Inferring meaningful pathways in weighted metabolic networks

An approach is presented for computing meaningful pathways in the network of small molecule metabolism comprising the chemical reactions characterized in all organisms. The metabolic network is described as a weighted graph in which all the compounds are included, but each compound is assigned a weight equal to the number of reactions in which it participates. Path finding is performed in this graph by searching for one or more paths with lowest weight. Performance is evaluated systematically by computing paths between the first and last reactions in annotated metabolic pathways, and comparing the intermediate reactions in the computed pathways to those in the annotated ones. For the sake of comparison, paths are computed also in the un-weighted raw (all compounds and reactions) and filtered (highly connected pool metabolites removed) metabolic graphs, respectively. The correspondence between the computed and annotated pathways is very poor (<30%) in the raw graph; increasing to approximately 65% in the filtered graph; reaching approximately 85% in the weighted graph. Considering the best-matching path among the five lightest paths increases the correspondence to 92%, on average. We then show that the average distance between pairs of metabolites is significantly larger in the weighted graph than in the raw unfiltered graph, suggesting that the small-world properties previously reported for metabolic networks probably result from irrelevant shortcuts through pool metabolites. In addition, we provide evidence that the length of the shortest path in the weighted graph represents a valid measure of the "metabolic distance" between enzymes. We suggest that the success of our simplistic approach is rooted in the high degree of specificity of the reactions in metabolic pathways, presumably reflecting thermodynamic constraints operating in these pathways. We expect our approach to find useful applications in inferring metabolic pathways in newly sequenced genomes.     

Inferring qualitative relations in genetic networks and metabolic pathways

MOTIVATION: Inferring genetic network architecture from time series data of gene expression patterns is an important topic in bioinformatics. Although inference algorithms based on the Boolean network were proposed, the Boolean network was not sufficient as a model of a genetic network. RESULTS: First, a Boolean network model with noise is proposed, together with an inference algorithm for it. Next, a qualitative network model is proposed, in which regulation rules are represented as qualitative rules and embedded in the network structure. Algorithms are also presented for inferring qualitative relations from time series data. Then, an algorithm for inferring S-systems (synergistic and saturable systems) from time series data is presented, where S-systems are based on a particular kind of nonlinear differential equation and have been applied to the analysis of various biological systems. Theoretical results are shown for Boolean networks with noises and simple qualitative networks. Computational results are shown for Boolean networks with noises and S-systems, where real data are not used because the proposed models are still conceptual and the quantity and quality of currently available data are not enough for the application of the proposed methods.     

Systemic properties of ensembles of metabolic networks: application of graphical and statistical methods to simple unbranched pathways

MOTIVATION: Mathematical models are the only realistic method for representing the integrated dynamic behavior of complex biochemical networks. However, it is difficult to obtain a consistent set of values for the parameters that characterize such a model. Even when a set of parameter values exists, the accuracy of the individual values is questionable. Therefore, we were motivated to explore statistical techniques for analyzing the properties of a given model when knowledge of the actual parameter values is lacking. RESULTS: The graphical and statistical methods presented in the previous paper are applied here to simple unbranched biosynthetic pathways subject to control by feedback inhibition. We represent these pathways within a canonical nonlinear formalism that provides a regular structure that is convenient for randomly sampling the parameter space. After constructing a large ensemble of randomly generated sets of parameter values, the structural and behavioral properties of the model with these parameter sets are examined statistically and classified. The results of our analysis demonstrate that certain properties of these systems are strongly correlated, thereby revealing aspects of organization that are highly probable independent of selection. Finally, we show how specification of a given behavior affects the distribution of acceptable parameter values.     

From local to global dilemmas in social networks

Social networks affect in such a fundamental way the dynamics of the population they support that the global, population-wide behavior that one observes often bears no relation to the individual processes it stems from. Up to now, linking the global networked dynamics to such individual mechanisms has remained elusive. Here we study the evolution of cooperation in networked populations and let individuals interact via a 2-person Prisoner's Dilemma--a characteristic defection dominant social dilemma of cooperation. We show how homogeneous networks transform a Prisoner's Dilemma into a population-wide evolutionary dynamics that promotes the coexistence between cooperators and defectors, while heterogeneous networks promote their coordination. To this end, we define a dynamic variable that allows us to track the self-organization of cooperators when co-evolving with defectors in networked populations. Using the same variable, we show how the global dynamics--and effective dilemma--co-evolves with the motifs of cooperators in the population, the overall emergence of cooperation depending sensitively on this co-evolution.     

Estimation of the number of extreme pathways for metabolic networks

Background  

The set of extreme pathways (ExPa), {p _i }, defines the convex basis vectors used for the mathematical characterization of the null space of the stoichiometric matrix for biochemical reaction networks. ExPa analysis has been used for a number of studies to determine properties of metabolic networks as well as to obtain insight into their physiological and functional states in silico. However, the number of ExPas, p = |{p _i }|, grows with the size and complexity of the network being studied, and this poses a computational challenge. For this study, we investigated the relationship between the number of extreme pathways and simple network properties.     

Some global properties of neural networks

Some properties of the global behaviour of a model of neural network are considered.The geometric concept of quadrant-degeneration is studied and it is shown to be independent of the algebraic concept of rank-degeneration. The results obtained are employed to solve some global problems of synthesis (i.e., independent of the initial state of the network) without the use of the theory of linear inequalities.     

Simple and fast alignment of metabolic pathways by exploiting local diversity

MOTIVATION: An important tool for analyzing biological networks is the ability to perform homology searches, i.e. given a pattern network one would like to be able to search for occurrences of similar (sub)networks within a set of host networks. In the context of metabolic pathways, Pinter et al. [Bioinformatics, 2005] proposed to solve this computationally hard problem by restricting it to the case where both the pattern and host networks are trees. This restriction, however, severely limits the applicability of their algorithm. RESULTS: We propose a very fast and simple algorithm for the alignment of metabolic pathways that does not restrict the topology of the host or pattern network in any way; instead, our algorithm exploits a natural property of metabolic networks that we call 'local diversity property'. Experiments on a test bed of metabolic pathways from the BioCyc database indicate that our algorithm is much faster than the restricted algorithm of Pinter et al.-the metabolic pathways of two organisms can be aligned in mere seconds-and yet has a wider range of applicability and yields new biological insights. Our ideas can likely be extended to work for the alignment of various types of biological networks other than metabolic pathways. AVAILABILITY: Our algorithm has been implemented in C++ as a user-friendly metabolic pathway alignment tool called METAPAT. The tool runs under Linux or Windows and can be downloaded at http://theinf1.informatik.uni-jena.de/metapat/     

Visualizing global properties of large complex networks

For complex biological networks, graphical representations are highly desired for understanding some design principles, but few drawing methods are available that capture topological features of a large and highly heterogeneous network, such as a protein interaction network. Here we propose the circular perspective drawing (CPD) method to visualize global structures of large complex networks. The presented CPD combines the quasi-continuous search (QCS) analogous to the steepest descent method with a random node swapping strategy for an enhanced calculation speed. The CPD depicts a network in an aesthetic manner by showing connection patterns between different parts of the network instead of detailed links between nodes. Global structural features of networks exhibited by CPD provide clues toward a comprehensive understanding of the network organizations. Availability: Software is freely available at http://www.cadlive.jp.     

Foundations of metabolic organization: coherence as a basis of computational properties in metabolic networks

Biological organization is based on the coherent energy transfer allowing for macromolecules to operate with high efficiency and realize computation. Computation is executed with virtually 100% efficiency via the coherent operation of molecular machines in which low-energy recognitions trigger energy-driven non-equilibrium dynamic processes. The recognition process is of quantum mechanical nature being a non-demolition measurement. It underlies the enzymatic conversion of a substrate into the product (an elementary metabolic phenomenon); the switching via separation of the direct and reverse routes in futile cycles provides the generation and complication of metabolic networks (coherence within cycles is maintained by the supramolecular organization of enzymes); the genetic level corresponding to the appearance of digital information is based on reflective arrows (catalysts realize their own self-reproduction) and operation of hypercycles. Every metabolic cycle via reciprocal regulation of both its halves can generate rhythms and spatial structures (resulting from the temporally organized depositions from the cycles). Via coherent events which percolate from the elementary submolecular level to organismic entities, self-assembly based on the molecular complementarity is realized and the dynamic informational field operating within the metabolic network is generated.     

A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks

A set of linear pathways often does not capture the full range of behaviors of a metabolic network. The concept of 'elementary flux modes' provides a mathematical tool to define and comprehensively describe all metabolic routes that are both stoichiometrically and thermodynamically feasible for a group of enzymes. We have used this concept to analyze the interplay between the pentose phosphate pathway (PPP) and glycolysis. The set of elementary modes for this system involves conventional glycolysis, a futile cycle, all the modes of PPP function described in biochemistry textbooks, and additional modes that are a priori equally entitled to pathway status. Applications include maximizing product yield in amino acid and antibiotic synthesis, reconstruction and consistency checks of metabolism from genome data, analysis of enzyme deficiencies, and drug target identification in metabolic networks.     

Observing and interpreting correlations in metabolomic networks

MOTIVATION: Metabolite profiling aims at an unbiased identification and quantification of all the metabolites present in a biological sample. Based on their pair-wise correlations, the data obtained from metabolomic experiments are organized into metabolic correlation networks and the key challenge is to deduce unknown pathways based on the observed correlations. However, the data generated is fundamentally different from traditional biological measurements and thus the analysis is often restricted to rather pragmatic approaches, such as data mining tools, to discriminate between different metabolic phenotypes. METHODS AND RESULTS: We investigate to what extent the data generated networks reflect the structure of the underlying biochemical pathways. The purpose of this work is 2-fold: Based on the theory of stochastic systems, we first introduce a framework which shows that the emergent correlations can be interpreted as a 'fingerprint' of the underlying biophysical system. This result leads to a systematic relationship between observed correlation networks and the underlying biochemical pathways. In a second step, we investigate to what extent our result is applicable to the problem of reverse engineering, i.e. to recover the underlying enzymatic reaction network from data. The implications of our findings for other bioinformatics approaches are discussed.     

Exploring local structural organization of metabolic networks using subgraph patterns

Metabolic networks of many cellular organisms share global statistical features. Their connectivity distributions follow the long-tailed power law and show the small-world property. In addition, their modular structures are organized in a hierarchical manner. Although the global topological organization of metabolic networks is well understood, their local structural organization is still not clear. Investigating local properties of metabolic networks is necessary to understand the nature of metabolism in living organisms. To identify the local structural organization of metabolic networks, we analysed the subgraphs of metabolic networks of 43 organisms from three domains of life. We first identified the network motifs of metabolic networks and identified the statistically significant subgraph patterns. We then compared metabolic networks from different domains and found that they have similar local structures and that the local structure of each metabolic network has its own taxonomical meaning. Organisms closer in taxonomy showed similar local structures. In addition, the common substrates of 43 metabolic networks were not randomly distributed, but were more likely to be constituents of cohesive subgraph patterns.     

Growth phase-dependent changes in the expression of global regulatory genes and associated metabolic pathways in Escherichia coli

Metabolic regulation in Escherichia coli was studied in terms of the changes in the expression of the global regulatory genes rpoD, rpoS, soxRS, cra, fadR, iclR and arcA at three different growth phases, in batch culture. The expression of rpoS and several rpoS-dependent metabolic pathway genes, such as tktB, talA, fumC, acnA, sucA, acs and sodC, were increased (∼1.5 to 2-fold) as the cells entered the late phase of growth. The changes in the expression of other global regulators and their effects on different metabolic pathway genes were less significant, as compared to rpoS, during the later phases of growth.     

Observing metabolic functions at the genome scale

Background  

High-throughput techniques have multiplied the amount and the types of available biological data, and for the first time achieving a global comprehension of the physiology of biological cells has become an achievable goal. This aim requires the integration of large amounts of heterogeneous data at different scales. It is notably necessary to extend the traditional focus on genomic data towards a truly functional focus, where the activity of cells is described in terms of actual metabolic processes performing the functions necessary for cells to live.     

Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

The regulation of biological networks relies significantly on convergent feedback signaling loops that render a global output locally accessible. Ideally, the recurrent connectivity within these systems is self-organized by a time-dependent phase-locking mechanism. This study analyzes recurrent fractal neural networks (RFNNs), which utilize a self-similar or fractal branching structure of dendrites and downstream networks for phase-locking of reciprocal feedback loops: output from outer branch nodes of the network tree enters inner branch nodes of the dendritic tree in single neurons. This structural organization enables RFNNs to amplify re-entrant input by over-the-threshold signal summation from feedback loops with equivalent signal traveling times. The columnar organization of pyramidal neurons in the neocortical layers V and III is discussed as the structural substrate for this network architecture. RFNNs self-organize spike trains and render the entire neural network output accessible to the dendritic tree of each neuron within this network. As the result of a contraction mapping operation, the local dendritic input pattern contains a downscaled version of the network output coding structure. RFNNs perform robust, fractal data compression, thus coping with a limited number of feedback loops for signal transport in convergent neural networks. This property is discussed as a significant step toward the solution of a fundamental problem in neuroscience: how is neuronal computation in separate neurons and remote brain areas unified as an instance of experience in consciousness? RFNNs are promising candidates for engaging neural networks into a coherent activity and provide a strategy for the exchange of global and local information processing in the human brain, thereby ensuring the completeness of a transformation from neuronal computation into conscious experience.     

Description of local and global shape properties of protein helices

A new method, dubbed “HAXIS” is introduced to describe local and global shape properties of a protein helix via its axis. HAXIS is based on coarse-graining and spline-fitting of the helix backbone. At each Cα anchor point of the backbone, a Frenet frame is calculated, which directly provides the local vector presentation of the helix. After cubic spline-fitting of the axis line, its curvature and torsion are calculated. This makes a rapid comparison of different helix forms and the determination of helix similarity possible. Distortions of the helix caused by individual residues are projected onto the helix axis and presented either by the rise parameter per residue or by the local curvature of the axis. From a non-redundant set of 2,017 proteins, 15,068 helices were investigated in this way. Helix start and helix end as well as bending and kinking of the helix are accurately described. The global properties of the helix are assessed by a polynomial fit of the helix axis and the determination of its overall curving and twisting. Long helices are more regular shaped and linear whereas short helices are often strongly bent and twisted. The distribution of different helix forms as a function of helix length is analyzed.