The small world of metabolism

Assessing experimentally derived interactions in a small world

Experimentally determined networks are susceptible to errors, yet important inferences can still be drawn from them. Many real networks have also been shown to have the small-world network properties of cohesive neighborhoods and short average distances between vertices. Although much analysis has been done on small-world networks, small-world properties have not previously been used to improve our understanding of individual edges in experimentally derived graphs. Here we focus on a small-world network derived from high-throughput (and error-prone) protein-protein interaction experiments. We exploit the neighborhood cohesiveness property of small-world networks to assess confidence for individual protein-protein interactions. By ascertaining how well each protein-protein interaction (edge) fits the pattern of a small-world network, we stratify even those edges with identical experimental evidence. This result promises to improve the quality of inference from protein-protein interaction networks in particular and small-world networks in general.

How the global structure of protein interaction networks evolves

Two processes can influence the evolution of protein interaction networks: addition and elimination of interactions between proteins, and gene duplications increasing the number of proteins and interactions. The rates of these processes can be estimated from available Saccharomyces cerevisiae genome data and are sufficiently high to affect network structure on short time-scales. For instance, more than 100 interactions may be added to the yeast network every million years, a fraction of which adds previously unconnected proteins to the network. Highly connected proteins show a greater rate of interaction turnover than proteins with few interactions. From these observations one can explain (without natural selection on global network structure) the evolutionary sustenance of the most prominent network feature, the distribution of the frequency P(d) of proteins with d neighbours, which is broad-tailed and consistent with a power law, that is: P(d) proportional, variant d (-gamma).

Patterns in randomly evolving networks: idiotypic networks

We present a model for the evolution of networks of occupied sites on undirected regular graphs. At every iteration step in a parallel update, I randomly chosen empty sites are occupied and occupied sites having occupied neighbor degree outside of a given interval (t(l),t(u)) are set empty. Depending on the influx I and the values of both lower threshold and upper threshold of the occupied neighbor degree, different kinds of behavior can be observed. In certain regimes stable long-living patterns appear. We distinguish two types of patterns: static patterns arising on graphs with low connectivity and dynamic patterns found on high connectivity graphs. Increasing I patterns become unstable and transitions between almost stable patterns, interrupted by disordered phases, occur. For still larger I the lifetime of occupied sites becomes very small and network structures are dominated by randomness. We develop methods to analyze the nature and dynamics of these network patterns, give a statistical description of defects and fluctuations around them, and elucidate the transitions between different patterns. Results and methods presented can be applied to a variety of problems in different fields and a broad class of graphs. Aiming chiefly at the modeling of functional networks of interacting antibodies and B cells of the immune system (idiotypic networks), we focus on a class of graphs constructed by bit chains. The biological relevance of the patterns and possible operational modes of idiotypic networks are discussed.

What do we mean by biological complexity?

The purpose of the present paper is to offer a precise definition of the concepts of integration, emergence and complexity in biological networks through the use of the information theory. If two distinct properties of a network are expressed by two discrete variables, the classical subadditivity principle of Shannon's information theory applies when all the nodes of the network are associated with these properties. If not, the subadditivity principle may not apply. This situation is often to be encountered with enzyme and metabolic networks, for some nodes may well not be associated with these two properties. This is precisely what is occurring with an enzyme that binds randomly its two substrates. This situation implies that an enzyme, or a metabolic network, may display a joint entropy equal, smaller, or larger than the corresponding sum of individual entropies of component sub-systems. In the first case, the collective properties of the network can be reduced to the individual properties of its components. Moreover, the network is devoid of any information. In the second case, the system displays integration effects, behaves as a coherent whole, and has positive information. But if the joint entropy of the network is smaller than the sum of the individual entropies of its components, then the system has emergent collective properties and can be considered complex. Moreover, under these conditions, its information is negative. The extent of negative information is enhanced if the enzyme, or the metabolic network, is far away from equilibrium.

Small worlds in RNA structures

I consider conformational spaces of tRNA(phe) defined by sets of suboptimal structures from the perspective of small-world networks. Herein, the influence of modifications on typical small-world network properties and the shape of energy landscapes is discussed. Results indicate that natural modifications influence the degree of local clustering and mean path lengths far more than random or no modifications. High frequencies in the thermodynamic ensemble coincide with high numbers of neighboring structures that one conformation can adopt by one elementary move. Conformation spaces indicate the existence of modular substructures. It can be shown that modifications leave the nature of small-world topology untouched albeit natural modifications have a reasonable enhancing and streamlining effect on the degree of clustering and therefore on the substructures of the conformational space.

Deciphering metabolic networks

All higher organisms divide major biochemical steps into different cellular compartments and often use tissue-specific division of metabolism for the same purpose. Such spatial resolution is accompanied with temporal changes of metabolite synthesis in response to environmental stimuli or developmental needs. Although analyses of primary and secondary gene products, i.e. transcripts, proteins, and metabolites, regularly do not cope with this spatial and temporal resolution, these gene products are often observed to be highly coregulated forming complex networks. Methods to study such networks are reviewed with respect to data acquisition, network statistics, and biochemical interpretation.

An automated method for finding molecular complexes in large protein interaction networks

BACKGROUND

Recent advances in proteomics technologies such as two-hybrid, phage display and mass spectrometry have enabled us to create a detailed map of biomolecular interaction networks. Initial mapping efforts have already produced a wealth of data. As the size of the interaction set increases, databases and computational methods will be required to store, visualize and analyze the information in order to effectively aid in knowledge discovery.

RESULTS

This paper describes a novel graph theoretic clustering algorithm, "Molecular Complex Detection" (MCODE), that detects densely connected regions in large protein-protein interaction networks that may represent molecular complexes. The method is based on vertex weighting by local neighborhood density and outward traversal from a locally dense seed protein to isolate the dense regions according to given parameters. The algorithm has the advantage over other graph clustering methods of having a directed mode that allows fine-tuning of clusters of interest without considering the rest of the network and allows examination of cluster interconnectivity, which is relevant for protein networks. Protein interaction and complex information from the yeast Saccharomyces cerevisiae was used for evaluation.

CONCLUSION

Dense regions of protein interaction networks can be found, based solely on connectivity data, many of which correspond to known protein complexes. The algorithm is not affected by a known high rate of false positives in data from high-throughput interaction techniques. The program is available from ftp://ftp.mshri.on.ca/pub/BIND/Tools/MCODE.

An automated method for finding molecular complexes in large protein interaction networks

BACKGROUND

Recent advances in proteomics technologies such as two-hybrid, phage display and mass spectrometry have enabled us to create a detailed map of biomolecular interaction networks. Initial mapping efforts have already produced a wealth of data. As the size of the interaction set increases, databases and computational methods will be required to store, visualize and analyze the information in order to effectively aid in knowledge discovery.

RESULTS

This paper describes a novel graph theoretic clustering algorithm, "Molecular Complex Detection" (MCODE), that detects densely connected regions in large protein-protein interaction networks that may represent molecular complexes. The method is based on vertex weighting by local neighborhood density and outward traversal from a locally dense seed protein to isolate the dense regions according to given parameters. The algorithm has the advantage over other graph clustering methods of having a directed mode that allows fine-tuning of clusters of interest without considering the rest of the network and allows examination of cluster interconnectivity, which is relevant for protein networks. Protein interaction and complex information from the yeast Saccharomyces cerevisiae was used for evaluation.

CONCLUSION

Dense regions of protein interaction networks can be found, based solely on connectivity data, many of which correspond to known protein complexes. The algorithm is not affected by a known high rate of false positives in data from high-throughput interaction techniques. The program is available from ftp://ftp.mshri.on.ca/pub/BIND/Tools/MCODE.

Metabolomics in systems biology

The primary aim of "omic" technologies is the nontargeted identification of all gene products (transcripts, proteins, and metabolites) present in a specific biological sample. By their nature, these technologies reveal unexpected properties of biological systems. A second and more challenging aspect of omic technologies is the refined analysis of quantitative dynamics in biological systems. For metabolomics, gas and liquid chromatography coupled to mass spectrometry are well suited for coping with high sample numbers in reliable measurement times with respect to both technical accuracy and the identification and quantitation of small-molecular-weight metabolites. This potential is a prerequisite for the analysis of dynamic systems. Thus, metabolomics is a key technology for systems biology. The aim of this review is to (a) provide an in-depth overview about metabolomic technology, (b) explore how metabolomic networks can be connected to the underlying reaction pathway structure, and (c) discuss the need to investigate integrative biochemical networks.

Edge overload breakdown in evolving networks

We investigate growing networks based on Barabási and Albert's algorithm for generating scale-free networks, but with edges sensitive to overload breakdown. The load is defined through edge betweenness centrality. We focus on the situation where the average number of connections per vertex is, like the number of vertices, linearly increasing in time. After an initial stage of growth, the network undergoes avalanching breakdowns to a fragmented state from which it never recovers. This breakdown is much less violent if the growth is by random rather than by preferential attachment (as defines the Barabási and Albert model). We briefly discuss the case where the average number of connections per vertex is constant. In this case no breakdown avalanches occur. Implications to the growth of real-world communication networks are discussed.

Evolving networks with disadvantaged long-range connections

We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to d(-alpha), where alpha is a tunable parameter of the model. We show that the properties of the networks grown with alpha<1 are close to those of the genuine scale-free construction, while for alpha>1 the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of alpha.

Spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are nonuniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.

Community structure in social and biological networks

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known--a collaboration network and a food web--and find that it detects significant and informative community divisions in both cases.

Vertex overload breakdown in evolving networks

We study evolving networks based on the Barabási-Albert scale-free network model with vertices sensitive to overload breakdown. The load of a vertex is defined as the betweenness centrality of the vertex. Two cases of load limitation are considered, corresponding to the fact that the average number of connections per vertex is increasing with the network's size ("extrinsic communication activity"), or that it is constant ("intrinsic communication activity"). Avalanchelike breakdowns for both load limitations are observed. In order to avoid such avalanches we argue that the capacity of the vertices has to grow with the size of the system. An interesting irregular dynamics of the formation of the giant component (for the intrinsic communication activity case) is also studied. Implications on the growth of the Internet are discussed.

Specificity and stability in topology of protein networks

Molecular networks guide the biochemistry of a living cell on multiple levels: Its metabolic and signaling pathways are shaped by the network of interacting proteins, whose production, in turn, is controlled by the genetic regulatory network. To address topological properties of these two networks, we quantified correlations between connectivities of interacting nodes and compared them to a null model of a network, in which all links were randomly rewired. We found that for both interaction and regulatory networks, links between highly connected proteins are systematically suppressed, whereas those between a highly connected and low-connected pairs of proteins are favored. This effect decreases the likelihood of cross talk between different functional modules of the cell and increases the overall robustness of a network by localizing effects of deleterious perturbations.

Topological and causal structure of the yeast transcriptional regulatory network

Interpretation of high-throughput biological data requires a knowledge of the design principles underlying the networks that sustain cellular functions. Of particular importance is the genetic network, a set of genes that interact through directed transcriptional regulation. Genes that exert a regulatory role encode dedicated transcription factors (hereafter referred to as regulating proteins) that can bind to specific DNA control regions of regulated genes to activate or inhibit their transcription. Regulated genes may themselves act in a regulatory manner, in which case they participate in a causal pathway. Looping pathways form feedback circuits. Because a gene can have several connections, circuits and pathways may crosslink and thus represent connected components. We have created a graph of 909 genetically or biochemically established interactions among 491 yeast genes. The number of regulating proteins per regulated gene has a narrow distribution with an exponential decay. The number of regulated genes per regulating protein has a broader distribution with a decay resembling a power law. Assuming in computer-generated graphs that gene connections fulfill these distributions but are otherwise random, the local clustering of connections and the number of short feedback circuits are largely underestimated. This deviation from randomness probably reflects functional constraints that include biosynthetic cost, response delay and differentiative and homeostatic regulation.

Can we discover novel pathways using metabolomic analysis?

Metabolomic analysis aims at the identification and quantitation of all metabolites in a given biological sample. Current data acquisition and network analysis strategies are classified on the basis of pathway elucidation and characteristics of theoretical networks. The development of metabolomic methods and tools is progressing rapidly, but an understanding of the resulting data is limited owing to a fundamental lack of biochemical and physiological knowledge about network organization in plants.

Wrestling with pleiotropy: genomic and topological analysis of the yeast gene expression network

The vast majority (>95%) of single-gene mutations in yeast affect not only the expression of the mutant gene, but also the expression of many other genes. These data suggest the presence of a previously uncharacterized "gene expression network"--a set of interactions between genes which dictate gene expression in the native cell environment. Here, we quantitatively analyze the gene expression network revealed by microarray expression data from 273 different yeast gene deletion mutants.(1) We find that gene expression interactions form a robust, error-tolerant "scale-free" network, similar to metabolic pathways(2) and artificial networks such as power grids and the internet.(3-5) Because the connectivity between genes in the gene expression network is unevenly distributed, a scale-free organization helps make organisms resistant to the deleterious effects of mutation, and is thus highly adaptive. The existence of a gene expression network poses practical considerations for the study of gene function, since most mutant phenotypes are the result of changes in the expression of many genes. Using principles of scale-free network topology, we propose that fragmenting the gene expression network via "genome-engineering" may be a viable and practical approach to isolating gene function.

Random graph models of social networks

We describe some new exactly solvable models of the structure of social networks, based on random graphs with arbitrary degree distributions. We give models both for simple unipartite networks, such as acquaintance networks, and bipartite networks, such as affiliation networks. We compare the predictions of our models to data for a number of real-world social networks and find that in some cases, the models are in remarkable agreement with the data, whereas in others the agreement is poorer, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Small world patterns in food webs

The analysis of some species-rich, well-defined food webs shows that they display the so-called small world behavior shared by a number of disparate complex systems. The three systems analysed (Ythan estuary web, Silwood web and the Little Rock lake web) have different levels of taxonomic resolution, but all of them involve high clustering and short path lengths (near two degrees of separation) between species. Additionally, the distribution of connections P(k) which is skewed in all the webs analysed shows long tails indicative of power-law scaling. These features suggest that communities might be self-organized in a non-random fashion that might have important consequences in their resistance to perturbations (such as species removal). The consequences for ecological theory are outlined.

Estimating coarse gene network structure from large-scale gene perturbation data

Large scale gene perturbation experiments generate information about the number of genes whose activity is directly or indirectly affected by a gene perturbation. From this information, one can numerically estimate coarse structural network features such as the total number of direct regulatory interactions and the number of isolated subnetworks in a transcriptional regulation network. Applied to the results of a large-scale gene knockout experiment in the yeast Saccharomyces cerevisiae, the results suggest that the yeast transcriptional regulatory network is very sparse, containing no more direct regulatory interactions than genes. The network comprises >100 independent subnetworks.

Metabolomics--the link between genotypes and phenotypes

Metabolites are the end products of cellular regulatory processes, and their levels can be regarded as the ultimate response of biological systems to genetic or environmental changes. In parallel to the terms 'transcriptome' and proteome', the set of metabolites synthesized by a biological system constitute its 'metabolome'. Yet, unlike other functional genomics approaches, the unbiased simultaneous identification and quantification of plant metabolomes has been largely neglected. Until recently, most analyses were restricted to profiling selected classes of compounds, or to fingerprinting metabolic changes without sufficient analytical resolution to determine metabolite levels and identities individually. As a prerequisite for metabolomic analysis, careful consideration of the methods employed for tissue extraction, sample preparation, data acquisition, and data mining must be taken. In this review, the differences among metabolite target analysis, metabolite profiling, and metabolic fingerprinting are clarified, and terms are defined. Current approaches are examined, and potential applications are summarized with a special emphasis on data mining and mathematical modelling of metabolism.

Complexity and fragility in ecological networks

A detailed analysis of three species-rich ecosystem food webs has shown that they display skewed distributions of connections. Such graphs of interaction are, in fact, shared by a number of biological and technological networks, which have been shown to display a very high homeostasis against random removals of nodes. Here, we analyse the responses of these ecological graphs to both random and selective perturbations (directed against the most-connected species). Our results suggest that ecological networks are very robust against random removals but can be extremely fragile when selective attacks are used. These observations have important consequences for biodiversity dynamics and conservation issues, current estimations of extinction rates and the relevance and definition of keystone species.

Scale-free behavior in protein domain networks

Several technical, social, and biological networks were recently found to demonstrate scale-free and small-world behavior instead of random graph characteristics. In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied. It was found that these networks exhibited small-world and scale-free topologies with a high degree of local clustering accompanied by a few long-distance connections. Moreover, these observations apply not only to the complete databases, but also to the domain distributions in proteomes of different organisms. The extent of connectivity among domains reflects the evolutionary complexity of the organisms considered.

Random graphs with arbitrary degree distributions and their applications

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes

In this paper, the structure and evolution of the protein interaction network of the yeast Saccharomyces cerevisiae is analyzed. The network is viewed as a graph whose nodes correspond to proteins. Two proteins are connected by an edge if they interact. The network resembles a random graph in that it consists of many small subnets (groups of proteins that interact with each other but do not interact with any other protein) and one large connected subnet comprising more than half of all interacting proteins. The number of interactions per protein appears to follow a power law distribution. Within approximately 200 Myr after a duplication, the products of duplicate genes become almost equally likely to (1) have common protein interaction partners and (2) be part of the same subnetwork as two proteins chosen at random from within the network. This indicates that the persistence of redundant interaction partners is the exception rather than the rule. After gene duplication, the likelihood that an interaction gets lost exceeds 2.2 x 10(-3)/Myr. New interactions are estimated to evolve at a rate that is approximately three orders of magnitude smaller. Every 300 Myr, as many as half of all interactions may be replaced by new interactions.

Metabolic control analysis: biological applications and insights

Metabolic control analysis provides a robust mathematical and theoretical framework for describing metabolic and signaling pathways and networks, and for quantifying the controls over these processes. Its application has already shed light on some of the principles underlying the regulation of metabolic pathways, and it is well suited to the analysis of the types of data emerging from genomic studies.