BACKGROUND
evolution has produced organisms that are robust to various perturbations, yet the specific knockout of a single gene can be lethal to the organism.
similarly, organisms have some redundancy in their metabolic pathways, but single reactions whose knockout brings the growth of a cell to a halt – called 'essential' reactions – are also known to exist in metabolic networks  <cit> .
what properties of a specific gene or reaction, within the context of the overall structure and organization of biochemical networks, make it essential for the organism?
we show that most essential metabolic reactions in escherichia coli  <cit> , saccharomyces cerevisiae  <cit>  and staphylococcus aureus  <cit>  can be explained by the fact that they are associated with a low degree metabolite.
metabolic and protein interaction networks contain nodes with a large variation in their degree of connectivity  <cit> .
in case of protein interaction networks it has been suggested that essentiality of a protein is correlated with its degree  <cit> .
hence, protein interaction networks are vulnerable to removal of highly connected proteins called 'hubs'.
in contrast, for metabolic networks, one is usually interested in the essentiality of reactions rather than metabolites.
recently, mahadevan and palsson  <cit>  have shown that low degree metabolites are almost as likely to be associated with essential reactions as high degree metabolites.
we show here that in fact almost all essential reactions are explained by virtue of being tagged to some low degree metabolite.
another theme in systems and computational biology has been to identify genetic regulatory modules  <cit> , functional clusters  <cit>  and graph-theoretic modules  <cit>  in metabolic networks.
modularity of complex biological networks contributes to the robustness, flexibility, and evolvability of organisms, and also towards making their organization more comprehensible  <cit> .
what structural features of metabolic networks cause specific subsets of metabolic reactions to have strongly correlated fluxes?
we observe that low degree metabolites lead to one such structure in the metabolic network.
such metabolites contribute to a rigidity or coherence of reaction fluxes in the network resulting in clusters of highly correlated reactions.
for example, in any steady state, where the concentrations of all metabolites are constant, a metabolite that can be produced in only one reaction and consumed in only one causes both reactions to have equal  fluxes.
maintaining the metabolic network close to a steady state then requires enzymes for both reactions to be simultaneously active, and hence the corresponding genes to be co-expressed, resulting in a transcription module containing those genes.
in this work we first locate metabolites based purely on their low degree in the metabolic network.
then we show that clusters of their reactions predict genetic regulatory modules, as captured in the structure of operons  <cit> , with a high probability in e. coli.
furthermore, the composition of most functional clusters is also explained via the low degree clusters embedded inside them.
biological networks have two properties that are currently regarded as unrelated: one, they have functional modules, and two, they have single genes or metabolic reactions whose knockout is lethal.
an implication of the present work is that in metabolic networks, both properties can arise as consequences of the same structural property: the existence of low degree metabolites.
our work provides an explanation, rather than just identification, of essential reactions and metabolic modules.
lowest degree metabolites and their clusters
a metabolite may be designated as 'uniquely produced' or 'up'  if, in the bipartite graph of reactions and metabolites, the node corresponding to the metabolite has in-degree  equal to unity; in other words, if there is only one reaction in the network that produces  the metabolite.
a metabolite that is both up and uc  has the lowest degree in the network.
examples of up-uc metabolites taken from the metabolic networks  <cit>  of e. coli and s. aureus can be seen in fig.
<dig>  in any metabolic steady state the concentration of a metabolite is fixed; its rate of production is equal to that of consumption.
hence for a up-uc metabolite in any steady state, the flux of the reaction producing it is proportional to that of the reaction consuming it, with the proportionality constant determined by the stoichiometric coefficients of the metabolite in the two reactions.
a 'up-uc cluster' of reactions may be defined as a set of reactions connected by up-uc metabolites.
in a steady state fixing the flux of any reaction in the up-uc cluster fixes the fluxes of all other reactions in the cluster .
these clusters include linear pathways but can also have branched or cyclic structure.
up-uc clusters are special cases of reaction/enzyme subsets  <cit>  and fully coupled reactions or co-sets  <cit> .
up-uc metabolites give rise to a situation wherein the inclusion of one reaction in a set implies the inclusion of another; such situations have also been considered  <cit>  in the context of identification and generation of feasible pathways and applied to e. coli metabolism.
each up-uc cluster of reactions can be replaced by an effective reaction without affecting the steady state performance and can be used to coarse-grain metabolic networks  <cit> .
a reaction was designated as 'uniquely producing' or 'up'  if it produced  a up  metabolite.
the number of up  reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus were found to be  <dig> ,  <dig>  and  <dig> , respectively, while the number of reactions that are either up or uc or both  is  <dig>   <dig> and  <dig>  we will show below that such reactions play a special role in metabolic networks.
RESULTS
essential reactions are largely explained by up/uc structure
we used the flux balance analysis   <cit>  approach to determine essential reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus.
we computed the steady state optimal flux vectors for each of these organisms in aerobic conditions for all permissible single organic carbon sources in a minimal medium.
we found a feasible solution  for  <dig>   <dig> and  <dig> sources in e. coli, s. cerevisiae and s. aureus respectively.
the list of feasible carbon sources under minimal media in these organisms is provided in supplementary tables s <dig>  s <dig> and s <dig> .
we considered the effect of 'switching off' reactions  one by one, on the optimal growth rate for each food source.
a reaction was designated as 'essential' for a particular food source if switching it off resulted in a zero optimal growth rate under that input condition.
we designated a reaction as 'globally essential' for an organism if it was essential for all its feasible minimal media under aerobic conditions.
the number of essential reactions for each minimal media varied between  <dig> and  <dig> reactions and the number of globally essential reactions was  <dig> for the e. coli metabolic network.
similarly, we found that the number of globally essential reactions in metabolic networks of s. cerevisiae and s. aureus were  <dig> and  <dig> respectively.
most essential reactions either produce or consume a up or uc metabolite
of the  <dig> globally essential reactions in the e. coli metabolic network,  <dig> were found to be either up or uc.
the probability of such a high overlap occurring by pure chance is very small.
we can quantify this by comparing to a null model in which the essentiality and the up/uc property of a reaction are considered to be independent of each other.
the probability that out of a set of  <dig> reactions two independently chosen subsets of size  <dig>  and  <dig>  will have an intersection of  <dig> or greater is p <10- <dig> .
similarly, we found a high fraction of globally essential reactions in metabolic networks of s. cerevisiae and s. aureus to be up or uc .
this explains why this subset is essential: there is simply no other path around these reactions in the entire network to produce or consume some metabolite that is presumably required for the eventual production of biomass.
in a recent paper  <cit>  mahadevan and palsson have determined, for each metabolite in the network, the fraction of its reactions that are essential.
they have observed that this 'lethality fraction' of the low degree metabolites is on average comparable to high degree metabolites, and in particular, some metabolites with in and out degree unity  have lethality fraction unity.
we present here a stronger result regarding the role of low degree metabolites: most essential reactions involve at least one up or uc metabolite.
these reactions may involve other metabolites of higher degree, but their essentiality is due to their uniqueness in producing or consuming a low degree metabolite.
the correspondence between essential and up/uc reactions is even tighter in the 'reduced network'
to understand the remaining globally essential reactions, we considered a reduced or pruned version of the network.
certain reactions in various reconstructed metabolic networks are such that they have a zero flux value under all steady states for stoichiometric reasons.
these reactions are referred to as 'strictly detailed balanced' reactions  <cit>  or 'blocked' reactions  <cit> , and can be removed from the network for any steady state analysis.
we used a previously described algorithm  <cit>  to determine blocked reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus.
we found  <dig>  of the  <dig>  reactions in the e. coli  metabolic network to be blocked.
we removed the blocked reactions from each network to obtain the 'reduced network' for each organism .
note that the essential reactions obtained by implementing fba on the reduced network are exactly the same as those obtained from the original network for each input condition.
hence, instead of requiring a metabolite to be up or uc across the entire metabolic network, we asked if it was up or uc in the reduced network.
the set of up metabolites and reactions so obtained turns out to be somewhat smaller than the original set.
in e. coli, s. cerevisiae and s. aureus the new set of up/uc reactions has  <dig>   <dig> and  <dig> reactions.
this is so because several reactions that were up/uc in the original network happen to be blocked and are now removed.
conversely some metabolite that was earlier not up can now become up after the removal of a reaction.
this adds new reactions to the up/uc set but this number turns out to be smaller than the number removed .
the new up metabolites have, by definition, their in  degree unity in the reduced network; even in the original network they have a low degree  degree in the original network is  <dig>  ).
we emphasize that the reduced network as defined above and hence the set of new up reactions is uniquely determined by the original network.
we found that  <dig> out of the  <dig> globally essential reactions  in the e. coli metabolic network to be up or uc in the reduced network .
similarly, we found that almost all globally essential reactions in s. cerevisiae and s. aureus were either up or uc in the reduced network  thereby underscoring the fact that nodes with a low degree of connectivity play an 'essential' role in metabolism.
the importance of low-degree nodes in the essential functionality of complex autocatalytic networks has also been observed elsewhere  <cit>  in a different context.
this finding provides some insight into the structural or topological origin of essential reactions in metabolic networks.
it is, of course, obvious that if a certain metabolite is an essential intermediate for the production of some biomass metabolite, and if this metabolite is uniquely produced or uniquely consumed, then the corresponding production or consumption reaction will be essential for the growth of the cell.
however the converse of this statement – that all essential reactions in the network should have this topological property – is far from obvious.
our finding that about 5– <dig> % of essential reactions do not have this property proves that the converse statement is indeed false.
thus the fact that the overwhelming majority  of essential reactions have this topological property is a characterization of the nature of metabolic networks found in organisms.
we remark that we do not as yet understand why the remaining essential reactions happen to be essential.
most up/uc reactions are essential in some condition or other
we found that there are  <dig> up or uc metabolic reactions in the e. coli reduced network.
<dig> of these  <dig> reactions were globally essential, while  <dig> of these  <dig> reactions  were essential for at least one of the  <dig> possible minimal media in e. coli.
such a large overlap is very unlikely , given that the number of reactions that are essential for at least one of the input conditions in the reduced e. coli metabolic network is  <dig>  some of these up/uc reactions were part of the input pathways of only one carbon source, hence they were essential only for that input.
in s. cerevisiae  <dig> out of  <dig> up/uc reactions  in the reduced network are essential in at least one input condition, while in s. aureus  <dig> out of  <dig>  have this property.
the p values for such large overlaps in the two organisms are, respectively, p <10- <dig> and p <10- <dig>  given that the number of reactions that are essential for at least one of the input conditions in those networks is  <dig> and  <dig>  the substantial difference between s. cerevisiae, a eukaryote, and the two bacteria may reflect a more evolved metabolic structure that needs to be further investigated.
comparison between computationally determined essential reactions and lethal single gene knockouts
to check the agreement of essential reactions in the e. coli metabolic network with a database  <cit>  of experimentally determined essential genes in a rich medium, we implemented fba for a rich medium containing all food sources for the e. coli metabolic network  <cit> .
we found  <dig> reactions to be essential in this medium for e. coli.
<dig> of these  <dig> reactions were found to be either up or uc in the reduced network.
of the  <dig> essential reactions in rich medium, information about the corresponding genes was available for only  <dig> reactions.
of these  <dig>   <dig> reactions had known isozymes, i.e, multiple enzymes catalyzing the reaction, hence the corresponding genes are not expected to be essential.
of the remaining  <dig> reactions,  <dig> had associated genes whose essentiality was undetermined in the database.
of the remaining  <dig> reactions,  <dig> reactions had associated genes that had been found to be essential in the database  <cit> , which is a fairly high fraction.
conversely, of the  <dig> essential genes determined for e. coli by gerdes et al,  <dig> genes were also part of the e. coli metabolic network  <cit>  used for our study.
<dig> of the above  <dig> essential genes had their products catalyzing only a single reaction in the e. coli metabolic network.
of these  <dig> essential genes,  <dig> were associated with a up or uc reaction.
further, using the reduced network, we found that  <dig> of the  <dig> essential genes were associated with a up or uc reaction.
the discrepancy between theoretical prediction and experimental data may be reconciled by the incomplete knowledge about possible isozymes for certain reactions or uncharacterized alternative metabolic pathways in the present in-silico metabolic model  <cit> .
low degree clusters predict regulatory modules
we found that the e. coli metabolic network  <cit>  contained  <dig> up-uc metabolites.
we determined all up-uc clusters in the network .
the total number of up-uc clusters in e. coli metabolic network was found to be 85; their size distribution is shown by the grey bars in fig.
<dig>  the list of all reactions in each up-uc cluster for the e. coli metabolic network is given in supplementary table s <dig> .
we then investigated whether the genes coding for the enzymes of the reactions in a up-uc cluster are part of the same operon in e. coli.
genes on the same operon are by definition part of a genetic module since they are coregulated.
at the moment genes corresponding to enzymes of reactions of the network have been identified for only part of the network.
of the  <dig> up-uc clusters in the e. coli metabolic network, only  <dig> clusters had two or more reactions with known corresponding genes.
we looked at the regulation of these  <dig> up-uc clusters using the known operon information from regulondb  <cit>  and ecocyc  <cit>  databases.
genes  that belong to the same operon in e. coli are indicated in supplementary table s <dig> .
for  <dig> of the  <dig> up-uc clusters, we found that two or more genes of the cluster were part of the same operon.
further,  <dig> of these  <dig> up-uc clusters had at least half of their genes belonging to the same operon.
we also found that  <dig> up-uc clusters have at least one possible set of constituent genes catalyzing all reactions in the cluster belonging to the same operon.
to show that two genes belonging to a up-uc cluster in e. coli have greater probability of lying on the same operon than otherwise expected, we performed the following test.
we found  <dig> unique genes catalyzing various reactions in the  <dig> up-uc clusters.
if we randomly pick any two of these  <dig> genes, the probability that the two genes lie on the same operon is  <dig> .
if we randomly pick a pair of genes that belong to the same up-uc cluster from this set of  <dig> genes, the probability that the two genes lie on the same operon is  <dig> .
thus regulatory modules are predicted correctly with a high probability by this method.
it is possible that up-uc clusters will find even greater correspondence with regulatory modules when expression data is analysed; our comparison rests only on operon data, and only about  <dig> percent of the transcriptional regulatory network of e. coli is presently believed to have been identified  <cit> .
it would also be interesting to extend this analysis to the other two organisms.
large up-uc clusters are analogous to network motifs
we asked the question: is it expected that a network like the e. coli metabolic network of  <dig> metabolites and  <dig> reactions with  <dig> up-uc metabolites will have a distribution of up-uc clusters as given in figure 2?
to answer this question, we compared the distribution of up-uc clusters in the real e. coli metabolic network with a suitably randomized version of the network  <cit> .
the randomized network has the same number of metabolite nodes and reaction nodes and the same number of incoming and outgoing links at each node as the real e. coli metabolic network .
averaging over  <dig> realizations of the randomized metabolic network we found a cluster distribution as shown by the black line in fig.
<dig>  this shows that the actual metabolic network of e. coli has its up-uc metabolites bunched up next to each other, forming larger clusters than expected in random networks with the same local connectivity properties.
thus, larger size  up-uc clusters are over-represented in the real e. coli metabolic network, and may be collectively considered as analogous to a network motif , while smaller size  up-uc clusters are under-represented in the real network, and may be collectively considered analogous to an 'anti-motif'  <cit> .
we also found qualitatively similar results for the metabolic networks of s. cerevisiae and s. aureus .
low degree metabolites explain perfect clusters
correlated reaction sets are sets of reactions in the metabolic network that are always used together in functional states of the network.
each flux vector obtained using fba represents one possible functional state of the network.
for each feasible minimal medium we obtained one flux vector with a nonzero growth rate.
we defined an 'active' reaction as one that had a nonzero flux in at least one of the latter flux vectors.
then we computed the correlation coefficient among fluxes of the active reactions across these flux vectors in a manner analogous to the correlation of gene activity from microarray data across different conditions  <cit>  .
a 'perfect cluster' is a set of reactions whose pairwise correlation coefficients with each other are all unity across all sets of conditions.
reactions in perfect clusters have fluxes that are proportional to each other with the same proportionality constant under all the flux vectors considered.
we found that in the e. coli metabolic network, most of the  <dig> active reactions under  <dig> input conditions were contained in several perfect clusters of size  <dig> or more .
these clusters, reported earlier in  <cit>  overlap highly with the clusters of  <cit> .
one might ask: why are particular subsets of reactions perfectly clustered to each other.
up-uc clusters provide a structural explanation for these perfect clusters.
of the  <dig> up-uc clusters in the entire e. coli network,  <dig> up-uc clusters are in the set of active reactions.
all the  <dig> active up-uc clusters are subsets of perfect clusters.
to further explain the observed clustering of reactions in the e. coli metabolic network, we considered up metabolites in the reduced network.
we found  <dig> up-uc clusters in the reduced network for e. coli.
table  <dig> shows that most of the perfect clusters in e. coli are explained in terms of up-uc clusters in the reduced network in the sense that up-uc clusters account for the bulk of reactions in the perfect clusters.
most of the co-sets reported in  <cit>  for e. coli are also explained by up-uc clusters in the reduced network .
further, we found that most perfect clusters in the metabolic networks of s. cerevisiae and s. aureus are also explained by up-uc clusters in their respective reduced networks .
very recently almaas et al  <cit>  have observed a set of  <dig> reactions that are always active under a diverse set of  <dig> conditions in the e. coli metabolic network that they designate as the 'core' of the network.
they also find these reactions to be highly correlated and most of them  to be essential.
we find that  <dig> of these  <dig> reactions are up/uc .
discussion and
CONCLUSIONS
in this paper we have observed that the lowest degree metabolites are implicated in two distinct properties of the metabolic networks, one, the existence of essential metabolic reactions , and two, existence of functional clusters in the metabolic networks .
to some extent the identification of up/uc metabolites depends on the way the metabolic network is curated.
for example, the networks we have used leave out certain non-enzymatic reactions such as protonation-deprotonation reactions.
since their inclusion would render some of the presently up metabolites non-up, our definition of up could be criticized as being somewhat arbitrary.
in this context it is worth noting that for the networks as they stand, our definition of up allows us to establish a connection between distinct properties of the network , and that our main findings hold for metabolic networks of three distinct organisms.
this suggests that up/uc reactions as defined by us do capture a certain pattern.
in our view the important point is not that other definitions of the network would obscure the pattern, but rather, that there do exist systematic definitions of the network in which a pattern is visible.
in metabolic networks the very existence of essential reactions is an indicator of the fragility of the system: even though the network has many reaction nodes, the removal of a single essential reaction node destroys the functionality of the network completely by blocking the flow of an essential intermediate.
isozymes are a way of dealing with this fragility.
however, not all essential reactions have isozymes  <cit> ; this means that evolution has tolerated this fragility.
our finding that essential reactions are tagged by low degree metabolites may provide some insight into why this is the case.
metabolites that participate in very few reactions perhaps do so in part because some feature of their chemical structure prohibits ready association with other molecules, i.e., their low degree is a consequence of constraints coming from chemistry.
then evolution tolerates the reactions that produce or consume such metabolites as essential because chemistry leaves it no choice.
alternatively, it could be that this fragility happens to be a byproduct of some other desirable structural property that contributes to robustness or evolvability, such as modularity.
we have drawn attention to the fact that low degree metabolites also play a role in functional clustering of reactions in the metabolic network.
we have further provided evidence that the up-uc clusters at the metabolic level correspond, with a high probability, to sets of genes forming modules at the regulatory level in e. coli.
this raises the question: if low degree metabolites contribute to modularity, could it be that the evolutionary advantages of that have outweighed the disadvantage of the above mentioned fragility caused by the same low degree metabolites?
is it the case that evolution has preferred 'chemically constrained' low degree metabolites in spite of the fragility they cause because they contribute to modularity?
a goal in biology is to understand highly evolved biological organization in terms of simpler and more inevitable structures  <cit> .
here we have presented evidence that certain genetic regulatory modules, in particular certain operons, mirror the low degree structure of the metabolites whose production and consumption they regulate.
this could be an example of how the origin of certain regulatory structure can be traced to simple chemical constraints.

 evolution has produced organisms that are robust to various perturbations, yet the specific knockout of a single gene can be lethal to the organism. similarly, organisms have some redundancy in their metabolic pathways, but single reactions whose knockout brings the growth of a cell to a halt – called'essential' reactions – are also known to exist in metabolic networks. what properties of a specific gene or reaction, within the context of the overall structure and organization of biochemical networks, make it essential for the organism? we show that most essential metabolic reactions in escherichia coli, saccharomyces cerevisiae and staphylococcus aureus can be explained by the fact that they are associated with a low degree metabolite. metabolic and protein interaction networks contain nodes with a large variation in their degree of connectivity. in case of protein interaction networks it has been suggested that essentiality of a protein is correlated with its degree. hence, protein interaction networks are vulnerable to removal of highly connected proteins called'hubs'. in contrast, for metabolic networks, one is usually interested in the essentiality of reactions rather than metabolites. recently, mahadevan and palsson have shown that low degree metabolites are almost as likely to be associated with essential reactions as high degree metabolites. we show here that in fact almost all essential reactions are explained by virtue of being tagged to some low degree metabolite. another theme in systems and computational biology has been to identify genetic regulatory modules, functional clusters and graph-theoretic modules in metabolic networks. modularity of complex biological networks contributes to the robustness, flexibility, and evolvability of organisms, and also towards making their organization more comprehensible. what structural features of metabolic networks cause specific subsets of metabolic reactions to have strongly correlated fluxes? we observe that low degree metabolites lead to one such structure in the metabolic network. such metabolites contribute to a rigidity or coherence of reaction fluxes in the network resulting in clusters of highly correlated reactions. for example, in any steady state, where the concentrations of all metabolites are constant, a metabolite that can be produced in only one reaction and consumed in only one causes both reactions to have equal fluxes. maintaining the metabolic network close to a steady state then requires enzymes for both reactions to be simultaneously active, and hence the corresponding genes to be co-expressed, resulting in a transcription module containing those genes. in this work we first locate metabolites based purely on their low degree in the metabolic network. then we show that clusters of their reactions predict genetic regulatory modules, as captured in the structure of operons, with a high probability in e. coli. furthermore, the composition of most functional clusters is also explained via the low degree clusters embedded inside them. biological networks have two properties that are currently regarded as unrelated: one, they have functional modules, and two, they have single genes or metabolic reactions whose knockout is lethal. an implication of the present work is that in metabolic networks, both properties can arise as consequences of the same structural property: the existence of low degree metabolites. our work provides an explanation, rather than just identification, of essential reactions and metabolic modules. lowest degree metabolites and their clusters a metabolite may be designated as'uniquely produced' or'up' if, in the bipartite graph of reactions and metabolites, the node corresponding to the metabolite has in-degree equal to unity; in other words, if there is only one reaction in the network that produces the metabolite. a metabolite that is both up and uc has the lowest degree in the network. examples of up-uc metabolites taken from the metabolic networks of e. coli and s. aureus can be seen in fig. in any metabolic steady state the concentration of a metabolite is fixed; its rate of production is equal to that of consumption. hence for a up-uc metabolite in any steady state, the flux of the reaction producing it is proportional to that of the reaction consuming it, with the proportionality constant determined by the stoichiometric coefficients of the metabolite in the two reactions. a'up-uc cluster' of reactions may be defined as a set of reactions connected by up-uc metabolites. in a steady state fixing the flux of any reaction in the up-uc cluster fixes the fluxes of all other reactions in the cluster. these clusters include linear pathways but can also have branched or cyclic structure. up-uc clusters are special cases of reaction/enzyme subsets and fully coupled reactions or co-sets. up-uc metabolites give rise to a situation wherein the inclusion of one reaction in a set implies the inclusion of another; such situations have also been considered in the context of identification and generation of feasible pathways and applied to e. coli metabolism. each up-uc cluster of reactions can be replaced by an effective reaction without affecting the steady state performance and can be used to coarse-grain metabolic networks. a reaction was designated as'uniquely producing' or'up' if it produced a up metabolite. the number of up reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus were found to be, and, respectively, while the number of reactions that are either up or uc or both is and we will show below that such reactions play a special role in metabolic networks.  essential reactions are largely explained by up/uc structure we used the flux balance analysis approach to determine essential reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus. we computed the steady state optimal flux vectors for each of these organisms in aerobic conditions for all permissible single organic carbon sources in a minimal medium. we found a feasible solution for and sources in e. coli, s. cerevisiae and s. aureus respectively. the list of feasible carbon sources under minimal media in these organisms is provided in supplementary tables s s and s. we considered the effect of'switching off' reactions one by one, on the optimal growth rate for each food source. a reaction was designated as'essential' for a particular food source if switching it off resulted in a zero optimal growth rate under that input condition. we designated a reaction as'globally essential' for an organism if it was essential for all its feasible minimal media under aerobic conditions. the number of essential reactions for each minimal media varied between and reactions and the number of globally essential reactions was for the e. coli metabolic network. similarly, we found that the number of globally essential reactions in metabolic networks of s. cerevisiae and s. aureus were and respectively. most essential reactions either produce or consume a up or uc metabolite of the globally essential reactions in the e. coli metabolic network, were found to be either up or uc. the probability of such a high overlap occurring by pure chance is very small. we can quantify this by comparing to a null model in which the essentiality and the up/uc property of a reaction are considered to be independent of each other. the probability that out of a set of reactions two independently chosen subsets of size and will have an intersection of or greater is p <10-. similarly, we found a high fraction of globally essential reactions in metabolic networks of s. cerevisiae and s. aureus to be up or uc. this explains why this subset is essential: there is simply no other path around these reactions in the entire network to produce or consume some metabolite that is presumably required for the eventual production of biomass. in a recent paper mahadevan and palsson have determined, for each metabolite in the network, the fraction of its reactions that are essential. they have observed that this'lethality fraction' of the low degree metabolites is on average comparable to high degree metabolites, and in particular, some metabolites with in and out degree unity have lethality fraction unity. we present here a stronger result regarding the role of low degree metabolites: most essential reactions involve at least one up or uc metabolite. these reactions may involve other metabolites of higher degree, but their essentiality is due to their uniqueness in producing or consuming a low degree metabolite. the correspondence between essential and up/uc reactions is even tighter in the'reduced network' to understand the remaining globally essential reactions, we considered a reduced or pruned version of the network. certain reactions in various reconstructed metabolic networks are such that they have a zero flux value under all steady states for stoichiometric reasons. these reactions are referred to as'strictly detailed balanced' reactions or'blocked' reactions, and can be removed from the network for any steady state analysis. we used a previously described algorithm to determine blocked reactions in the metabolic networks of e. coli, s. cerevisiae and s. aureus. we found of the reactions in the e. coli metabolic network to be blocked. we removed the blocked reactions from each network to obtain the'reduced network' for each organism. note that the essential reactions obtained by implementing fba on the reduced network are exactly the same as those obtained from the original network for each input condition. hence, instead of requiring a metabolite to be up or uc across the entire metabolic network, we asked if it was up or uc in the reduced network. the set of up metabolites and reactions so obtained turns out to be somewhat smaller than the original set. in e. coli, s. cerevisiae and s. aureus the new set of up/uc reactions has and reactions. this is so because several reactions that were up/uc in the original network happen to be blocked and are now removed. conversely some metabolite that was earlier not up can now become up after the removal of a reaction. this adds new reactions to the up/uc set but this number turns out to be smaller than the number removed. the new up metabolites have, by definition, their in degree unity in the reduced network; even in the original network they have a low degree degree in the original network is ). we emphasize that the reduced network as defined above and hence the set of new up reactions is uniquely determined by the original network. we found that out of the globally essential reactions in the e. coli metabolic network to be up or uc in the reduced network. similarly, we found that almost all globally essential reactions in s. cerevisiae and s. aureus were either up or uc in the reduced network thereby underscoring the fact that nodes with a low degree of connectivity play an'essential' role in metabolism. the importance of low-degree nodes in the essential functionality of complex autocatalytic networks has also been observed elsewhere in a different context. this finding provides some insight into the structural or topological origin of essential reactions in metabolic networks. it is, of course, obvious that if a certain metabolite is an essential intermediate for the production of some biomass metabolite, and if this metabolite is uniquely produced or uniquely consumed, then the corresponding production or consumption reaction will be essential for the growth of the cell. however the converse of this statement – that all essential reactions in the network should have this topological property – is far from obvious. our finding that about 5– % of essential reactions do not have this property proves that the converse statement is indeed false. thus the fact that the overwhelming majority of essential reactions have this topological property is a characterization of the nature of metabolic networks found in organisms. we remark that we do not as yet understand why the remaining essential reactions happen to be essential. most up/uc reactions are essential in some condition or other we found that there are up or uc metabolic reactions in the e. coli reduced network. of these reactions were globally essential, while of these reactions were essential for at least one of the possible minimal media in e. coli. such a large overlap is very unlikely, given that the number of reactions that are essential for at least one of the input conditions in the reduced e. coli metabolic network is some of these up/uc reactions were part of the input pathways of only one carbon source, hence they were essential only for that input. in s. cerevisiae out of up/uc reactions in the reduced network are essential in at least one input condition, while in s. aureus out of have this property. the p values for such large overlaps in the two organisms are, respectively, p <10- and p <10- given that the number of reactions that are essential for at least one of the input conditions in those networks is and the substantial difference between s. cerevisiae, a eukaryote, and the two bacteria may reflect a more evolved metabolic structure that needs to be further investigated. comparison between computationally determined essential reactions and lethal single gene knockouts to check the agreement of essential reactions in the e. coli metabolic network with a database of experimentally determined essential genes in a rich medium, we implemented fba for a rich medium containing all food sources for the e. coli metabolic network. we found reactions to be essential in this medium for e. coli. of these reactions were found to be either up or uc in the reduced network. of the essential reactions in rich medium, information about the corresponding genes was available for only reactions. of these reactions had known isozymes, i.e, multiple enzymes catalyzing the reaction, hence the corresponding genes are not expected to be essential. of the remaining reactions, had associated genes whose essentiality was undetermined in the database. of the remaining reactions, reactions had associated genes that had been found to be essential in the database, which is a fairly high fraction. conversely, of the essential genes determined for e. coli by gerdes et al, genes were also part of the e. coli metabolic network used for our study. of the above essential genes had their products catalyzing only a single reaction in the e. coli metabolic network. of these essential genes, were associated with a up or uc reaction. further, using the reduced network, we found that of the essential genes were associated with a up or uc reaction. the discrepancy between theoretical prediction and experimental data may be reconciled by the incomplete knowledge about possible isozymes for certain reactions or uncharacterized alternative metabolic pathways in the present in-silico metabolic model. low degree clusters predict regulatory modules we found that the e. coli metabolic network contained up-uc metabolites. we determined all up-uc clusters in the network. the total number of up-uc clusters in e. coli metabolic network was found to be 85; their size distribution is shown by the grey bars in fig. the list of all reactions in each up-uc cluster for the e. coli metabolic network is given in supplementary table s. we then investigated whether the genes coding for the enzymes of the reactions in a up-uc cluster are part of the same operon in e. coli. genes on the same operon are by definition part of a genetic module since they are coregulated. at the moment genes corresponding to enzymes of reactions of the network have been identified for only part of the network. of the up-uc clusters in the e. coli metabolic network, only clusters had two or more reactions with known corresponding genes. we looked at the regulation of these up-uc clusters using the known operon information from regulondb and ecocyc databases. genes that belong to the same operon in e. coli are indicated in supplementary table s. for of the up-uc clusters, we found that two or more genes of the cluster were part of the same operon. further, of these up-uc clusters had at least half of their genes belonging to the same operon. we also found that up-uc clusters have at least one possible set of constituent genes catalyzing all reactions in the cluster belonging to the same operon. to show that two genes belonging to a up-uc cluster in e. coli have greater probability of lying on the same operon than otherwise expected, we performed the following test. we found unique genes catalyzing various reactions in the up-uc clusters. if we randomly pick any two of these genes, the probability that the two genes lie on the same operon is. if we randomly pick a pair of genes that belong to the same up-uc cluster from this set of genes, the probability that the two genes lie on the same operon is. thus regulatory modules are predicted correctly with a high probability by this method. it is possible that up-uc clusters will find even greater correspondence with regulatory modules when expression data is analysed; our comparison rests only on operon data, and only about percent of the transcriptional regulatory network of e. coli is presently believed to have been identified. it would also be interesting to extend this analysis to the other two organisms. large up-uc clusters are analogous to network motifs we asked the question: is it expected that a network like the e. coli metabolic network of metabolites and reactions with up-uc metabolites will have a distribution of up-uc clusters as given in figure 2? to answer this question, we compared the distribution of up-uc clusters in the real e. coli metabolic network with a suitably randomized version of the network. the randomized network has the same number of metabolite nodes and reaction nodes and the same number of incoming and outgoing links at each node as the real e. coli metabolic network. averaging over realizations of the randomized metabolic network we found a cluster distribution as shown by the black line in fig. this shows that the actual metabolic network of e. coli has its up-uc metabolites bunched up next to each other, forming larger clusters than expected in random networks with the same local connectivity properties. thus, larger size up-uc clusters are over-represented in the real e. coli metabolic network, and may be collectively considered as analogous to a network motif, while smaller size up-uc clusters are under-represented in the real network, and may be collectively considered analogous to an'anti-motif'. we also found qualitatively similar results for the metabolic networks of s. cerevisiae and s. aureus. low degree metabolites explain perfect clusters correlated reaction sets are sets of reactions in the metabolic network that are always used together in functional states of the network. each flux vector obtained using fba represents one possible functional state of the network. for each feasible minimal medium we obtained one flux vector with a nonzero growth rate. we defined an'active' reaction as one that had a nonzero flux in at least one of the latter flux vectors. then we computed the correlation coefficient among fluxes of the active reactions across these flux vectors in a manner analogous to the correlation of gene activity from microarray data across different conditions. a'perfect cluster' is a set of reactions whose pairwise correlation coefficients with each other are all unity across all sets of conditions. reactions in perfect clusters have fluxes that are proportional to each other with the same proportionality constant under all the flux vectors considered. we found that in the e. coli metabolic network, most of the active reactions under input conditions were contained in several perfect clusters of size or more. these clusters, reported earlier in overlap highly with the clusters of. one might ask: why are particular subsets of reactions perfectly clustered to each other. up-uc clusters provide a structural explanation for these perfect clusters. of the up-uc clusters in the entire e. coli network, up-uc clusters are in the set of active reactions. all the active up-uc clusters are subsets of perfect clusters. to further explain the observed clustering of reactions in the e. coli metabolic network, we considered up metabolites in the reduced network. we found up-uc clusters in the reduced network for e. coli. table shows that most of the perfect clusters in e. coli are explained in terms of up-uc clusters in the reduced network in the sense that up-uc clusters account for the bulk of reactions in the perfect clusters. most of the co-sets reported in for e. coli are also explained by up-uc clusters in the reduced network. further, we found that most perfect clusters in the metabolic networks of s. cerevisiae and s. aureus are also explained by up-uc clusters in their respective reduced networks. very recently almaas et al have observed a set of reactions that are always active under a diverse set of conditions in the e. coli metabolic network that they designate as the'core' of the network. they also find these reactions to be highly correlated and most of them to be essential. we find that of these reactions are up/uc. discussion and  in this paper we have observed that the lowest degree metabolites are implicated in two distinct properties of the metabolic networks, one, the existence of essential metabolic reactions, and two, existence of functional clusters in the metabolic networks. to some extent the identification of up/uc metabolites depends on the way the metabolic network is curated. for example, the networks we have used leave out certain non-enzymatic reactions such as protonation-deprotonation reactions. since their inclusion would render some of the presently up metabolites non-up, our definition of up could be criticized as being somewhat arbitrary. in this context it is worth noting that for the networks as they stand, our definition of up allows us to establish a connection between distinct properties of the network, and that our main findings hold for metabolic networks of three distinct organisms. this suggests that up/uc reactions as defined by us do capture a certain pattern. in our view the important point is not that other definitions of the network would obscure the pattern, but rather, that there do exist systematic definitions of the network in which a pattern is visible. in metabolic networks the very existence of essential reactions is an indicator of the fragility of the system: even though the network has many reaction nodes, the removal of a single essential reaction node destroys the functionality of the network completely by blocking the flow of an essential intermediate. isozymes are a way of dealing with this fragility. however, not all essential reactions have isozymes; this means that evolution has tolerated this fragility. our finding that essential reactions are tagged by low degree metabolites may provide some insight into why this is the case. metabolites that participate in very few reactions perhaps do so in part because some feature of their chemical structure prohibits ready association with other molecules, i.e., their low degree is a consequence of constraints coming from chemistry. then evolution tolerates the reactions that produce or consume such metabolites as essential because chemistry leaves it no choice. alternatively, it could be that this fragility happens to be a byproduct of some other desirable structural property that contributes to robustness or evolvability, such as modularity. we have drawn attention to the fact that low degree metabolites also play a role in functional clustering of reactions in the metabolic network. we have further provided evidence that the up-uc clusters at the metabolic level correspond, with a high probability, to sets of genes forming modules at the regulatory level in e. coli. this raises the question: if low degree metabolites contribute to modularity, could it be that the evolutionary advantages of that have outweighed the disadvantage of the above mentioned fragility caused by the same low degree metabolites? is it the case that evolution has preferred'chemically constrained' low degree metabolites in spite of the fragility they cause because they contribute to modularity? a goal in biology is to understand highly evolved biological organization in terms of simpler and more inevitable structures. here we have presented evidence that certain genetic regulatory modules, in particular certain operons, mirror the low degree structure of the metabolites whose production and consumption they regulate. this could be an example of how the origin of certain regulatory structure can be traced to simple chemical constraints. 
the methods developed here could be used in predicting essential reactions and metabolic modules in other organisms from the list of metabolic reactions. we find that most reactions identified as essential in these networks turn out to be those involving the production or consumption of low degree metabolites. applying graph theoretic methods to these metabolic networks, we identified connected clusters of these low degree metabolites. we find that the composition of the latter'functional clusters' is also largely explained in terms of clusters of low degree metabolites in each of these organisms. our findings mean that most metabolic reactions that are essential can be tagged by one or more low degree metabolites. furthermore, reactions whose fluxes are strongly correlated can be thought of as'glued together' by these low degree metabolites. a goal of computational and systems biology is to go beyond identification towards an explanation of specific modules and essential genes and reactions in terms of specific structural or evolutionary constraints. furthermore, we find that larger sized clusters are over-represented in the real network and are analogous to a'network motif. the genes involved in several operons in e. coli are correctly predicted as those of enzymes catalyzing the reactions of these clusters. recently there has been a lot of interest in identifying modules at the level of genetic and metabolic networks of organisms, as well as in identifying single genes and reactions that are essential for the organism. in the metabolic networks of escherichia coli, saccharomyces cerevisiae and staphylococcus aureus, we identified metabolites with a low degree of connectivity, particularly those that are produced and/or consumed in just a single reaction. using flux balance analysis we also determined reactions essential for growth in these metabolic networks.
 recently there has been a lot of interest in identifying modules at the level of genetic and metabolic networks of organisms, as well as in identifying single genes and reactions that are essential for the organism. a goal of computational and systems biology is to go beyond identification towards an explanation of specific modules and essential genes and reactions in terms of specific structural or evolutionary constraints.  in the metabolic networks of escherichia coli, saccharomyces cerevisiae and staphylococcus aureus, we identified metabolites with a low degree of connectivity, particularly those that are produced and/or consumed in just a single reaction. using flux balance analysis we also determined reactions essential for growth in these metabolic networks. we find that most reactions identified as essential in these networks turn out to be those involving the production or consumption of low degree metabolites. applying graph theoretic methods to these metabolic networks, we identified connected clusters of these low degree metabolites. the genes involved in several operons in e. coli are correctly predicted as those of enzymes catalyzing the reactions of these clusters. furthermore, we find that larger sized clusters are over-represented in the real network and are analogous to a'network motif. using fba for the above mentioned three organisms we independently identified clusters of reactions whose fluxes are perfectly correlated. we find that the composition of the latter'functional clusters' is also largely explained in terms of clusters of low degree metabolites in each of these organisms.  our findings mean that most metabolic reactions that are essential can be tagged by one or more low degree metabolites. those reactions are essential because they are the only ways of producing or consuming their respective tagged metabolites. furthermore, reactions whose fluxes are strongly correlated can be thought of as'glued together' by these low degree metabolites. the methods developed here could be used in predicting essential reactions and metabolic modules in other organisms from the list of metabolic reactions. 