Mass Conservation and Inference of Metabolic Networks
from High-Throughput Mass Spectrometry Data
Pradeep Bandaru1,2, Mukesh Bansal1, and Ilya Nemenman3*
1Center for Computational Biology and Bioinformatics, Joint Centers for Systems Biology, and
Columbia Initiative in Systems Biology, 2Department of Biomedical Informatics, Columbia
University, New York, NY 10032, USA
3Departments of Physics and Biology and Computational and Life Sciences Strategic Initiative,
Emory University, Atlanta, GA 30322
We present a step towards the metabolome-wide computational inference of cellular
metabolic reaction networks from metabolic profiling data, such as mass spectrometry. The
reconstruction is based on identification of irreducible statistical interactions among the
metabolite activities using the ARACNE reverse-engineering algorithm and on constraining
possible metabolic transformations to satisfy the conservation of mass. The resulting algorithms
are validated on synthetic data from an abridged computational model of Escherichia coli
metabolism. Precision rates upwards of 50% are routinely observed for identification of full
metabolic reactions, and recalls upwards of 20% are also seen.
Prior to the widespread availability of annotated metabolic databases, metabolic network
reconstruction was carried out primarily with biochemical assays of enzymatic activity (1-3).,
resulting in a pathway-centric depiction of chemical reactions occurring in a cell. For some
organisms, it has been possible to assemble these data into genome-wide metabolic networks by
means of various Metabolic Flux Analysis (MFA) techniques (4-9). Now, recent developments
in the burgeoning field of metabolic profiling (10-12) and especially mass-spectrometry
metabolomics (13-17), which aims at high-throughput, real-time characterization of the entire
cellular metabolic state, have opened up yet another way of approaching the metabolic network
reconstruction problem, focusing on statistical interactions among metabolites. This parallels the
transition that had happened in the analysis of transcriptional regulatory networks with the
advent of gene expression profiling (18-22), which similarly characterizes the genome-wide
transcriptional state of the cell.
Specifically, distinct cellular phenotypes, phases of the cell cycle, or intrinsic and
extrinsic perturbations result in changes in cellular metabolite concentrations. However, even
with such changes, concentrations of metabolites that transform into each other should stay
correlated, and the observed dependencies can be used to predict metabolic reactions
computationally even if the identities of the metabolites are unknown, preventing the application
of MFA methods. We attempted this approach (23) using the ARACNE statistical reverse
engineering method, first developed in the context of inferring transcriptional networks from
mRNA expression profiles (19, 24, 25). However, mass-spectrometry methods provide a wealth
of information about the metabolites in addition to their abundances: MS-MS methods and
isotopic labeling (13-17) can recover the molecular structure, and, especially crucial for this
paper, masses of metabolites are typically measured to the accuracy of 104K 105. Since mass
must be conserved in any metabolic transformation, this information presents an additional
source of data for reverse engineering methods that has not been widely used. Namely, a
statistical dependence among metabolites can indicate an actual metabolic transformation only if
the putative chemical reaction constructed from these metabolites conserves mass. This rule can
be applied even when identities of metabolites are unknown and, therefore, it is especially useful
for high-throughput global metabolic profiling where spectral peaks cannot necessarily be
In this paper, we present a Mass-Constrained adaptation of the ARACNE algorithm,
ARACNE-MC, which should be considered as a first foray into the field, laying the foundation
for future studies. Note that, in the case of metabolism, we are not content with knowing just the
statistical dependencies among the metabolites, even if they correspond to bona-fide metabolic
transformations. Instead we aim at a substantially more complicated task of reconstructing
complete metabolic reactions, identifying all of their substrates and products.
We test the algorithm on reduced toy models of the Escherichia coli metabolome with in
silico simulated metabolic profiles. Applications to real-world biological problems will have to
wait for larger experimental datasets.
Outlines of the algorithms
The ARACNE Algorithm
The basis of the reverse engineering approach we undertake is the notion that molecular
species that participate in biochemical reactions have statistically dependent expressions (23).
Within the ARACNE framework (24, 25), one views metabolite expressions as random variables
sampled from stationary probability distributions. This randomness accounts for effects of
unknown states of unobserved metabolites and other chemical species and of the experimental
noise. Chemical transformations correspond to nonzero multivariate statistical dependencies
among metabolite concentrations (26). In particular, the relevant measure of statistical
dependency between two variables c1,c2
is their mutual information (MI) (27)
P c1 P c2
where P c1,c2
denotes their joint probability distribution, P ci are the marginals, and
the average over the distribution P.
The MI is generally nonzero for bona fide interacting metabolites, but it may also be
nonzero for chemicals that are connected through an intermediate and do not transform into each
other directly. In fact such false positives are generally a bigger problem than false negatives
(i.e., missing a true interaction) in computational networks reverse engineering: false positives
are plentiful and lower the confidence in the validity of every specific prediction.
The ARACNE algorithm (24, 25) eliminates some of the false positives by using the data
processing inequality (DPI) (27) to isolate statistical interactions that have the highest chance of
corresponding to true biological transformations. Specifically, under certain assumptions that are
often applicable in transcriptional (24) and metabolic (23) contexts, if
min I c1;c2
then the c1 c3 interaction is indirect. Hence ARACNE starts with a fully connected graph of
the measured chemical species as putative interaction partners, compares MIs for every triplet of
chemical species in the dataset, and removes the weakest pairwise interaction in every such
triplet from further consideration. Practical complications in the application of ARACNE revolve
around accurate, unbiased estimation of MI and of the threshold (5-15% for typical applications)
above which a difference between two MI values becomes significant for the DPI application
(19, 24). Though developed for reverse engineering transcriptional networks, ARACNE has also
been validated in the context of synthetic metabolic networks (23).
The ARACNE-MC Algorithm
As we have emphasized, mass-spectrometry provides additional information about the
metabolic state of a cell, namely, masses of the metabolites. This creates extra means for
eliminating false positive metabolic transformations: even if statistical dependencies suggest an
interaction, the implied putative chemical reaction may not conserve mass and hence be
impossible. To use this additional constraint, we propose the ARACNE-MC algorithm (MC
stands for Mass Constrained). Like the original ARACNE, ARACNE-MC aims at reducing false
positives, potentially at the cost of increasing the false negatives.
For a list of metabolites a with masses ma and metabolic activities in i’th spectrometer
run cai, we start by building a list of putative metabolic reactions allowed by mass conservation,
focusing on a limited set of template reactions that are allowed in the analysis. In this paper, we
consider only three templates (a) 1x1, l1 r2; (b) 1x2, l1 r1 r2; and (c) 2x2,
l1 l2 r1 r2 (indexes l and r stand for left and right, respectively). See Fig. 1 for
example reactions of each template. Eliminating more complicated reactions from consideration
will result in the elimination of bona fide statistical interactions, and hence in extra false
negatives, which we accept. We build a list of all putative reactions that fall into the allowed
template classes and satisfy the mass conservation,
, where is the
mass equality tolerance, set to 104 throughout this paper. We refer to such reactions as
conforming reactions. Identifying them has a computational complexity of O(M), where is
the maximum number of metabolites on either side of the template (2 in this paper).
While possible in principle, a conforming reaction may not exist in a real cell due to a
multitude of factors. If present, it should result in statistical dependencies among its reactants and
products. There will be up to one such dependence for a 1x1 reaction (two choose two), three for
a 1x2 reaction (three choose two), and six for a 2x2 reaction (four choose two). Therefore, we
prune the list of conforming reactions by identifying those that are supported by statistical
dependencies. To do so, we apply ARACNE to metabolite activity profiles, cai. Specifically, we
estimate the pairwise MIs I ca,cb
using the algorithms of (19) and then apply the DPI to the list
of MIs to select the interactions that have the highest chance of being direct. Then the
conforming reactions are ranked by how many of their pairwise member metabolite interactions
are identified as direct by ARACNE. We expect that reactions that are conforming and supported
statistically will have a high chance to be bona fide metabolic reactions. The ARACNE-MC1
algorithm thus requires selection of the threshold for the number of ARACNE-supported
metabolite pairwise interactions, and identifies all of the conforming reactions passing the
threshold as putative reactions.
We notice that the same metabolic statistical interaction may be a part of multiple
reactions. One “strong” reaction may be responsible for much of the MI associated with a
particular link, and thus for the corresponding interaction surviving the ARACNE DPI
application. However, ARACNE-MC1 would count this interaction in support of every
conforming reaction to which it belongs. To avoid this multiple counting, we sort all conforming
reactions by their “strength” as measured by the cumulative mutual information in all of its
interactions. We then ensure that an interaction is counted as supporting only the strongest of its
associated reactions; this is the ARACNE-MC2 algorithm.
Flowcharts of both algorithms are illustrated in Fig 2.
The algorithms, implemented in MATLAB, are available at
http://menem.com/~ilya/wiki/index.php/Bandaru_et_al,_2010. The performance of the
algorithms will depend on a variety of choices, such as the DPI and mass comparison tolerances,
MI estimation parameters, or thresholds for the number of interaction supports needed to
proclaim a reaction as existing. Some of these choices are explored later in the paper. For others,
which we believe may differ dramatically for synthetic and for experimental data, we leave the
detailed analysis to future publications.
Synthetic Tests of ARACNE-MC
Data Generation and Performance Metrics
To validate performance of ARACNE-MC, we used the Kyoto Encyclopedia of Genes
and Genomes (KEGG) (31) to create synthetic metabolic networks, which then served as a
source of simulated data for tests. KEGG provides a detailed description of metabolites and
reactions found in the metabolic pathways of various model organisms. The entirety of the
metabolic pathways of Escherichia coli were downloaded and pruned to include only mass-
balanced reactions of the three types shown in Figure 1. Two different synthetic networks were
constructed to test the performance of ARACNE-MC on metabolic networks of different sizes: a
small network, containing 86 unique metabolites and 50 metabolic reactions, and a large
network with 218 metabolites and 136 reactions. Detailed specifications of the networks are
available at http://menem.com/~ilya/wiki/index.php/Publications/Bandaru_et_al.,_2010.
Since a majority of kinetic rates are unknown in KEGG, for each of the reactions we
randomly generated forward and backward rates, with the forward to backward ratio being, on
average, a hundred. We used the open-source COPASI software (28) to simulate the dynamics of
the network multiple times, perturbing the kinetic rates each time by a random multiplicative
factor with a standard deviation of 15%. This is similar to the approach in (24) and is supposed
to represent changes to the rates due to different extracellular environments and phenotypic and
metabolic states of the cells. Each simulation started with equal metabolite concentrations of
1uM and was run to a steady state. One thousand steady states were simulated this way and used
as synthetic metabolic profiles for input to ARACNE-MC.
To measure the algorithm performance, we choose the metrics of precision and recall.,
Precision, NTP/ NTP NFP
, measures the fraction of true predictions among all predictions
(where indices T, F, P, and N stand for true, false, positives, and negatives). Precision
corresponds to the expected success rate in the experimental validation of computational
predictions. Similarly, recall, NTP/ NTP NFN
, indicates the fraction of all reactions
recovered by the algorithm. Precision and recall values of 1 indicate perfect performance.
However, there is generally a tradeoff between the two. We emphasize that all of these metrics
are calculated based on recovery of complete metabolic reactions, rather than simple statistical
correlations among the metabolites, as in (23).
We performed two tests to verify that accurate knowledge of both the metabolite profiles
and the mass constraints is necessary for ARACNE-MC1 and 2 performance. Firstly, we
randomized entries in the mass-conforming reactions while leaving the statistical relations intact.
Secondly, we randomized metabolic profiles while preserving the correct mass constraints. In
both cases, ARACNE-MC1 and 2 failed to predict metabolic reactions beyond chance, indicating
an equal reliance on the two types of data (results not shown).
Table 1 details the results from the ARACNE-MC1 algorithm in the reconstruction of the
large (218 metabolites) and small (86 metabolites) synthetic networks. Precisions well over 50%
with significant recall rates are seen for many parameter combinations. The results suggest that
the ARACNE algorithm is robust to changes in network size and topology. We emphasize again
that, for our performance metrics, all substrates and products of a reaction must be predicted
correctly for the reaction to be counted as correct. This is a very stringent performance test.
Performance of ARACNE-MC2 is illustrated in Table 2. We see that removing double
counting of interactions has an effect of increasing the precision (sometimes to the maximum
level of 1) with only a marginal loss in the recall.
To illustrate the dependence of the ARACNE-MC reconstruction on the DPI tolerance
parameter, Table 3 shows performance of ARACNE-MC1 for the tolerance of 1 (i.e., no DPI
applied). While performance is weaker compared to the tolerance of 0, the effect is not dramatic,
suggesting weak sensitivity to the parameters. The specific best value of the parameter will likely
depend on the size of the dataset and on the experimental noise, and will need to be established
for each particular application independently as in (29)
Finally, a crucial feature of any computational reverse engineering algorithm is the
dependence of its performance on the size of the experimental dataset. We test this for
ARACNE-MC2 in Table 4. Specifically, both the precision and the recall degrade gracefully as
the data set size decreases from 1000 to 100, and no meaningful reconstruction is possible when
the size becomes comparable to the number of the analyzed metabolic species. This explains, in
particular, why our application of the algorithms to the existing experimental data set of Ishii, et
al. (17), which includes approximately 30 steady-state metabolic profiles of Escherichia coli and
195 metabolites in each profile, has failed.
The ARACNE-MC1 and 2 algorithms represent the first computational step towards
identification of metabolic reaction networks from high-throughput mass-spectrometry profile
data, armed with detailed knowledge of metabolite masses. Performance of the algorithm on
synthetic data sets is encouraging, warranting further development and application to real-life
data sets, when available. Selection of optimal values of many parameters of the algorithm,
which we expect to depend on the details of the experimental data, will need to be performed at
that time. Further, depending on the experimental resolution for many small, common
metabolites, additional modifications of ARACNE-MC will need to be considered. In particular,
to reduce the rate of false negatives, frequent interactions among common substrates (ATP,
water, NADP, etc.) can be treated as supported statistically for every conforming reaction.
As implemented now, the algorithm is data-intensive, requiring more metabolic profiles
than the number of considered metabolites. Current absence of such large datasets is the biggest
obstacle in application of the algorithm to real-world problems. However, we expect that ion-
mobility mass spectrometry with nanoliter chemostat cultures (30) will be able to provide the
necessary amounts of data in the immediate future.
Synthetic Networks Generation
Synthetic metabolic networks were created from the KEGG database. Using Escherichia
coli as a model system for these synthetic networks, we downloaded mass-balanced reactions
randomly, so that the final analyzed network is representative of the metabolism of E. coli. A
small synthetic network containing 86 unique metabolites and 50 metabolic reactions and a large
synthetic network containing 218 metabolites and 136 metabolic reactions were generated. See
the appendices for detailed descriptions of the metabolites in each network and their
Analysis and Simulation Parameters
Two main parameters of ARACNE algorithm are the p-value for accepting an MI
estimate as nonzero and the DPI tolerance threshold. For the purposes of this study, the DPI
tolerance was varied between 1 (no DPI application) and 0 (stringent edge elimination), and the
p-value threshold was set to the default level of 1e-4. Additionally, the mass comparison relative
tolerance was 1e-4.
We especially thank Andrea Califano, who has been an invaluable help in the early stages
of this research. We also thank William Hlavacek, Fangping Mu, Joel Berendzen, and
Manjunath Kustagi for important contributions. PB and IN were partially supported by
NIH/NIGMS under 1R21GM080216.
Figure 1. Examples of three types of metabolic reactions included in the synthetic network and
generated by the mass constraints algorithm. Type 1 reactions are termed two-by-two reactions,
Type 2a and 2b reactions are termed one-by-two reactions, and Type 3 reactions are termed one-
by-one reactions. Chemical structures from KEGG are provided for illustration.
Figure 2. Flowchart of the ARACNE-MC algorithm.
TP FP Precision
Table 1: Performance of ARACNE-MC1 for different networks and DPI tolerances. The first
value in each cell in the four right columns corresponds to the small networks with zero
tolerance; the second value is for the small network with the tolerance of one, and the third value
is for the large network. All data is for 1000 simulated metabolic profiles. When the DPI
tolerance is 0, precisions over 80% are possible with recalls in the teens when large ARACNE
support for a conforming reaction is requested, and recall of 25-50% still leaves the precision
Interactions TP FP Precision Recall
Table 2. Performance of ARACNE-MC2 for the small and the large networks. The DPI
tolerance is 0.
Table 3: Precision and recall results for selected reaction template / ARACNE supports /
surviving ARACNE supports from the large network against a varying number of input
metabolic profiles using the ARACNE-MC2 algorithm.
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