Abstract

In this study we analyze the topology and structure of Airport Network of India (ANI) using the graph theoretic approach. We investigate the transmission flow in the network using graph centrality measures. We observe that though ANI exhibits scale-free behaviour, it differs from Barabasi-Albert scale-free model and we suggest a scale-free model that results in high clustering coefficient to be most suitable to model ANI. The analysis is not only useful in planning the infrastructure and expansion of the air-traffic connectivity, but also in managing the flow of transportation during emergencies such as accidental failure of the airport, closing down the airport due to unexpected climate changes, terrorist attacks etc. It has been observed that densely connected air-transportation networks play a major role in the spread of infectious diseases, viz., Avian-influenza, Swine-Flu, etc., turning from epidemic into pandemic. Knowledge of the connectivity pattern for reduction of flights on certain routes can help to reduce the spread of the disease.

Abstract

Two classes of networks that have been extensively studied in the analysis of physical systems are: (i) regular networks, wherein each node interacts with a specified number of neighboring nodes on geometrical lattices, and (ii) random networks, wherein every pair of nodes have a fixed probability of interacting with each other. Recently much attention is being focused on a class of network models that are neither strictly regular nor completely random, but are somewhere in-between and exhibit properties of both, called small world networks. These networks are shown to exhibit high clustering (i.e., nodes sharing a common neighbor have a higher probability of being connected to each other than to other nodes) and a low average path length (the average of the shortest distance/path between every pair of nodes in the network). Examples for small-world networks are found to occur widely across the biological (e.g., neural connection patterns), social (e.g., friendship network, co-authorship) and technological (e.g., the world wide web) domains. Also it has been observed that a large number of real complex systems exhibit power-law behaviour in their degree distribution, i.e., a few nodes have a very high degree. Such networks are referred to as scale-free networks. Thus, non-standard topologies with long-range connections (i.e., non-local diffusion) and uneven degree distributions are not uncommon in real-life systems and may provide different kinds of spatiotemporal dynamics depending on the extent of non-local diffusion. Here we discuss the characterization and control of spatiotemporal dynamics on four different network topologies, viz., (i) regular, (ii) random, (iii) small-world, and (iv) scale-free by external perturbation or pinning a few nodes in the network. This would provide us insight into the role of the network topology (the underlying connectivity structure) on the dynamics of the systems defined on such networks and the efficacy with which the dynamics can be the controlled or manipulated. An extensively studied example of a nonlinear system exhibiting a wide variety of complex dynamics ranging from simple periodic behavior to chaos is the logistic map. Here we define coupled logistic maps on the four different topologies and systematically investigate the control by external perturbation/pinning for two chaotic regimes: (i) r = 3.6 (weak-chaos), and (ii) r = 3.9 (strong chaos). Our preliminary results show that pinning nodes at regularly spaced intervals, 2nd, 4th, etc., the pinning strength required on a small-world network is similar to that in case of regular networks. For 25% of the nodes pinned, the dynamics on the scale-free topology is controllable but on random networks, it is not. However, on pinning nodes having high centrality measures, viz., degree, betweenness and closeness, we observe that control of the spatiotemporal dynamics is achieved, by pinning only 10% of high degree/ betweenness nodes in low chaotic regime on both small-world and scale-free networks. Complete control of the network is not observed on pinning nodes with high closeness values. For strongly chaotic dynamics, control is achievable only on scale-free networks on pinning 20% of nodes having either high degree, betweenness, or closeness values. Key Words: Dynamics, Control, Topologies, Coupled Logistic Maps, Chaos, Regular, Smallworld, Random, Scale-free networks, Centrality.

Abstract

In the present study, we have constructed and analysed the metabolic networks of virulent and non-virulent strain of E. coli. Concepts from graph theory are used to systematically determine how differences in basic metabolism of different strains of E. coliare reflected at the systems level. Analysis of metabolic networks can help us to understand the evolutionary history of cellular metabolic process and we can also identify the potential drug target by comparing the networks.

Abstract

Abstract. Here we present a simple method based on graph spectral properties to automatically partition multi-domain proteins into individual domains. The identification of structural domains in proteins is based on the assumption that the interactions between the amino acids are higher within a domain than across the domains. These interactions and the topological details of protein structures can be effectively captured by the protein contact graph, constructed by considering each amino acid as a node with an edge drawn between two nodes if the C alpha atoms of the amino acids are within 7. Here we show that Newmans community detection approach in social networks can be used to identify domains in protein structures. We have implemented this approach on protein contact networks and analyze the eigenvectors of the largest eigenvalue of modularity matrix, which is a modified form of the Adjacency matrix, using a quality function called modularity to identify optimal divisions of the network into domains. The proposed approach works even when the domains are formed with amino acids not occurring sequentially along the polypeptide chain and no a priori information regarding the number of nodes is required.

Abstract

In most real world networks the connectivity topology between individual entities is not governed by regular, nearest neighbour interactions but also includes long range interactions, e.g., small-world topology of interacting neurons, scale-free behaviour of gene regulatory networks, etc. In this study we have analyzed the synchronization and control of the dynamics in such networks having non-regular topology, viz., small-world, random and scale-free topologies, compared to that in regular networks modeled on coupled map lattices. For the analysis, logistic map, which exhibits rich dynamical behaviour, is modeled on each node and connectivity between the nodes is defined based on the Watt-Strogatz model for small-world and random networks and Albert-Barabasi model for the scale-free network. The synchronization and control of the complex dynamics is analyzed for varying coupling strengths and connectivity delays in the four network topologies. We observe that the synchronization of the network depends on the connection topology and also on connection delays, with poor synchronization observed in regular and small-world networks compared to random and scale-free topology. The connection delays between constituent units are capable of sustaining complex dynamical behaviour not seen in the undelayed system.

Abstract

The synchronization and control of dynamical systems on a regular lattice has been extensively studied. However, it has been observed that the underlying topology of a real-system need not always be regular, e.g., the brain has been observed to have a connection structure with small-world properties. Thus, non-standard topologies with long-range connections (i.e., non-local diffusion) are not uncommon in real-life systems and may provide different kinds of spatiotemporal dynamics depending on the extent of non-local diffusion. This led us to investigate the spatiotemporal dynamics of coupled logistic map on four different topological networks, viz., regular, small-world, random and scale-free. In particular, we are interested in the synchronization and control of the spatiotemporal dynamics. It has been shown earlier that the transmission delays between constituent units are capable of sustaining complex dynamical behavior, a phenomenon not observed in the undelayed systems. In fact, the synchronization of the dynamics is observed for a much larger parameter space when the underlying topology is scale-free or random. In our earlier work on the control of spatiotemporal dynamics, we have shown the effect of externally applied perturbation or 'pinning' on regular network. By varying the strength and sign of the pinning strength we were able to target the system to any desired dynamical state. Here we investigate the effect of pinning the dynamics on various topologies. Using graph centrality measures, viz., degree, betweenness and closeness for pinning the nodes, our preliminary results suggest that high betweenness nodes perform better in controlling the spatiotemporal dynamics with as few as 10% of nodes required for pinning in case of system exhibiting weak chaotic dynamics. This study has important practical applications in the control of dynamical diseases such as epileptic seizures and bursts on a small-world brain topology.

Abstract

Here we present a simple method based on graph spectral properties to automatically partition multi-domain proteins into individual domains. The identification of structural domains in proteins is based on the assumption that the interactions between the amino acids are higher within a domain than across the domains. These interactions and the topological details of protein structures can be effectively captured by the protein contact graph, constructed by considering each amino acid as a node with an edge drawn between two nodes if the Cα atoms of the amino acids are within 7Å. Here we show that Newman’s community detection approach in social networks can be used to identify domains in protein structures. We have implemented this approach on protein contact networks and analyze the eigenvectors of the largest eigenvalue of modularity matrix, which is a modified form of the Adjacency matrix, using a quality function called “modularity” to identify optimal divisions of the network into domains. The proposed approach works even when the domains are formed with amino acids not occurring sequentially along the polypeptide chain and no a priori information regarding the number of nodes is required.

Abstract

Here we apply the graph-theoretic concept of Betweenness centrality to a class of protein repeats, e.g., Armadillo (ARM) and HEAT. The Betweenness of a node represents how often a node is traversed on the shortest path between all pairs of nodes i, j in the network and thus gives the contribution of each node in the network. These repeats are not easily detectable at the sequence level because of low conservation between independent repeated units, e.g., HEAT repeats are known to have less than 13% identity. Their identification at the structure level typically involves self structure-structure comparison, which can be computationally very intensive. Our analysis of a set of proteins from ARM and HEAT repeat family shows that the repeat regions exhibit similar connectivity patterns for the repeating units. Since it is generally accepted that in many networks, the larger the degree of a node, the larger the chance that many of the shortest paths will pass through this node, computing vertex Betweenness provides a simple and elegant approach for identifying tandem structural repeats in proteins.

Abstract

Recently much attention is being focused on the analysis of the three dimensional structure of proteins using graph and network concepts. The simplest protein network graph is constructed by defining the amino acids as nodes and the edges drawn between amino acids (within a threshold) which attempt to capture not only covalent but also non-covalent interactions. Analysis of the topological details of proteins with known structures, such as clustering of specific types of amino acids important for structure, folding and function, is of great value and is an active field of research. Since structures of a large number of proteins is now available, automatic methods of analysis are required to analyze them and recently, the tools from graph theory are being explored for such analysis. Here, we propose to use graph properties and their spectral analysis for identifying patterns in protein structures, in particular, structural repeats or motifs. These repeats are not easily detectable at the sequence level because of low conservation or high divergence between independent repeated units. The importance of such repeats in understanding biological function resides not only in their high frequency among known sequences, but also in their abilities to confer multiple binding and structural roles on proteins, e.g., zinc finger domain, a constituent of transcription factors involved in DNA binding, where the composition and copy number of individual tandem repeats confers selectivity and activity of DNA binding. This functional versatility is apparent not only among different repeat types, but also for similar repeats from the same family. Our understanding of repeats, with respect to their structures, functions, and evolution, therefore represents a considerable challenge. Our analysis of the degree distribution and spectral analysis of a set of proteins from various repeat families show that the repeat regions exhibit similar connectivity patterns in the graph. The network property, Betweenness, was found to identify accurately the repeat boundaries in protein families. Thus, this approach can help in identifying repeated motifs in proteins which are difficult to identify at the sequence level.

Abstract

About 14% of the protein sequences in the Swissprot database contain repetitive region, viz., tandem repeats, multiple copies of motifs/profiles, multiple copies of domain. And eukaryotic proteins are more likely to have repeats than Bacteria and Archaea. Our main focus is only on tandem repeats. Tandem repeat can be defined as contiguous repeat pattern of two or more copies. These copies can be exact or approximate. Many proteins with these repeat are involved in functions like transcription, translation, protein-protein interaction. Proteins with tandem repeats are involved in various neurodegenerative diseases like Huntington's disease. These proteins are also found to occur in sequences which are poorly conserved in evolution. We have developed a database of tandem repeats in protein sequences: Protein Tandem Repeats DataBase(PTRDB). The data for this database is extracted using our in-house tool PEPPER, a tool for identifying PEPtide PEriodic Repeats. This database is built on SwissProt (ver 51.7). PTRDB have 3145 proteins with 4713 tandem repeats. 77.74% are found in Eukaryota, 16.63% in Bacteria and 0.67% in Archaea. About 5% of proteins in this database are associated with disease. We have classified the database organism wise. This database can be queried by various attributes like SwissProt ID, organism name, PDBID, repeat pattern, repeat length, keyword and copy number. It gives the detail information of tandem repeats like SwissProt ID (hyperlinked to SwissProt to give Fasta sequence), organism name, PDBID (hyperlinked to PDB), one line description of the protein, gene name, protein name, taxonomy ID (hyperlinked to EBI), Family information (hyperlinked to Pfam), Description of associated disease, OMIM ID, scoring matrix with gap extension for the repeat, repeat pattern with alignment score, copy number, repeat length, start point, end point, alignment of repeat pattern with repeat region, secondary structure information of the repeat region. With the increasing abundance of tandem protein repeats in various proteomes this database will be increasingly important in proteome comparison.