Identification of biochemical network modules based on shortest retroactive distances

Modularity analysis offers a route to better understand the organization of cellular biochemical networks as well as to derive practically useful, simplified models of these complex systems. While there is general agreement regarding the qualitative properties of a biochemical module, there is no clear consensus on the quantitative criteria that may be used to systematically derive these modules. In this work, we investigate cyclical interactions as the defining characteristic of a biochemical module. We utilize a round trip distance metric, termed Shortest Retroactive Distance (ShReD), to characterize the retroactive connectivity between any two reactions in a biochemical network and to group together network components that mutually influence each other. We evaluate the metric on two types of networks that feature feedback interactions: (i) epidermal growth factor receptor (EGFR) signaling and (ii) liver metabolism supporting drug transformation. For both networks, the ShReD partitions found hierarchically arranged modules that confirm biological intuition. In addition, the partitions also revealed modules that are less intuitive. In particular, ShReD-based partition of the metabolic network identified a 'redox' module that couples reactions of glucose, pyruvate, lipid and drug metabolism through shared production and consumption of NADPH. Our results suggest that retroactive interactions arising from feedback loops and metabolic cycles significantly contribute to the modularity of biochemical networks. For metabolic networks, cofactors play an important role as allosteric effectors that mediate the retroactive interactions.     

Determining modular organization of protein interaction networks by maximizing modularity density

Background

With ever increasing amount of available data on biological networks, modeling and understanding the structure of these large networks is an important problem with profound biological implications. Cellular functions and biochemical events are coordinately carried out by groups of proteins interacting each other in biological modules. Identifying of such modules in protein interaction networks is very important for understanding the structure and function of these fundamental cellular networks. Therefore, developing an effective computational method to uncover biological modules should be highly challenging and indispensable.

Results

The purpose of this study is to introduce a new quantitative measure modularity density into the field of biomolecular networks and develop new algorithms for detecting functional modules in protein-protein interaction (PPI) networks. Specifically, we adopt the simulated annealing (SA) to maximize the modularity density and evaluate its efficiency on simulated networks. In order to address the computational complexity of SA procedure, we devise a spectral method for optimizing the index and apply it to a yeast PPI network.

Conclusions

Our analysis of detected modules by the present method suggests that most of these modules have well biological significance in context of protein complexes. Comparison with the MCL and the modularity based methods shows the efficiency of our method.

     

Spotlight: Assembly of protein complexes by integrating graph clustering methods

As is generally assumed, clusters in protein–protein interaction (PPI) networks perform specific, crucial functions in biological systems. Various network community detection methods have been developed to exploit PPI networks in order to identify protein complexes and functional modules. Due to the potential role of various regulatory modes in biological networks, a single method may just apply a single graph property and neglect communities highlighted by other network properties.     

An iterative network partition algorithm for accurate identification of dense network modules

A key step in network analysis is to partition a complex network into dense modules. Currently, modularity is one of the most popular benefit functions used to partition network modules. However, recent studies suggested that it has an inherent limitation in detecting dense network modules. In this study, we observed that despite the limitation, modularity has the advantage of preserving the primary network structure of the undetected modules. Thus, we have developed a simple iterative Network Partition (iNP) algorithm to partition a network. The iNP algorithm provides a general framework in which any modularity-based algorithm can be implemented in the network partition step. Here, we tested iNP with three modularity-based algorithms: multi-step greedy (MSG), spectral clustering and Qcut. Compared with the original three methods, iNP achieved a significant improvement in the quality of network partition in a benchmark study with simulated networks, identified more modules with significantly better enrichment of functionally related genes in both yeast protein complex network and breast cancer gene co-expression network, and discovered more cancer-specific modules in the cancer gene co-expression network. As such, iNP should have a broad application as a general method to assist in the analysis of biological networks.     

Modular response analysis of cellular regulatory networks

The sheer complexity of intracellular regulatory networks, which involve signal transducing, metabolic, and genetic circuits, hampers our ability to carry out a quantitative analysis of their functions. Here, we describe an approach that greatly simplifies this type of analysis by capitalizing on the modular organization of such networks. Steady-state responses of the network as a whole are accounted for in terms of intermodular interactions between the modules alone; processes operating solely within modules need not be considered when analysing signal transfer through the entire network. The intermodular interactions are quantified through (local) response coefficients which populate an interaction map (matrix). This matrix can be derived from a biochemical or molecular biological analysis of (macro) molecular interactions that constitute the regulatory network. The approach is illustrated by two examples: (i) mitogenic signalling through the mitogen-activated protein kinase cascade in the epidermal growth factor receptor network and (ii) regulation of ammonium assimilation in Escherichia coli.     

Toward a classification of isodynamic feed-forward motifs

A preceding study analysed how the topology of network motifs affects the overall rate of the underlying biochemical processes. Surprisingly, it was shown that topologically non-isomorphic motifs can still be isodynamic in the sense that they exhibit the exact same performance rate. Because of the high prevalence of feed-forward functional modules in biological networks, one may hypothesize that evolution tends to favour motifs with faster dynamics. As a step towards ranking the efficiency of feed-forward network motifs, we use a linear flow model to prove theorems establishing that certain classes of motifs are isodynamic. In partitioning the class of all motifs on n nodes into equivalence classes based upon their dynamics, we establish a basis for comparing the efficiency/performance rates of different motifs. The potential biological importance of the theorems is briefly discussed and is the subject of an ongoing large-scale project.     

Toward a classification of isodynamic feed-forward motifs

A preceding study analysed how the topology of network motifs affects the overall rate of the underlying biochemical processes. Surprisingly, it was shown that topologically non-isomorphic motifs can still be isodynamic in the sense that they exhibit the exact same performance rate. Because of the high prevalence of feed-forward functional modules in biological networks, one may hypothesize that evolution tends to favour motifs with faster dynamics. As a step towards ranking the efficiency of feed-forward network motifs, we use a linear flow model to prove theorems establishing that certain classes of motifs are isodynamic. In partitioning the class of all motifs on n nodes into equivalence classes based upon their dynamics, we establish a basis for comparing the efficiency/performance rates of different motifs. The potential biological importance of the theorems is briefly discussed and is the subject of an ongoing large-scale project.     

Motif search in graphs: application to metabolic networks

The classic view of metabolism as a collection of metabolic pathways is being questioned with the currently available possibility of studying whole networks. Novel ways of decomposing the network into modules and motifs that could be considered as the building blocks of a network are being suggested. In this work, we introduce a new definition of motif in the context of metabolic networks. Unlike in previous works on (other) biochemical networks, this definition is not based only on topological features. We propose instead to use an alternative definition based on the functional nature of the components that form the motif, which we call a reaction motif. After introducing a formal framework motivated by biological considerations, we present complexity results on the problem of searching for all occurrences of a reaction motif in a network and introduce an algorithm that is fast in practice in most situations. We then show an initial application to the study of pathway evolution. Finally, we give some general features of the observed number of occurrences in order to highlight some structural features of metabolic networks     

Probing the extent of randomness in protein interaction networks

Protein-protein interaction (PPI) networks are commonly explored for the identification of distinctive biological traits, such as pathways, modules, and functional motifs. In this respect, understanding the underlying network structure is vital to assess the significance of any discovered features. We recently demonstrated that PPI networks show degree-weighted behavior, whereby the probability of interaction between two proteins is generally proportional to the product of their numbers of interacting partners or degrees. It was surmised that degree-weighted behavior is a characteristic of randomness. We expand upon these findings by developing a random, degree-weighted, network model and show that eight PPI networks determined from single high-throughput (HT) experiments have global and local properties that are consistent with this model. The apparent random connectivity in HT PPI networks is counter-intuitive with respect to their observed degree distributions; however, we resolve this discrepancy by introducing a non-network-based model for the evolution of protein degrees or "binding affinities." This mechanism is based on duplication and random mutation, for which the degree distribution converges to a steady state that is identical to one obtained by averaging over the eight HT PPI networks. The results imply that the degrees and connectivities incorporated in HT PPI networks are characteristic of unbiased interactions between proteins that have varying individual binding affinities. These findings corroborate the observation that curated and high-confidence PPI networks are distinct from HT PPI networks and not consistent with a random connectivity. These results provide an avenue to discern indiscriminate organizations in biological networks and suggest caution in the analysis of curated and high-confidence networks.     

BicNET: Flexible module discovery in large-scale biological networks using biclustering

Background

Despite the recognized importance of module discovery in biological networks to enhance our understanding of complex biological systems, existing methods generally suffer from two major drawbacks. First, there is a focus on modules where biological entities are strongly connected, leading to the discovery of trivial/well-known modules and to the inaccurate exclusion of biological entities with subtler yet relevant roles. Second, there is a generalized intolerance towards different forms of noise, including uncertainty associated with less-studied biological entities (in the context of literature-driven networks) and experimental noise (in the context of data-driven networks). Although state-of-the-art biclustering algorithms are able to discover modules with varying coherency and robustness to noise, their application for the discovery of non-dense modules in biological networks has been poorly explored and it is further challenged by efficiency bottlenecks.

Methods

This work proposes Biclustering NETworks (BicNET), a biclustering algorithm to discover non-trivial yet coherent modules in weighted biological networks with heightened efficiency. Three major contributions are provided. First, we motivate the relevance of discovering network modules given by constant, symmetric, plaid and order-preserving biclustering models. Second, we propose an algorithm to discover these modules and to robustly handle noisy and missing interactions. Finally, we provide new searches to tackle time and memory bottlenecks by effectively exploring the inherent structural sparsity of network data.

Results

Results in synthetic network data confirm the soundness, efficiency and superiority of BicNET. The application of BicNET on protein interaction and gene interaction networks from yeast, E. coli and Human reveals new modules with heightened biological significance.

Conclusions

BicNET is, to our knowledge, the first method enabling the efficient unsupervised analysis of large-scale network data for the discovery of coherent modules with parameterizable homogeneity.

     

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.     

In silico evolution of functional modules in biochemical networks

Understanding the large reaction networks found in biological systems is a daunting task. One approach is to divide a network into more manageable smaller modules, thus simplifying the problem. This is a common strategy used in engineering. However, the process of identifying biological modules is still in its infancy and very little is understood about the range and capabilities of motif structures found in biological modules. In order to delineate these modules, a library of functional motifs has been generated via in silico evolution techniques. On the basis of their functional forms, networks were evolved from four broad areas: oscillators, bistable switches, homeostatic systems and frequency filters. Some of these motifs were constructed from simple mass action kinetics, others were based on Michaelis-Menten kinetics as found in protein/protein networks and the remainder were based on Hill equations as found in gene/protein interaction networks. The purpose of the study is to explore the capabilities of different network architectures and the rich variety of functional forms that can be generated. Ultimately, the library may be used to delineate functional motifs in real biological networks.     

Computer-aided design of biological circuits using TinkerCell

Synthetic biology is an engineering discipline that builds on modeling practices from systems biology and wet-lab techniques from genetic engineering. As synthetic biology advances, efficient procedures will be developed that will allow a synthetic biologist to design, analyze, and build biological networks. In this idealized pipeline, computer-aided design (CAD) is a necessary component. The role of a CAD application would be to allow efficient transition from a general design to a final product. TinkerCell is a design tool for serving this purpose in synthetic biology. In TinkerCell, users build biological networks using biological parts and modules. The network can be analyzed using one of several functions provided by TinkerCell or custom programs from third-party sources. Since best practices for modeling and constructing synthetic biology networks have not yet been established, TinkerCell is designed as a flexible and extensible application that can adjust itself to changes in the field.     

Structural Analysis to Determine the Core of Hypoxia Response Network

The advent of sophisticated molecular biology techniques allows to deduce the structure of complex biological networks. However, networks tend to be huge and impose computational challenges on traditional mathematical analysis due to their high dimension and lack of reliable kinetic data. To overcome this problem, complex biological networks are decomposed into modules that are assumed to capture essential aspects of the full network''s dynamics. The question that begs for an answer is how to identify the core that is representative of a network''s dynamics, its function and robustness. One of the powerful methods to probe into the structure of a network is Petri net analysis. Petri nets support network visualization and execution. They are also equipped with sound mathematical and formal reasoning based on which a network can be decomposed into modules. The structural analysis provides insight into the robustness and facilitates the identification of fragile nodes. The application of these techniques to a previously proposed hypoxia control network reveals three functional modules responsible for degrading the hypoxia-inducible factor (HIF). Interestingly, the structural analysis identifies superfluous network parts and suggests that the reversibility of the reactions are not important for the essential functionality. The core network is determined to be the union of the three reduced individual modules. The structural analysis results are confirmed by numerical integration of the differential equations induced by the individual modules as well as their composition. The structural analysis leads also to a coarse network structure highlighting the structural principles inherent in the three functional modules. Importantly, our analysis identifies the fragile node in this robust network without which the switch-like behavior is shown to be completely absent.     

In search of the biological significance of modular structures in protein networks

Many complex networks such as computer and social networks exhibit modular structures, where links between nodes are much denser within modules than between modules. It is widely believed that cellular networks are also modular, reflecting the relative independence and coherence of different functional units in a cell. While many authors have claimed that observations from the yeast protein–protein interaction (PPI) network support the above hypothesis, the observed structural modularity may be an artifact because the current PPI data include interactions inferred from protein complexes through approaches that create modules (e.g., assigning pairwise interactions among all proteins in a complex). Here we analyze the yeast PPI network including protein complexes (PIC network) and excluding complexes (PEC network). We find that both PIC and PEC networks show a significantly greater structural modularity than that of randomly rewired networks. Nonetheless, there is little evidence that the structural modules correspond to functional units, particularly in the PEC network. More disturbingly, there is no evolutionary conservation among yeast, fly, and nematode modules at either the whole-module or protein-pair level. Neither is there a correlation between the evolutionary or phylogenetic conservation of a protein and the extent of its participation in various modules. Using computer simulation, we demonstrate that a higher-than-expected modularity can arise during network growth through a simple model of gene duplication, without natural selection for modularity. Taken together, our results suggest the intriguing possibility that the structural modules in the PPI network originated as an evolutionary byproduct without biological significance.     

Evolution of complex modular biological networks

Biological networks have evolved to be highly functional within uncertain environments while remaining extremely adaptable. One of the main contributors to the robustness and evolvability of biological networks is believed to be their modularity of function, with modules defined as sets of genes that are strongly interconnected but whose function is separable from those of other modules. Here, we investigate the in silico evolution of modularity and robustness in complex artificial metabolic networks that encode an increasing amount of information about their environment while acquiring ubiquitous features of biological, social, and engineering networks, such as scale-free edge distribution, small-world property, and fault-tolerance. These networks evolve in environments that differ in their predictability, and allow us to study modularity from topological, information-theoretic, and gene-epistatic points of view using new tools that do not depend on any preconceived notion of modularity. We find that for our evolved complex networks as well as for the yeast protein–protein interaction network, synthetic lethal gene pairs consist mostly of redundant genes that lie close to each other and therefore within modules, while knockdown suppressor gene pairs are farther apart and often straddle modules, suggesting that knockdown rescue is mediated by alternative pathways or modules. The combination of network modularity tools together with genetic interaction data constitutes a powerful approach to study and dissect the role of modularity in the evolution and function of biological networks.     

Using the topology of metabolic networks to predict viability of mutant strains

Understanding the relationships between the structure (topology) and function of biological networks is a central question of systems biology. The idea that topology is a major determinant of systems function has become an attractive and highly disputed hypothesis. Although structural analysis of interaction networks demonstrates a correlation between the topological properties of a node (protein, gene) in the network and its functional essentiality, the analysis of metabolic networks fails to find such correlations. In contrast, approaches utilizing both the topology and biochemical parameters of metabolic networks, e.g., flux balance analysis, are more successful in predicting phenotypes of knockout strains. We reconcile these seemingly conflicting results by showing that the topology of the metabolic networks of both Escherichia coli and Saccharomyces cerevisiae are, in fact, sufficient to predict the viability of knockout strains with accuracy comparable to flux balance analysis on large, unbiased mutant data sets. This surprising result is obtained by introducing a novel topology-based measure of network transport: synthetic accessibility. We also show that other popular topology-based characteristics such as node degree, graph diameter, and node usage (betweenness) fail to predict the viability of E. coli mutant strains. The success of synthetic accessibility demonstrates its ability to capture the essential properties of the metabolic network, such as the branching of chemical reactions and the directed transport of material from inputs to outputs. Our results strongly support a link between the topology and function of biological networks and, in agreement with recent genetic studies, emphasize the minimal role of flux rerouting in providing robustness of mutant strains.     

EXTINCTIONS IN HETEROGENEOUS ENVIRONMENTS AND THE EVOLUTION OF MODULARITY

Extinctions of local subpopulations are common events in nature. Here, we ask whether such extinctions can affect the design of biological networks within organisms over evolutionary timescales. We study the impact of extinction events on modularity of biological systems, a common architectural principle found on multiple scales in biology. As a model system, we use networks that evolve toward goals specified as desired input–output relationships. We use an extinction–recolonization model, in which metapopulations occupy and migrate between different localities. Each locality displays a different environmental condition (goal), but shares the same set of subgoals with other localities. We find that in the absence of extinction events, the evolved computational networks are typically highly optimal for their localities with a nonmodular structure. In contrast, when local populations go extinct from time to time, we find that the evolved networks are modular in structure. Modular circuitry is selected because of its ability to adapt rapidly to the conditions of the free niche following an extinction event. This rapid adaptation is mainly achieved through genetic recombination of modules between immigrants from neighboring local populations. This study suggests, therefore, that extinctions in heterogeneous environments promote the evolution of modular biological network structure, allowing local populations to effectively recombine their modules to recolonize niches.     

Modular organization of protein interaction networks

MOTIVATION: Accumulating evidence suggests that biological systems are composed of interacting, separable, functional modules. Identifying these modules is essential to understand the organization of biological systems. RESULT: In this paper, we present a framework to identify modules within biological networks. In this approach, the concept of degree is extended from the single vertex to the sub-graph, and a formal definition of module in a network is used. A new agglomerative algorithm was developed to identify modules from the network by combining the new module definition with the relative edge order generated by the Girvan-Newman (G-N) algorithm. A JAVA program, MoNet, was developed to implement the algorithm. Applying MoNet to the yeast core protein interaction network from the database of interacting proteins (DIP) identified 86 simple modules with sizes larger than three proteins. The modules obtained are significantly enriched in proteins with related biological process Gene Ontology terms. A comparison between the MoNet modules and modules defined by Radicchi et al. (2004) indicates that MoNet modules show stronger co-clustering of related genes and are more robust to ties in betweenness values. Further, the MoNet output retains the adjacent relationships between modules and allows the construction of an interaction web of modules providing insight regarding the relationships between different functional modules. Thus, MoNet provides an objective approach to understand the organization and interactions of biological processes in cellular systems. AVAILABILITY: MoNet is available upon request from the authors.