eigenpathways. First, the principal singular mode (first PRM) is a valid flux distribution, adhering to mass balance and thermodynamic constraints just as is the case with the principal eigenpathway. Second, the regulatory correspondence that was achieved with the eigenpathways is also possible with the PRMs. For example, the dominant feature of the second PRM is also the flux split between glycolysis and the pentose phosphate pathway. The third singular mode describes, similarly to the third eigenpathway, the glycolytic pathway with the production of ATP, but instead of proceeding through to pyruvate with the production of NADH, it proceeds through to lactate with the utilization of NADH to produce NAD. The fourth PRM, as can be seen in Figure 5, is closer to eigenpathway 5, and this is due to the dominant feature of pyruvate uptake and subsequent conversion to lactate. Finally, the fifth PRM shares the same dominant characterization as eigenpathway 4, this being the flux split between lower glycolysis and the RapoportLuebering shunt.
The PRMs of W have a slightly different interpretation than the eigenpathways derived from P. Mathematically, the modes of W are the orthogonal directions of maximal variance in the steadystate solution space of the stoichiometric matrix. Biochemically, they correspond to the directions of greatest flexibility in achieving a networkwide flux distribution. The unequal variances in these orthogonal directions imply a hierarchy, wherein control of the more dominant modes gives the cell more control of achieving a desired flux distribution.
Conclusion
From these results we conclude that SVD of a sample of uniformly obtained steady state flux distributions for a metabolic network leads to a similar means of characterizing the regulatory problem as was obtained previously with ExPas. This was a necessary demonstration on a small system, before applying the technique to genomescale networks. With a number of genomescale metabolic networks now significantly reconstructed for a number of organisms, the results presented in this work indicate that it should now be possible – given that feasible genomescale sampling has been demonstrated [29, 30] – to realize the dimensionality and mechanisms of metabolic regulation implicit in the topology of these networks.
Methods
Human red blood cell model
For this work we utilized a previouslyconstructed kinetic model [31] and a stoichiometric model [22, 24] of the human red blood cell. The stoichiometric model accounts for 39 metabolites and 32 internal metabolic reactions, and also contains 12 primary exchange fluxes and 7 currency exchange fluxes.
Physiologic steadystate flux distributions
Four steadystate flux distributions were calculated using an in silico kinetic model [31], as detailed in earlier work [24], and used to represent physiological states of interest in the solution space. The first distribution was for an unstressed state or normal physiological state, corresponding to no metabolic load. The other three corresponded to the primary metabolic demands placed upon the hRBC based on its physiologic role in the human body. These demands were for ATP (energy load), NADPH (oxidative load), and NADH (methemoglobin load). In each case, the load was maximized in the kinetic model to the highest value possible where the model could still be solved. Thus, they represented as high a load as the network could handle. All these states were previously computed and used in a previous study [24]. These maximum values were 0.37 mM/h for ATP load (above a mandatory ~1 mM/hr running of the Na^{+}/K^{+} pump), 2.5 mM/h for the NADPH load, and 1.7 mM/h for the NADH load.
Physiologic flux constraints
The null space of the stoichiometric matrix of the hRBC metabolic network is a (mathematically) unbounded space. To transform this space into one that is physiologically relevant, we applied, as before, constraints by bounding the minimum and maximum fluxes for each reaction. All irreversible reactions were constrained with a minimum flux of 0 mM/h and a maximum flux of 2000 mM/h unless experimentallydetermined maximum flux values were available. Such maximum reaction rates were available for 22 reactions, and have been published previously [22, 32]. Additionally, certain fluxes must have a minimum, nonzero value in order to sustain the hRBC. These values were obtained from a previous study on the hRBC [32, 33], and since they are few in number are given here: DPGase, 0.3 mM/h; ATP production, 1 mM/h; NADH production, 0.4 mM/h; and NADPH production, 0.05 mM/h.
HitandRun sampling of the solution space
We use a variant of an artificial centering hitandrun algorithm [34] to sample random flux distributions in the steadystate solution space of the hRBC. The algorithm is a random walk algorithm entailing three steps. The first step is to generate a set of "warmup" points in the solution space. This generation of warmup points was performed by slightly reducing the maximum flux and increasing the minimum flux for all reactions and then calculating a candidate flux distribution along a reaction coordinate using linear programming. Ten thousand such warmup points were created. The second step of the algorithm is to select an initial point from one of the "warm up" points generated. The warm up points were stored in a matrix W and were used to calculate an approximate center s of the solution space. The third step calculates the sample points. Beginning with an initial, randomly selected point x_{ m }from W, a direction vector is constructed by choosing a randomly selected point y from W and moving along the direction vector d = ys from x_{ m }resulting in a new point x_{m+1}. After each such move x_{m+1}was randomly substituted into W and the approximate center recalculated. For the results presented herein, we saved a sample point every 1000 moves in the space. It should be noted that the red blood cell metabolic network does not have any nontrivial biochemical loops (type III ExPas) [35], and thus the samples generated do not violate "loop law" thermodynamic constraints [36, 37] that would otherwise need to be explicitly addressed in sampling [30]. Note that in previous work [22], it was stated that the hRBC contains 16 Type III ExPas. The computation of ExPas in that work entailed the decoupling of reversible reactions into forward and backward reactions, and these resulting trivial loops were the 16 Type III pathways. In this work, reversible reactions were not decoupled, and so trivial Type III pathways were not an issue. One issue not encountered in this work which will is important for sampling genomescale networks is the issue of nontrivial Type III ExPas (more than 2 reactions). Since Type III ExPas correspond to thermodynamically infeasible flux distributions, the generation of samples from genomescale systems will need to be cognizant of this issue. This issue has been addressed elsewhere and will not be an impediment to sampling genomescale metabolic networks [30].
Extreme pathways
The extreme pathways were calculated from the stoichiometric matrix of the hRBC using a previously published algorithm [2]. V_{max} values for the reactions in the hRBC have been previously published [22, 32]. Each extreme pathway was scaled by the most limiting V_{max} amongst all reactions that it contained. All of the extreme pathways had at least one reaction with an experimentally defined V_{max}.
Angle between vectors
Angles between modes were computed using standard linear algebra. For two vectors u and v, the angle was computed as Θ = cos^{1} (u·v/u v).
Regulatory importance of reactions
The result of applying the SVD to either the matrix of ExPas or the matrix of uniformly sampled flux distributions is a set of eigenvectors. Each reaction has a corresponding loading vector, which is the vector of the reaction's contribution to each of the eigenvectors. Specifically, if v_{ij} is the contribution of reaction j to eigenvector i, and there are m eigenvectors, then the loading vector for reaction j is l_{ j }= (v_{1j}, v_{2j}, ..., v_{ mj }). The score for reaction j is then given by the l^{2}norm of the loading vector:  l_{ j } = (v_{1j}^{2} + v_{2j}^{2} + ... + v_{ mj }^{2})^{1/2}.
Abbreviations
 SVD:

singular value decomposition
 ExPa:

extreme pathway
 PRM:

principal regulatory mode
Declarations
Acknowledgements
This work was funded in part by NIHGMS grant number GM068837. BOP serves on the Scientific Advisory Board of Genomatica Inc.
Authors’ Affiliations
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