Journal of Theoretical Biology 2
sp
n
a
t U
haft
UM
962
rm
4 N
k a
, a
at c
(see Reinhart Heinrich’s annotated bibliography in this
reactions and their products are added to an expanding
network. This iterative process will end if no further
This concept has been applied in recent papers, including
determining sets of seed compounds required for the
synthesis of a specific compound or set of compounds. In
ARTICLE IN PRESSparticular the latter set may comprise metabolic precursors
that the cell requires for maintenance or growth. Therefore
solving this inverse problem may indicate minimal nutri-
0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2007.10.036

Corresponding author. Tel.: +49 30 2093 8325; fax: +49 30 2093 8813.
E-mail address: handorf@physik.hu-berlin.de (T. Handorf).issue). He considered such principles as essential prerequi-
sites for understanding evolution of metabolism. It was in
this line of thought that Reinhart Heinrich and some of us
at the Humboldt University developed the concept of
network expansion in the early 2000s (Ebenho¨h et al., 2004;
Handorf et al., 2005).
The basic principle is that a reaction can only operate if
all of its substrates are available as nutrients or can be
provided by other reactions of the network. This condition
is applied in an iterative manner. Starting from the
nutrients, which are called seed compounds, operable
a discussion on hierarchical structuring of metabolic
networks (Handorf et al., 2006), a comparison of metabolic
capabilities of organism specific networks (Ebenho¨h et al.,
2005), a model of metabolic evolution (Ebenho¨h et al.,
2006) and the analysis of changes of metabolic capacities in
response to environmental perturbations (Ebenho¨h and
Liebermeister, 2006). Further, scopes have been utilized
to study the effect of oxygen in metabolic networks
(Raymond and Segre´ , 2006) and to predict the viability
of mutant strains (Wunderlich and Mirny, 2006).
In this work we consider the inverse problem ofexternal resources in order to sustain the growth or maintenance of an organism, given the structure of its metabolic network? Although
this problem is highly combinatorial, we show that it is possible to calculate locally minimal nutrient sets that can be interpreted in terms
of resource types. Using these types we predict broad nutritional requirements for 447 organisms, providing clues for possible
environments from the knowledge of their metabolic networks.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Nutrient; Scope; Metabolic network
1. Introduction
Among the numerous approaches developed by
Reinhart Heinrich on metabolism, one important focus
was the identification of design and optimality principles
reaction fulfills the above condition. The set of metabolites
in the expanded network is called the scope of the seed
compounds and represents all metabolites that can in
principle be synthesized from the seed by the analyzed
metabolic network (Handorf et al., 2005).metabolic network. Here we investigate the inverse problem: which chemical compounds or sets of compounds must be provided asAn environmental per
Thomas Handorf
a,
, Nils Christia
a
Theoretical Biophysics, Department of Biology, Humbold
b
Max Planck Institute for Molecular Plant Physiology, Wissensc
c
Universite´ de Lyon, Universite´ Lyon 1, CNRS, INRIA,
43 Bd du 11 Novembre 1918, 6
Received 22 June 2007; received in revised fo
Available online
Abstract
In principle the knowledge of an organism’s metabolic networ
Expanding metabolic networks: scopes of compounds, robustness
network expansion generating the set of all possible metabolites th52 (2008) 530–537
ective on metabolism
, Oliver Ebenho¨h
b
, Daniel Kahn
c
niversity Berlin, Invalidenstr. 42, 10115 Berlin, Germany
spark Golm, Am Mu¨hlenberg 1, 14476 Potsdam-Golm, Germany
R 5558, Laboratoire de Biome´trie et Biologie Evolutive,
2 Villeurbanne cedex, France
25 October 2007; accepted 26 October 2007
ovember 2007
llows to infer its biosynthetic capabilities. Handorf et al. [2005.
nd evolution. J. Mol. Evol. 61, 498–512] developed a method of
an be produced from a set of compounds, given the structure of a
www.elsevier.com/locate/yjtbi

ARTICLE IN PRESS
oretional requirements that must be met to sustain main-
tenance or growth of an organism, based on the knowledge
of its metabolic network. We apply this methodology to a
number of organisms for which metabolic networks are
defined in the KEGG database (Kanehisa et al., 2006) and
show that this inverse methodology can indeed provide
clues on possible nutritional requirements of organisms
and their environment.
2. Methods
2.1. The target set of required metabolites
A key function of metabolism is to chemically convert
available nutrients into products which are required by
other cellular processes. Precursors for central cellular
functions like protein synthesis, DNA replication, energy
or cofactor production are ubiquitous. Since the detailed
requirements may vary from cell type to cell type, we apply
a systematic approach, combined with biological knowl-
edge, to identify a universal set of necessary metabolites,
referred to as the target set T.
We construct the target set by determining those
metabolites which occur in at least 90% of the analyzed
organisms. These include amino acids, nucleotides,
many cofactors, organic acids and sugar phosphates. We
manually refined this list by including plausible compounds
which are missing and removing compounds whose
presence in the target set seemed not reasonable. The
detailed list of target metabolites as well as the removed
compounds and the reasons for their removal can be found
in the supplementary material.
2.2. Identifying minimal resources
To identify minimal sets of required resources that
enable an organism to produce all metabolites contained in
the target set, we develop an algorithm that relies on the
method of expanding networks which was introduced in
Handorf et al. (2005). Starting from a given set of initial
metabolites, the seed S, the network expansion algorithm
determines all those metabolites which a particular
metabolic network is capable to produce when only the
seed compounds are available. These metabolites are called
the scope of the seed, denoted SðSÞ. The identification of
minimal resources is now described as the problem to
identify minimal sets of seed compounds for which the
scope contains all target metabolites. A seed S is minimal if
its scope contains the target T and no proper subset of S
fulfills this condition:
S is minimal seed if T  SðSÞ and 8S
0
 S : TgSðS
0
Þ.
(1)
For a given network, we determine minimal seeds with the
following greedy algorithm: (1) Initially, we define an
T. Handorf et al. / Journal of Theordered list containing all metabolites occurring in the
network. Clearly, the seed composed of all metabolitesfrom the list must produce a scope containing the target
set. (2) Beginning from the top, stepwise each metabolite is
removed from the list and the scope is recalculated for the
corresponding reduced seed. If now the scope does not
contain the full target set, the metabolite is written back to
the list, otherwise it remains permanently removed. (3) Step
(2) is repeated until the complete list has been traversed.
The metabolites contained in the resulting list represent a
minimal seed because the further removal of any metabo-
lite would result in a scope that does not contain all target
metabolites.
Since the ordering of the list in step (1) determines which
metabolites are preferentially removed (those near the top)
and which preferentially remain in the seed (those near the
end), differently ordered lists will result in different
minimal seeds. Clearly, it is impossible to test all possible
orderings of the list of metabolites and thus the complete
set, denoted M, containing all minimal seeds cannot be
calculated. However, significant information about the
structure of M can be obtained by calculating seeds for a
sufficient number of random orderings. Further, as not all
minimal seeds are equally biologically meaningful, the size
of the search space can further be reduced by incorporating
biological information on which metabolites can actually
be used as nutrients. Such metabolites should reside near
the end of the list, which can be achieved in the following
way: First, all metabolites are sorted by decreasing
molecular weight. This ordering leads to a preferential
removal of large metabolites from the list of possible seed
compounds and has been introduced to avoid minimal
seeds containing only a small number of chemically rich
but large metabolites that are unlikely to be transported
into a cell. Second, for several metabolites, the transport
processes over the membrane are well characterized.
For our calculations, we have identified from the
KEGG pathways KO02010 and KO02060 all those
metabolites which can be translocated by ABC transpor-
ters or the phosphotransferase system. This biological
knowledge has been considered by symbolically assigning
all these compounds a negative ‘‘mass’’ which shifts
them towards the end of the list. The resulting list
ensures that metabolites that are known to be transported
and small molecules are preferentially chosen as seed
compounds.
To identify and analyze a large number of possible
minimal resource sets, we construct a large number of
perturbed lists and apply the algorithm repeatedly for each
network. The random perturbation is designed in a way
ensuring that large metabolites remain as a tendency near
the top of the list. Specifically, we randomly chose two
metabolites from the list and exchange their positions with
a probability
p ¼
exp 
1
b
Dm

for Dm40;
8
>
<
(2)
tical Biology 252 (2008) 530–537 5311 for Dmp0;
>
: