Dense Percolation in Large-Scale Mean-Field Random Networks Is Provably “Explosive”

Recent reports suggest that evolving large-scale networks exhibit “explosive percolation”: a large fraction of nodes suddenly becomes connected when sufficiently many links have formed in a network. This phase transition has been shown to be continuous (second-order) for most random network formation processes, including classical mean-field random networks and their modifications. We study a related yet different phenomenon referred to as dense percolation, which occurs when a network is already connected, but a large group of nodes must be dense enough, i.e., have at least a certain minimum required percentage of possible links, to form a “highly connected” cluster. Such clusters have been considered in various contexts, including the recently introduced network modularity principle in biological networks. We prove that, contrary to the traditionally defined percolation transition, dense percolation transition is discontinuous (first-order) under the classical mean-field network formation process (with no modifications); therefore, there is not only quantitative, but also qualitative difference between regular and dense percolation transitions. Moreover, the size of the largest dense (highly connected) cluster in a mean-field random network is explicitly characterized by rigorously proven tight asymptotic bounds, which turn out to naturally extend the previously derived formula for the size of the largest clique (a cluster with all possible links) in such a network. We also briefly discuss possible implications of the obtained mathematical results on studying first-order phase transitions in real-world linked systems.     

A Network Approach to Analyzing Highly Recombinant Malaria Parasite Genes

The var genes of the human malaria parasite Plasmodium falciparum present a challenge to population geneticists due to their extreme diversity, which is generated by high rates of recombination. These genes encode a primary antigen protein called PfEMP1, which is expressed on the surface of infected red blood cells and elicits protective immune responses. Var gene sequences are characterized by pronounced mosaicism, precluding the use of traditional phylogenetic tools that require bifurcating tree-like evolutionary relationships. We present a new method that identifies highly variable regions (HVRs), and then maps each HVR to a complex network in which each sequence is a node and two nodes are linked if they share an exact match of significant length. Here, networks of var genes that recombine freely are expected to have a uniformly random structure, but constraints on recombination will produce network communities that we identify using a stochastic block model. We validate this method on synthetic data, showing that it correctly recovers populations of constrained recombination, before applying it to the Duffy Binding Like-α (DBLα) domain of var genes. We find nine HVRs whose network communities map in distinctive ways to known DBLα classifications and clinical phenotypes. We show that the recombinational constraints of some HVRs are correlated, while others are independent. These findings suggest that this micromodular structuring facilitates independent evolutionary trajectories of neighboring mosaic regions, allowing the parasite to retain protein function while generating enormous sequence diversity. Our approach therefore offers a rigorous method for analyzing evolutionary constraints in var genes, and is also flexible enough to be easily applied more generally to any highly recombinant sequences.     

Intrinsic properties of Boolean dynamics in complex networks

We study intrinsic properties of attractor in Boolean dynamics of complex networks with scale-free topology, comparing with those of the so-called Kauffman's random Boolean networks. We numerically study both frozen and relevant nodes in each attractor in the dynamics of relatively small networks (20?N?200). We investigate numerically robustness of an attractor to a perturbation. An attractor with cycle length of ?_c in a network of size N consists of ?_c states in the state space of 2^N states; each attractor has the arrangement of N nodes, where the cycle of attractor sweeps ?_c states. We define a perturbation as a flip of the state on a single node in the attractor state at a given time step. We show that the rate between unfrozen and relevant nodes in the dynamics of a complex network with scale-free topology is larger than that in Kauffman's random Boolean network model. Furthermore, we find that in a complex scale-free network with fluctuation of the in-degree number, attractors are more sensitive to a state flip for a highly connected node (i.e. input-hub node) than to that for a less connected node. By some numerical examples, we show that the number of relevant nodes increases, when an input-hub node is coincident with and/or connected with an output-hub node (i.e. a node with large output-degree) one another.     

Emergence and fragmentation of the alpha-band driven by neuronal network dynamics

Rhythmic neuronal network activity underlies brain oscillations. To investigate how connected neuronal networks contribute to the emergence of the α-band and to the regulation of Up and Down states, we study a model based on synaptic short-term depression-facilitation with afterhyperpolarization (AHP). We found that the α-band is generated by the network behavior near the attractor of the Up-state. Coupling inhibitory and excitatory networks by reciprocal connections leads to the emergence of a stable α-band during the Up states, as reflected in the spectrogram. To better characterize the emergence and stability of thalamocortical oscillations containing α and δ rhythms during anesthesia, we model the interaction of two excitatory networks with one inhibitory network, showing that this minimal topology underlies the generation of a persistent α-band in the neuronal voltage characterized by dominant Up over Down states. Finally, we show that the emergence of the α-band appears when external inputs are suppressed, while fragmentation occurs at small synaptic noise or with increasing inhibitory inputs. To conclude, α-oscillations could result from the synaptic dynamics of interacting excitatory neuronal networks with and without AHP, a principle that could apply to other rhythms.     

Vaccination with Leishmania infantum Acidic Ribosomal P0 but Not with Nucleosomal Histones Proteins Controls Leishmania infantum Infection in Hamsters

Background

Several intracellular Leishmania antigens have been identified in order to find a potential vaccine capable of conferring long lasting protection against Leishmania infection. Histones and Acid Ribosomal proteins are already known to induce an effective immune response and have successfully been tested in the cutaneous leishmaniasis mouse model. Here, we investigate the protective ability of L. infantum nucleosomal histones (HIS) and ribosomal acidic protein P0 (LiP0) against L. infantum infection in the hamster model of visceral leishmaniasis using two different strategies: homologous (plasmid DNA only) or heterologous immunization (plasmid DNA plus recombinant protein and adjuvant).

Methodology/Principal Findings

Immunization with both antigens using the heterologous strategy presented a high antibody production level while the homologous strategy immunized group showed predominantly a cellular immune response with parasite load reduction. The pcDNA-LiP0 immunized group showed increased expression ratio of IFN-γ/IL-10 and IFN-γ/TGF-β in the lymph nodes before challenge. Two months after infection hamsters immunized with the empty plasmid presented a pro-inflammatory immune response in the early stages of infection with increased expression ratio of IFN-γ/IL-10 and IFN-γ/TGF-β, whereas hamsters immunized with pcDNA-HIS presented an increase only in the ratio IFN-γ/ TGF-β. On the other hand, hamsters immunized with LiP0 did not present any increase in the IFN-γ/TGF-β and IFN-γ/IL-10 ratio independently of the immunization strategy used. Conversely, five months after infection, hamsters immunized with HIS maintained a pro-inflammatory immune response (ratio IFN-γ/ IL-10) while pcDNA-LiP0 immunized hamsters continued showing a balanced cytokine profile of pro and anti-inflammatory cytokines. Moreover we observed a significant reduction in parasite load in the spleen, liver and lymph node in this group compared with controls.

Conclusions/Significance

Our results suggest that vaccination with L. infantum LiP0 antigen administered in a DNA formulation could be considered a potential component in a vaccine formulation against visceral leishmaniasis.     

Spectrin Tetramer Formation Is Not Required for Viable Development in Drosophila

The dominant paradigm for spectrin function is that (αβ)₂-spectrin tetramers or higher order oligomers form membrane-associated two-dimensional networks in association with F-actin to reinforce the plasma membrane. Tetramerization is an essential event in such structures. We characterize the tetramerization interaction between α-spectrin and β-spectrins in Drosophila. Wild-type α-spectrin binds to both β- and β_H-chains with high affinity, resembling other non-erythroid spectrins. However, α-spec^R22S, a tetramerization site mutant homologous to the pathological α-spec^R28S allele in humans, eliminates detectable binding to β-spectrin and reduces binding to β_H-spectrin ∼1000-fold. Even though spectrins are essential proteins, α-spectrin^R22S rescues α-spectrin mutants to adulthood with only minor phenotypes indicating that tetramerization, and thus conventional network formation, is not the essential function of non-erythroid spectrin. Our data provide the first rigorous test for the general requirement for tetramer-based non-erythroid spectrin networks throughout an organism and find that they have very limited roles, in direct contrast to the current paradigm.     

Classification of Beta-Lactamases and Penicillin Binding Proteins Using Ligand-Centric Network Models

β-lactamase mediated antibiotic resistance is an important health issue and the discovery of new β-lactam type antibiotics or β-lactamase inhibitors is an area of intense research. Today, there are about a thousand β-lactamases due to the evolutionary pressure exerted by these ligands. While β-lactamases hydrolyse the β-lactam ring of antibiotics, rendering them ineffective, Penicillin-Binding Proteins (PBPs), which share high structural similarity with β-lactamases, also confer antibiotic resistance to their host organism by acquiring mutations that allow them to continue their participation in cell wall biosynthesis. In this paper, we propose a novel approach to include ligand sharing information for classifying and clustering β-lactamases and PBPs in an effort to elucidate the ligand induced evolution of these β-lactam binding proteins. We first present a detailed summary of the β-lactamase and PBP families in the Protein Data Bank, as well as the compounds they bind to. Then, we build two different types of networks in which the proteins are represented as nodes, and two proteins are connected by an edge with a weight that depends on the number of shared identical or similar ligands. These models are analyzed under three different edge weight settings, namely unweighted, weighted, and normalized weighted. A detailed comparison of these six networks showed that the use of ligand sharing information to cluster proteins resulted in modules comprising proteins with not only sequence similarity but also functional similarity. Consideration of ligand similarity highlighted some interactions that were not detected in the identical ligand network. Analysing the β-lactamases and PBPs using ligand-centric network models enabled the identification of novel relationships, suggesting that these models can be used to examine other protein families to obtain information on their ligand induced evolutionary paths.     

An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in σ_e², the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the θ-logistic model, where density dependence of population growth is a function of N^θ, long-term evolution maximizes E[N^θ]=[1−σ_e²/(2r)]K^θ. While σ_e² is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, θ has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.     

Structural Bridges through Fold Space

Several protein structure classification schemes exist that partition the protein universe into structural units called folds. Yet these schemes do not discuss how these units sit relative to each other in a global structure space. In this paper we construct networks that describe such global relationships between folds in the form of structural bridges. We generate these networks using four different structural alignment methods across multiple score thresholds. The networks constructed using the different methods remain a similar distance apart regardless of the probability threshold defining a structural bridge. This suggests that at least some structural bridges are method specific and that any attempt to build a picture of structural space should not be reliant on a single structural superposition method. Despite these differences all representations agree on an organisation of fold space into five principal community structures: all-α, all-β sandwiches, all-β barrels, α/β and α + β. We project estimated fold ages onto the networks and find that not only are the pairings of unconnected folds associated with higher age differences than bridged folds, but this difference increases with the number of networks displaying an edge. We also examine different centrality measures for folds within the networks and how these relate to fold age. While these measures interpret the central core of fold space in varied ways they all identify the disposition of ancestral folds to fall within this core and that of the more recently evolved structures to provide the peripheral landscape. These findings suggest that evolutionary information is encoded along these structural bridges. Finally, we identify four highly central pivotal folds representing dominant topological features which act as key attractors within our landscapes.     

Non-Homogeneous Fractal Hierarchical Weighted Networks

A model of fractal hierarchical structures that share the property of non-homogeneous weighted networks is introduced. These networks can be completely and analytically characterized in terms of the involved parameters, i.e., the size of the original graph N_k and the non-homogeneous weight scaling factors r ₁, r ₂, · · · r_M. We also study the average weighted shortest path (AWSP), the average degree and the average node strength, taking place on the non-homogeneous hierarchical weighted networks. Moreover the AWSP is scrupulously calculated. We show that the AWSP depends on the number of copies and the sum of all non-homogeneous weight scaling factors in the infinite network order limit.     

Matching–centrality decomposition and the forecasting of new links in networks

Networks play a prominent role in the study of complex systems of interacting entities in biology, sociology, and economics. Despite this diversity, we demonstrate here that a statistical model decomposing networks into matching and centrality components provides a comprehensive and unifying quantification of their architecture. The matching term quantifies the assortative structure in which node makes links with which other node, whereas the centrality term quantifies the number of links that nodes make. We show, for a diverse set of networks, that this decomposition can provide a tight fit to observed networks. Then we provide three applications. First, we show that the model allows very accurate prediction of missing links in partially known networks. Second, when node characteristics are known, we show how the matching–centrality decomposition can be related to this external information. Consequently, it offers us a simple and versatile tool to explore how node characteristics explain network architecture. Finally, we demonstrate the efficiency and flexibility of the model to forecast the links that a novel node would create if it were to join an existing network.     

A Family of Algorithms for Computing Consensus about Node State from Network Data

Biological and social networks are composed of heterogeneous nodes that contribute differentially to network structure and function. A number of algorithms have been developed to measure this variation. These algorithms have proven useful for applications that require assigning scores to individual nodes–from ranking websites to determining critical species in ecosystems–yet the mechanistic basis for why they produce good rankings remains poorly understood. We show that a unifying property of these algorithms is that they quantify consensus in the network about a node''s state or capacity to perform a function. The algorithms capture consensus by either taking into account the number of a target node''s direct connections, and, when the edges are weighted, the uniformity of its weighted in-degree distribution (breadth), or by measuring net flow into a target node (depth). Using data from communication, social, and biological networks we find that that how an algorithm measures consensus–through breadth or depth– impacts its ability to correctly score nodes. We also observe variation in sensitivity to source biases in interaction/adjacency matrices: errors arising from systematic error at the node level or direct manipulation of network connectivity by nodes. Our results indicate that the breadth algorithms, which are derived from information theory, correctly score nodes (assessed using independent data) and are robust to errors. However, in cases where nodes “form opinions” about other nodes using indirect information, like reputation, depth algorithms, like Eigenvector Centrality, are required. One caveat is that Eigenvector Centrality is not robust to error unless the network is transitive or assortative. In these cases the network structure allows the depth algorithms to effectively capture breadth as well as depth. Finally, we discuss the algorithms'' cognitive and computational demands. This is an important consideration in systems in which individuals use the collective opinions of others to make decisions.     

A generalized Flory-Stockmayer kinetic theory of connectivity percolation and rigidity percolation of cytoskeletal networks

Actin networks are essential for living cells to move, reproduce, and sense their environments. The dynamic and rheological behavior of actin networks is modulated by actin-binding proteins such as α-actinin, Arp2/3, and myosin. There is experimental evidence that actin-binding proteins modulate the cooperation of myosin motors by connecting the actin network. In this work, we present an analytical mean field model, using the Flory-Stockmayer theory of gelation, to understand how different actin-binding proteins change the connectivity of the actin filaments as the networks are formed. We follow the kinetics of the networks and estimate the concentrations of actin-binding proteins that are needed to reach connectivity percolation as well as to reach rigidity percolation. We find that Arp2/3 increases the actomyosin connectivity in the network in a non-monotonic way. We also describe how changing the connectivity of actomyosin networks modulates the ability of motors to exert forces, leading to three possible phases of the networks with distinctive dynamical characteristics: a sol phase, a gel phase, and an active phase. Thus, changes in the concentration and activity of actin-binding proteins in cells lead to a phase transition of the actin network, allowing the cells to perform active contraction and change their rheological properties.     

Mechanism of Auxiliary Subunit Modulation of Neuronal α1E Calcium Channels

Voltage-gated calcium channels are composed of a main pore-forming α₁ moiety, and one or more auxiliary subunits (β, α₂δ) that modulate channel properties. Because modulatory properties may vary greatly with different channels, expression systems, and protocols, it is advantageous to study subunit regulation with a uniform experimental strategy. Here, in HEK 293 cells, we examine the expression and activation gating of α_1E calcium channels in combination with a β (β₁–β₄) and/or the α₂δ subunit, exploiting both ionic- and gating-current measurements. Furthermore, to explore whether more than one auxiliary subunit can concomitantly specify gating properties, we investigate the effects of cotransfecting α₂δ with β subunits, of transfecting two different β subunits simultaneously, and of COOH-terminal truncation of α_1E to remove a second β binding site. The main results are as follows. (a) The α₂δ and β subunits modulate α_1E in fundamentally different ways. The sole effect of α₂δ is to increase current density by elevating channel density. By contrast, though β subunits also increase functional channel number, they also enhance maximum open probability (G_max/Q_max) and hyperpolarize the voltage dependence of ionic-current activation and gating-charge movement, all without discernible effect on activation kinetics. Different β isoforms produce nearly indistinguishable effects on activation. However, β subunits produced clear, isoform-specific effects on inactivation properties. (b) All the β subunit effects can be explained by a gating model in which subunits act only on weakly voltage-dependent steps near the open state. (c) We find no clear evidence for simultaneous modulation by two different β subunits. (d) The modulatory features found here for α_1E do not generalize uniformly to other α₁ channel types, as α_1C activation gating shows marked β isoform dependence that is absent for α_1E. Together, these results help to establish a more comprehensive picture of auxiliary-subunit regulation of α_1E calcium channels.     

Evolution Characteristics of the Network Core in the Facebook

Statistical properties of the static networks have been extensively studied. However, online social networks are evolving dynamically, understanding the evolving characteristics of the core is one of major concerns in online social networks. In this paper, we empirically investigate the evolving characteristics of the Facebook core. Firstly, we separate the Facebook-link(FL) and Facebook-wall(FW) datasets into 28 snapshots in terms of timestamps. By employing the k-core decomposition method to identify the core of each snapshot, we find that the core sizes of the FL and FW networks approximately contain about 672 and 373 nodes regardless of the exponential growth of the network sizes. Secondly, we analyze evolving topological properties of the core, including the k-core value, assortative coefficient, clustering coefficient and the average shortest path length. Empirical results show that nodes in the core are getting more interconnected in the evolving process. Thirdly, we investigate the life span of nodes belonging to the core. More than 50% nodes stay in the core for more than one year, and 19% nodes always stay in the core from the first snapshot. Finally, we analyze the connections between the core and the whole network, and find that nodes belonging to the core prefer to connect nodes with high k-core values, rather than the high degrees ones. This work could provide new insights into the online social network analysis.