Is the intrinsic disorder of proteins the cause of the scale-free architecture of protein-protein interaction networks?

In protein-protein interaction (PPI) networks certain topological properties appear to be recurrent: network maps are considered scale-free. It is possible that this topology is reflected in the protein structure. In this paper, we investigate the role of protein disorder in the network topology. We find that the disorder of a protein (or of its neighbors) is independent of its number of PPIs. This result suggests that protein disorder does not play a role in the scale-free architecture of protein networks.     

Protein interaction networks of Saccharomyces cerevisiae, Caenorhabditis elegans and Drosophila melanogaster: large-scale organization and robustness

High-throughput screens have begun to reveal protein interaction networks in several organisms. To understand the general properties of these protein interaction networks, a systematic analysis of topological structure and robustness was performed on the protein interaction networks of Saccharomyces cerevisiae, Caenorhabditis elegans and Drosophila melanogaster. It shows that the three protein interaction networks have a scale-free and high-degree clustering nature as the consequence of their hierarchical organization. It also shows that they have the small-world property with similar diameter at 4-5. Evaluation of the consequences of random removal of both proteins and interactions from the protein interaction networks suggests their high degree of robustness. Simulation of a protein's removal shows that the protein interaction network's error tolerance is accompanied by attack vulnerability. These fundamental analyses of the networks might serve as a starting point for further exploring complex biological networks and the coming research of "systems biology".     

The emergence of scaling in sequence-based physical models of protein evolution

It has recently been discovered that many biological systems, when represented as graphs, exhibit a scale-free topology. One such system is the set of structural relationships among protein domains. The scale-free nature of this and other systems has previously been explained using network growth models that, although motivated by biological processes, do not explicitly consider the underlying physics or biology. In this work we explore a sequence-based model for the evolution protein structures and demonstrate that this model is able to recapitulate the scale-free nature observed in graphs of real protein structures. We find that this model also reproduces other statistical feature of the protein domain graph. This represents, to our knowledge, the first such microscopic, physics-based evolutionary model for a scale-free network of biological importance and as such has strong implications for our understanding of the evolution of protein structures and of other biological networks.     

Emergence of scale-free distribution in protein–protein interaction networks based on random selection of interacting domain pairs

Recent analyses of biological and artificial networks have revealed a common network architecture, called scale-free topology. The origin of the scale-free topology has been explained by using growth and preferential attachment mechanisms. In a cell, proteins are the most important carriers of function, and are composed of domains as elemental units responsible for the physical interaction between protein pairs. Here, we propose a model for protein–protein interaction networks that reveals the emergence of two possible topologies. We show that depending on the number of randomly selected interacting domain pairs, the connectivity distribution follows either a scale-free distribution, even in the absence of the preferential attachment, or a normal distribution. This new approach only requires an evolutionary model of proteins (nodes) but not for the interactions (edges). The edges are added by means of random interaction of domain pairs. As a result, this model offers a new mechanistic explanation for understanding complex networks with a direct biological interpretation because only protein structures and their functions evolved through genetic modifications of amino acid sequences. These findings are supported by numerical simulations as well as experimental data.     

Some protein interaction data do not exhibit power law statistics

It has been claimed that protein-protein interaction (PPI) networks are scale-free, and that identifying high-degree "hub" proteins reveals important features of PPI networks. In this paper, we evaluate the claims that PPI node degree sequences follow a power law, a necessary condition for networks to be scale-free. We provide two PPI network examples which clearly do not have power laws when analyzed correctly, and thus at least these PPI networks are not scale-free. We also show that these PPI networks do appear to have power laws according to methods that have become standard in the existing literature. We explain the source of this error using numerically generated data from analytic formulas, where there are no sampling or noise ambiguities.     

Disease spread in small-size directed networks: Epidemic threshold, correlation between links to and from nodes, and clustering

Network epidemiology has mainly focused on large-scale complex networks. It is unclear whether findings of these investigations also apply to networks of small size. This knowledge gap is of relevance for many biological applications, including meta-communities, plant–pollinator interactions and the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. Moreover, many small-size biological networks are inherently asymmetrical and thus cannot be realistically modelled with undirected networks. We modelled disease spread and establishment in directed networks of 100 and 500 nodes at four levels of connectance in six network structures (local, small-world, random, one-way, uncorrelated, and two-way scale-free networks). The model was based on the probability of infection persistence in a node and of infection transmission between connected nodes. Regardless of the size of the network, the epidemic threshold did not depend on the starting node of infection but was negatively related to the correlation coefficient between in- and out-degree for all structures, unless networks were sparsely connected. In this case clustering played a significant role. For small-size scale-free directed networks to have a lower epidemic threshold than other network structures, there needs to be a positive correlation between number of links to and from nodes. When this correlation is negative (one-way scale-free networks), the epidemic threshold for small-size networks can be higher than in non-scale-free networks. Clustering does not necessarily have an influence on the epidemic threshold if connectance is kept constant. Analyses of the influence of the clustering on the epidemic threshold in directed networks can also be spurious if they do not consider simultaneously the effect of the correlation coefficient between in- and out-degree.     

Modularity in the evolution of yeast protein interaction network

Protein interaction networks are known to exhibit remarkable structures: scale-free and small-world and modular structures. To explain the evolutionary processes of protein interaction networks possessing scale-free and small-world structures, preferential attachment and duplication-divergence models have been proposed as mathematical models. Protein interaction networks are also known to exhibit another remarkable structural characteristic, modular structure. How the protein interaction networks became to exhibit modularity in their evolution? Here, we propose a hypothesis of modularity in the evolution of yeast protein interaction network based on molecular evolutionary evidence. We assigned yeast proteins into six evolutionary ages by constructing a phylogenetic profile. We found that all the almost half of hub proteins are evolutionarily new. Examining the evolutionary processes of protein complexes, functional modules and topological modules, we also found that member proteins of these modules tend to appear in one or two evolutionary ages. Moreover, proteins in protein complexes and topological modules show significantly low evolutionary rates than those not in these modules. Our results suggest a hypothesis of modularity in the evolution of yeast protein interaction network as systems evolution.     

Uncovering network systems within protein structures

Traditionally, proteins have been viewed as a construct based on elements of secondary structure and their arrangement in three-dimensional space. In a departure from this perspective we show that protein structures can be modelled as network systems that exhibit small-world, single-scale, and to some degree, scale-free properties. The phenomenological network concept of degrees of separation is applied to three-dimensional protein structure networks and reveals how amino acid residues can be connected to each other within six degrees of separation. This work also illuminates the unique features of protein networks in comparison to other networks currently studied. Recognising that proteins are networks provides a means of rationalising the robustness in the overall three-dimensional fold of a protein against random mutations and suggests an alternative avenue to investigate the determinants of protein structure, function and folding.     

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.     

Asymptotic states and topological structure of an activation-deactivation chemical network

The influence of the topology on the asymptotic states of a network of interacting chemical species has been studied by simulating its time evolution. Random and scale-free networks have been designed to support relevant features of activation-deactivation reactions networks (mapping signal transduction networks) and the system of ordinary differential equations associated to the dynamics has been numerically solved. We analysed stationary states of the dynamics as a function of the network's connectivity and of the distribution of the chemical species on the network; we found important differences between the two topologies in the regime of low connectivity. In particular, only for low connected scale-free networks it is possible to find zero activity patterns as stationary states of the dynamics which work as signal off-states. Asymptotic features of random and scale-free networks become similar as the connectivity increases.     

The role of log-normal dynamics in the evolution of biochemical pathways

The study of the scale-free topology in non-biological and biological networks and the dynamics that can explain this fascinating property of complex systems have captured the attention of the scientific community in the last years. Here, we analyze the biochemical pathways of three organisms (Methanococcus jannaschii, Escherichia coli, Saccharomyces cerevisiae) which are representatives of the main kingdoms Archaea, Bacteria and Eukaryotes during the course of the biological evolution. We can consider two complementary representations of the biochemical pathways: the enzymes network and the chemical compounds network. In this article, we propose a stochastic model that explains that the scale-free topology with exponent in the vicinity of gamma approximately 3/2 found across these three organisms is governed by the log-normal dynamics in the evolution of the enzymes network. Precisely, the fluctuations of the connectivity degree of enzymes in the biochemical pathways between evolutionary distant organisms follow the same conserved dynamical principle, which in the end is the origin of the stationary scale-free distribution observed among species, from Archaea to Eukaryotes. In particular, the log-normal dynamics guarantees the conservation of the scale-free distribution in evolving networks. Furthermore, the log-normal dynamics also gives a possible explanation for the restricted range of observed exponents gamma in the scale-free networks (i.e., gamma > or = 3/2). Finally, our model is also applied to the chemical compounds network of biochemical pathways and the Internet network.     

A scale-free structure prior for graphical models with applications in functional genomics

The problem of reconstructing large-scale, gene regulatory networks from gene expression data has garnered considerable attention in bioinformatics over the past decade with the graphical modeling paradigm having emerged as a popular framework for inference. Analysis in a full Bayesian setting is contingent upon the assignment of a so-called structure prior-a probability distribution on networks, encoding a priori biological knowledge either in the form of supplemental data or high-level topological features. A key topological consideration is that a wide range of cellular networks are approximately scale-free, meaning that the fraction, , of nodes in a network with degree is roughly described by a power-law with exponent between and . The standard practice, however, is to utilize a random structure prior, which favors networks with binomially distributed degree distributions. In this paper, we introduce a scale-free structure prior for graphical models based on the formula for the probability of a network under a simple scale-free network model. Unlike the random structure prior, its scale-free counterpart requires a node labeling as a parameter. In order to use this prior for large-scale network inference, we design a novel Metropolis-Hastings sampler for graphical models that includes a node labeling as a state space variable. In a simulation study, we demonstrate that the scale-free structure prior outperforms the random structure prior at recovering scale-free networks while at the same time retains the ability to recover random networks. We then estimate a gene association network from gene expression data taken from a breast cancer tumor study, showing that scale-free structure prior recovers hubs, including the previously unknown hub SLC39A6, which is a zinc transporter that has been implicated with the spread of breast cancer to the lymph nodes. Our analysis of the breast cancer expression data underscores the value of the scale-free structure prior as an instrument to aid in the identification of candidate hub genes with the potential to direct the hypotheses of molecular biologists, and thus drive future experiments.     

Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks

A growing number of studies are investigating the effect of contact structure on the dynamics of epidemics in large-scale complex networks. Whether findings thus obtained apply also to networks of small size, and thus to many real-world biological applications, is still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of network structure (local, small-world, random and scale-free), there is a linear threshold determined by the probability of infection transmission between connected nodes and the probability of infection persistence in an infected node. The threshold is significantly lower for scale-free networks compared to local, random and small-world ones only if super-connected nodes have a higher number of links both to and from other nodes. The starting point, the node at which the epidemic starts, does not affect the threshold conditions, but has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation.     

Scale‐free networks in biology: new insights into the fundamentals of evolution?*

Scale-free network models describe many natural and social phenomena. In particular, networks of interacting components of a living cell were shown to possess scale-free properties. A recent study((1)) compares the system-level properties of metabolic and information networks in 43 archaeal, bacterial and eukaryal species and claims that the scale-free organization of these networks is more conserved during evolution than their content.     

How scale-free are biological networks.

The concept of scale-free network has emerged as a powerful unifying paradigm in the study of complex systems in biology and in physical and social studies. Metabolic, protein, and gene interaction networks have been reported to exhibit scale-free behavior based on the analysis of the distribution of the number of connections of the network nodes. Here we study 10 published datasets of various biological interactions and perform goodness-of-fit tests to determine whether the given data is drawn from the power-law distribution. Our analysis did not identify a single interaction network that has a nonzero probability of being drawn from the power-law distribution.     

Are scale-free networks robust to measurement errors?

Background  

Many complex random networks have been found to be scale-free. Existing literature on scale-free networks has rarely considered potential false positive and false negative links in the observed networks, especially in biological networks inferred from high-throughput experiments. Therefore, it is important to study the impact of these measurement errors on the topology of the observed networks.     

A thermodynamic view of networks

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.     

Similarity networks of protein binding sites

An increasing attention has been dedicated to the characterization of complex networks within the protein world. This work is reporting how we uncovered networked structures that reflected the structural similarities among protein binding sites. First, a 211 binding sites dataset has been compiled by removing the redundant proteins in the Protein Ligand Database (PLD) (http://www-mitchell.ch.cam.ac.uk/pld/). Using a clique detection algorithm we have performed all-against-all binding site comparisons among the 211 available ones. Within the set of nodes representing each binding site an edge was added whenever a pair of binding sites had a similarity higher than a threshold value. The generated similarity networks revealed that many nodes had few links and only few were highly connected, but due to the limited data available it was not possible to definitively prove a scale-free architecture. Within the same dataset, the binding site similarity networks were compared with the networks of sequence and fold similarity networks. In the protein world, indications were found that structure is better conserved than sequence, but on its own, sequence was better conserved than the subset of functional residues forming the binding site. Because a binding site is strongly linked with protein function, the identification of protein binding site similarity networks could accelerate the functional annotation of newly identified genes. In view of this we have discussed several potential applications of binding site similarity networks, such as the construction of novel binding site classification databases, as well as the implications for protein molecular design in general and computational chemogenomics in particular.