Metabolism is a fascinating cell machinery underlying life and disease and genome-scale reconstructions provide us with a captivating view of its complexity. However, deciphering the relationship between metabolic structure and function remains a major challenge. In particular, turning observed structural regularities into organizing principles underlying systemic functions is a crucial task that can be significantly addressed after endowing complex network representations of metabolism with the notion of geometric distance. Here, we design a cartographic map of metabolic networks by embedding them into a simple geometry that provides a natural explanation for their observed network topology and that codifies node proximity as a measure of hidden structural similarities. We assume a simple and general connectivity law that gives more probability of interaction to metabolite/reaction pairs which are closer in the hidden space. Remarkably, we find an astonishing congruency between the architecture of E. coli and human cell metabolisms and the underlying geometry. In addition, the formalism unveils a backbone-like structure of connected biochemical pathways on the basis of a quantitative cross-talk. Pathways thus acquire a new perspective which challenges their classical view as self-contained functional units. Cells are self-organized entities that carry-out specialized tasks at different interrelated omic-levels [1] involving different actors, from codifying genes to energycarrier or constitutive metabolites. A key towards understanding this complex architecture at a systems level is provided by reliable genome-wide reconstructions of the set of biochemical reactions that underly the functional cell machinery [2] . Such reconstructions can be analyzed using tools and techniques from complex networks theory [3] [4] [5] , a discipline that is being used in the characterization of biological, chemical, infrastructural, technological or social-based systems of complex relationships [6, 7] . More precisely, nodes in metabolic networks account for either metabolites or reactions, while links represent the interactions among them. Apart from providing a largescale organizational picture, these network-based representations have permitted to analyze sensible issues in cellular metabolism, like flux balances [8, 9] , regulation [10] , robustness [11] , or reaction reliability [12] . The advantage of using network-based representations, in whatever context we employ them, may be arguably questioned by the fact that complex networks are customarily modeled as pure topological constructions lacking a true geometric measure of separation among nodes. This is aggravated by the fact that complex networks have the small-world property [13] , meaning that every pair of nodes in the system are very close in topological distance. This is an important and obvious degeneracy if we think in terms of optimizing routing or transportation strategies in man-engineered networks, but can be equally crucial when referring to the description of the metabolic functioning at a single cell level. As a matter of fact, the related attempt of separating nodes into communities, that has been already pursued in different contexts [14] and, in particular, applied to metabolic networks [15] , has proven to be an extremely difficult task. Classical community detection approaches turn out to be a posteriori classification methods, and do not provide insights into any potential connectivity law underlying the observed topology. These questions could be significantly addressed by quantifying the abstract concept of node proximity in terms of a metric distance which could be combined into a simple and general probabilistic connectivity law. Such a biochemical connectivity law, relying on metric distances, may provide a simple explanation of the large scale topological structure observed in metabolism [16] , and it can also be used, like in this work, to revisit the concept of biochemical pathways. In this paper, we uncover the hidden geometry of the E. coli and human metabolisms and find that their network topologies obey an extremely simple and powerful -metric-based-probabilistic connectivity law. In particular, given a pair metabolite/reaction separated by a geometric distance d mr in the underlying metric space, the probability of existence of a connection between them is here shown to be a decreasing function of the effective distance d ef f ≡ d mr /(k r k m ), where degrees k m and k r count the number of their respective neighboring nodes. The geometric distance d mr -a measure of structural affinity between metabolites and reactions-is in this way modulated by the product of degrees of the two involved nodes, so that the degree heterogeneity observed in the metabolic network is properly taken into account. Naturally, a key ingredient in our approach concerns the suitable geometry substantiating this distance. We find that a simple one dimensional closed Euclidean space, i.e. a circle, when combined with the network degree heterogeneity is enough to capture the global organization of the network. Using statistical inference techniques, we find angle-based coordinates in this space for the full set of metabolites and reactions, which expose the extraordinary congruency of our model. As a direct application of the proposed cartographic maps of metabolism, we compare the results of our arXiv:1109.1934v1 [q-bio.MN] 9 Sep 2011 2 embedding with the standard classification of reactions in terms of biochemical pathways. Such a reactionaggregated analysis reveals rather disparate trends when pathways are characterized in terms of the circle-based localizations of their constituents reactions. Some specific pathways appear concentrated over narrow sectors of polar angles, while more transversal ones are widespread over the circle. This points to a diversity of pathway topologies, with some of them displaying groups of densely interconnected reactions while some others evidencing a much more weakly connected internal structure. Moreover, pathways themselves admit to be linked using the discovered connectivity law. This strategy reveals different levels of cross-talk between pathways, leading to a coarse-grained view of metabolic networks or, in other words, to the build-up of networks of pathways. Such a higher level in the hierarchical organization of metabolic networks advises against the study of pathways as autonomous subsystems and should permit to calibrate more accurately how a pathway-localized perturbation spreads over the entire network. I. RESULTS A simple abstraction of a given metabolism is given by its bipartite network representation. This amounts to consider metabolites and reactions as belonging to different subsets of nodes, with metabolites (irrespectively considered as reactants and products) linked to all reactions they take part in, and thus avoiding connections between nodes of the same kind, see Fig. 1a . The first step towards mapping this network consists in defining a geometric model that can advantageously represent it. The simplest realization of a one-dimensional homogeneous and isotropic closed metric space that can globally embed a network is a circle of radius R. Nodes, in our case metabolites and reactions separately, are distributed on it according to specific angular coordinates to be determined. The whole strategy to find these coordinates rests on a precise definition of the interactions between nodes in terms of their ring-based distances. We prescribe a connection probability between a reaction r and a metabolite m, with respective bipartite degrees k r and k m and separated by a distance d mr on the circle (d mr = R∆θ mr , ∆θ mr being the angular separation between metabolite and reaction) to be a decreasing function of such distance rescaled by the product of node degrees [17] , It is worth-stressing that this is the central and unique law underlying the whole formalism. Notice that this choice is particularly suggestive since by identifying the node degree as a measure of its mass, this interaction mimics the Newtonian form of gravitational interaction. More precisely, the explicit form for the above interaction reads This particular prescription combines, in a simple way, the classical network topological concept of node degrees with the newly introduced notion of geometric distance. All in all, this functional form expresses an intuitive view, i.e. closer nodes in the metric space are more likely to be linked, while nodes with higher degrees sustain farther reaching connections regardless of their distances. Figure 1b shows a visual sketch summarizing the basic trends of the bipartite formalism just outlined. We refer to it with the notation S 1 × S 1 , see Methods. Besides, this model gives rise to a maximum-entropy ensemble of graphs that are therefore maximally random given their specific constraints [18, 19] . Finally, parameters µ and β are consistently determined to reproduce the statistical properties of the original network. Parameter µ fixes the total number of edges, whereas β controls clustering, i.e, a measure of short-range loops, see Appendix B. To infer the angle-based coordinates for metabolites and reactions in the ring we use a two-step procedure. Starting from the original bipartite network, we first perform a one-mode projection over the set of metabolites by connecting two metabolites whenever they participate in the same reaction. We then circle-embed such a unipartite metabolites network applying the unipartite version of the formalism as described earlier [20] . Finally, using this partial allocation as an initial fixed template, we complete the embedding of the reactions by invoking a maximum likelihood inference strategy (the detailed description of the embedding algorithm and the coordinates of metabolites and reactions are fully reported in Appendix C). We apply our formalism to the iAF1260 version of the K12 MG1655 strain of E. coli metabolism [21] and to human cell metabolism [22] , both provided in the BiGG database [23, 24] , see Appendix B. Before presenting the embedding for these metabolic networks, we comment on the validation of the proposed mapping procedure. We first perform a direct calibration which amounts to compare the set of observed metabolite-reaction connection probabilities in the original reconstructions with the theoretical connection probability given by Eq. (B9). Explicit results are presented in Fig. 1c , both for E. coli and human metabolisms. Beyond the striking agreement between observed and predicted connections, it is worth noticing that the two analyzed networks are perfectly represented with the same β exponent fitted to a value β = 1.3. We also check the discrimination power of our algorithm by computing the Receiver Operating Characteristics (ROC) curve of our model [25] , which compares the true positive rate (TPR) vs. the false positive rate (FPR) and informs about how good is our method at model. Nodes are randomly distributed in the circle and given expected degrees, symbolically represented by the sizes of the nodes. The distance between two nodes is computed as the length of the arc separating the nodes. Due to the peculiar rescaling of distances by degrees in Eq. (B8), a node can connect not only to nearby nodes but also to far apart nodes with large degree. c, The plot shows a comparison between the empirical connection probability for the E. coli and human metabolisms and the theoretical one given in Eq. (B9). The empirical connection probability is computed as the fraction between the number of actual connections at effective distance dmr/µkmkr and the total number of pairs at the same effective distance. d, The Receiver Operating Characteristic (ROC) curve computed for our model for the E. coli and human metabolisms is shown. To calculate the ROC curves, we rank (from highest to lowest) the connection probabilities given by the model for all possible pairs metabolite/reaction (either present or absent) using the previously inferred coordinates. We then define at each value a threshold probability that allows us to discriminate between positive interactions (those above the threshold) from negative ones (those below the threshold) and to compute the fraction of true positive connections (True Positive Rate TPR) and that of false positive connections (False Positive Rate FPR), with the understanding that a true positive connection is an observed link above the threshold, while a false positive is an non-existing one above the threshold. correctly discern real links. Results are shown in Fig. 1d . When representing the TPR in front of the FPR, a totally random guess would result in a straight line along the diagonal. In contrast, the ROC curve of our model lies far above the diagonal, which indicates a remarkable discrimination power. A convenient summary statistic can be defined as the area under the ROC curve (AUC statistic), which represents the probability that a randomly chosen observed link in the network has a higher probability of existence according to the model than a randomly chosen non-existing one. This statistic ranges in the interval [0.5, 1], being AUC = 0.5 a random prediction and AUC = 1 a perfect prediction. In our case, values are AUC = 0.96 for E. coli and AUC = 0.97 for human metabolism. Both validation tests confirm that our model adjusts nearly perfectly to the real data. Figure 2 shows the embedding representation of the E. coli metabolism (the mapping of the human metabolism is provided in Appendix F). For the sake of clarity, metabolites are displaced towards the center of the circle by an amount proportional to their degree so that hub metabolites are close to the center of the disk whereas low degree ones are placed in the periphery. The distribution over the circle is far from being uniform as it could be naively expected. Indeed, this is a distinctive signature of the delicate structural organization of metabolic networks. In particular, different levels of aggregation are readily visible, inasmuch as human settlements are unevenly distributed in population maps. Simultaneously with densely occupied areas, empty regions are visible and appear irregularly punctuated with occasional metabolite-reaction associations. As a whole, this landscape is an indication of some hierarchical trends existing in the analyzed networks and prompts us to look for eventual higher organizational levels. In this regard, we revise the biochemical concept of pathways, classically understood as chains of step-by-step reactions which transform a principal chemical into another either for immediate use, to propagate metabolic fluxes or for cell storage. In Fig. 2 , we identify pathways in the circle by plotting their names at the average angular position of all their constitutive reactions. In Figs. 3 and 4 , we propose two complementary representations of the metabolic pathways of E. coli as they appear annotated in the BiGG database. In Fig. 3 , we show the angular distribution on the ring of the whole list of pathways (up to 33, plus an Unassigned category of reactions not represented in the figure) , evaluated from the circle-based embedding of the reactions they involve. We recognize rather disparate spectra of angular distributions. Strongly localized pathways, e. g. the Folate pathway or Oxidative Phosphorylation, coexist with more distributed ones. The latter can adopt either a discrete bimodal, a multi-peaked form, e. g. the Histidine and Glycolisis pathways respectively, or can even transversally spread over the ring closer to a homogeneous distribution. The Alternate Carbon, the Transport Inner  Global geometric map of E. coli's metabolism. Angular distribution of reactions and metabolites inferred by the method. Yellow circles represent reactions whereas blue squares are metabolites. For each metabolite, the symbol size is proportional to the logarithm of the degree and radially placed according to the expression r = R − 2 ln km. Black (grey) connections are those that according to the model have a probability of existence larger (smaller) than 0.5. The names of the different pathways, radially-written, are located at the average angular position of all the reactions belonging to a given pathway, and the font size is proportional to the logarithm of the number of reactions in the pathway. Notice that we do not represent transversal pathways and that some pathways seem to be located in empty regions (e.g. Inorganic Ion Transport). This is due to the fact that some pathways display bimodal or multi-peaked distributions so that the average appears in between the peaks, see Fig. 3 and Table I in Appendix D. Membrane, or the Cofactor and Prosthetic Group pathways are representative examples of this latter category (see Table I in Appendix D for further details). Our method is, therefore, able to discriminate concentrated pathways, consistent with the classical view of modular subsystems, from others which are indeed formed of subunits, and even from those finally responsible of producing or consuming metabolites in turn extensively used by many other pathways. The embedding of reactions and metabolites in the circle can also be used to aggregate pathways into broader categories. To do so, the embedding circle is first divided into eight different angular sectors delimited by void regions in the ranked distribution of reaction angles, see Fig. 4a . The pathway concentration, i.e. the fraction of reactions of that pathway in each sector, is shown in Fig. 4b-i . Clearly, there are sectors monopolized by one or at most two pathways -e. g., Murein in Sector 7, Fig. 4h -, whereas other sectors are largely shared by many pathways -e.g. , different Aminoacid-based pathways in Sector 4, Fig. 4e . In all cases, the higher concentrations in each sector mostly correspond to pathways in related functional categories: Sector 1 and Sector 2 in Fig. 4b -c aggregate pathways related to Cell Membrane metabolism, Sector 3 and Sector 4 in Fig. 4d -e concentrate Central metabolism, with Sector 3 including Energy and part of the Nucleotide metabolism and Sector 4 including Amino Acid metabolism, Sector 5 in Fig. 4f condenses the remaining Nucleotide metabolism, and Sector 6 and Sector 7 in Fig. 4g -h account for the Glycan metabolism, with Sector 6 mixing basically mono and polysaccharide related pathways, while the pathways related to murein, a polymer that forms the cell wall, appearing well separated in Sector 7. Corresponding representations for human metabolism are shown in Fig. 10 . The number of pathways is considerably larger but common features to E. coli pathway show the pathway concentration, the fraction of reactions of that pathway, in each sector. The higher concentrations in each sector mostly correspond to pathways in related functional categories: S1 and S2 aggregate pathways related to Cell Membrane metabolism (plots b and c), S3 concentrates Central metabolism including Energy and part of the Nucleotide metabolism (plot d), S4 gathers Central metabolism including Amino Acids metabolism (plot e), S5 condenses the remaining Nucleotide metabolism (plot f) and S6 and S7 account for Glycan metabolism (plots g and h), with S6 mixing basically mono and polysaccharide related pathways and pathways related to murein, a polymer that forms the cell wall, well separated in S7. Pathway names have been abbreviated in standard forms whenever possible. localization patterns are evidenced in qualitative terms. Pathways can be divided again into different categories according to their angular concentration, with the difference that the general level of pathway localization in human metabolism is higher than in E. coli. The average angular concentration of pathways in human metabolism is 0.82, as compared to 0.79 in E. coli (see Methods) and the average size of maximum peaks in the pathways angular distributions in 0.36 for E. coli while for human metabolism it is 0.50. However, the higher level of localization seems to coexist with a higher entanglement of the different families of metabolic reactions, i.e carbon metabolism, lipid metabolism, etc.. Another observation is that transversal pathways in E. coli, like Cofactor and 6 Prosthetic group or Transport, are split into a number of more specialized pathways in human metabolism and, in fact, the category of transversal pathways itself, as defined in E. coli, is here minimally represented. In the first part of the paper, the S 1 ×S 1 formalism was applied to embed the observed metabolic networks into a circle, enabling to locate the reactions and metabolites related to every specific pathway in a simple onedimensional geometry. This information can be used to build a higher hierarchical level in the architecture of the metabolic network aimed at quantifying the interconnectivity between pathways. In turn this allows us to introduce the concept of network of pathways. Adjacencies between a pair of pathways are computed on the basis of the corresponding lists of reactions in each pathway and the set of metabolites shared by both lists. When the set of overlapping metabolites is not empty, the connection probabilities for the links between pathway reactions and common metabolites that correspond to observed interactions in the network are summed to give an absolute measure of the strenght of the interaction between the pair of pathways. Overlaps between pathways pairs assemble a higher order weighted network where pathways are nodes and links display heterogeneous intensities. However, the resulting network is very dense and needs to be conveniently filtered in order to provide meaningful information about the system. In E. coli, 460 out of a potential total of 561 pathways pairs overlap while for human cells 1689 pathways pairs out of 4278 have common metabolites. In practice we use a disparity-based threshold [26] (see Methods) that discards links whose intensities are compatible with random fluctuations at some specific significance level. As a result these pathway-based networks provide metabolic backbones i.e., subnetworks of pathways which display the statistically relevant interactions. As an illustration of the power of the metabolic backbone concept, panels in Fig. 5 reproduce the corresponding constructions for E. coli and human metabolisms. Interestingly, metabolic backbones offer a perspective that reveals functional constraints. Both for E. coli and human metabolism, star-like patterns are particularly neat. In E. coli, transversal pathways act as hub-like structures that interconnect different number of specific and more localized pathways, usually belonging to the same metabolic family. For instance, the Cofactor and Prosthetic Group Biosynthesis pathway connects many of the amino acid pathways to energy or nucleotide metabolism, and Alternate Carbon acts as the main intermediary of many Carbohydrate pathways with the rest of the backbone. Analogously, some pathways in the metabolic backbone of the human cell, like Folate or Fatty Acid Oxidation or Keratan Sulfate Biosynthesis, play a relevant role in providing systems' level connectivity to the network and connect a number of other specific pathways. From a broad perspective, a cartographic representation of complex networks supposes to map the positions of nodes in an underlying geometric space and shares some fundamental problems with traditional geographical cartography on what concerns techniques, generalizations or design: how to represent the topology of the mapped network on the metric space, which characteristics of the network are not relevant to the map's purpose and can be eliminated, how to reduce the complexity of the characteristics that will be mapped, etc.. Despite 7 the difficulties, cartographic maps based on geometrical spaces are crucial to identify dominant nodes, to understand how different subparts of the system, like pathways in our context, relate to each other, to back up more accurate methods of prediction of missing and spurious interactions [27, 28] , or to find optimal transport routes. In our metabolic maps, the astonishing congruency between the architecture of metabolic networks and the underlying geometry is supported by a biochemical interaction law that, irrespective of the studied organisms, of the nature and complexity of the reactions they account for, or of the different structural label of the metabolites they involve, seems to comply with a simple Newtonian-like form and allows us to make predictions about the probabilities of interaction among sets of metabolites forming reactions. Specifically, the sum of the probabilities running over all the metabolites participating in a certain biochemical reaction can be interpreted as a topological version of the well-known concept of reaction-based affinity, and each summand could thus be identified with the chemical potential of that particular metabolite in relation to its chemical partners in the particular reaction. Our results point to a systems level definition of chemical affinity in terms of network-based probabilities of interaction which depend on the distances in the underlying geometric space and on intrinsic properties of nodes which convert some of them in hubs. Such probabilistic network-based chemical affinities allow us to recover the established biochemical organization of pathways as connected metabolic families, but at the same time raise new questions claiming for the need of rethinking its classical definition as self-contained units. We find that different pathways may have disparate internal structures, some of them being more modular and conforming better to the classical definition, while subunits pointing to differentiated functionalities can be distinguished in others. We have also unveiled a higher level of systems' level interactions represented by metabolic backbones, defined on the basis of a quantitative crosstalk between pathways. This particular idea advises us against the use of very specific biochemical protocols aimed to single-out particular pathways as they might be prone to underestimate the delicate connections that underlay the net and secure its proper functioning. Such metabolic features are common to human cells and E. coli. However, a comparative study shows that pathways in human metabolism are in general more modular and display less overlap of common metabolites with other pathways. At the same time the different human metabolic families are more entangled and sectors are mode difficult to characterize, a possible signature of a higher functional complexity or merely a side effect of the kind of reconstruction that mixes in a single network reactions that happen in diversely differentiated cells. Summarizing, in this work we provide cartographic maps of two representative metabolisms that capture their specific complexities, explaining many of their system properties and provide a new perspective on the definition, cross-talk, and hierarchical organization of biochemical pathways. These maps, embedded in a simple geometric space, rely on a probabilistic biochemical connectivity law which emerges from the different physicochemical forces acting at a molecular level and that naturally conveys a higher interaction likelihood to elements which are closer in the underlying space. Similar maps for other biological networks are expected to be equally congruent and to help to transform data into knowledge and knowledge into understanding, paving the way for new discoveries in systems biology prediction and control.
This model has been recently generalized to bipartite net-works in REF , where it was called the S 1 × S 1 model, and utilized to study cell metabolism.
Uncovering the hidden geometry behind metabolic networks
arXiv: Molecular Networks
Biology Physics Medicine