Recent Progress on the Analysis of Power-law Features in Complex Cellular Networks

Complex interactions between different kinds of bio-molecules and essential nutrients are responsible for cellular functions. Rapid advances in theoretical modeling and experimental analyses have shown that drastically different biological and non-biological networks share a common architecture. That is, the probability that a selected node in the network has exactly k edges decays as a power-law. This finding has definitely opened an intense research and debate on the origin and implications of this ubiquitous pattern. In this review, we describe the recent progress on the emergence of power-law distributions in cellular networks. We first show the internal characteristics of the observed complex networks uncovered using graph theory. We then briefly review some works that have significantly contributed to the theoretical analysis of cellular networks and systems, from metabolic and protein networks to gene expression profiles. This prevalent topology observed in so many diverse biological systems suggests the existence of generic laws and organizing principles behind the cellular networks.     

The human extended mitochondrial metabolic network: new hubs from lipids

Even if systems thinking is not new in biology, rationalizing the explosively growing amount of knowledge has been the compelling reason for the sudden rise and spreading of systems biology. Based on 'omics' data, several genome-scale metabolic networks have been reconstructed and validated. One of the most striking aspects of complex metabolic networks is the pervasive power-law appearance of metabolite connectivity. However, the combinatorial diversity of some classes of compounds, such as lipids, has been scarcely considered so far. In this work, a lipid-extended human mitochondrial metabolic network has been built and analyzed. It is shown that, considering combinatorial diversity of lipids and multipurpose enzymes, an intimate connection between membrane lipids and oxidative phosphorilation appears. This finding leads to some biomedical considerations on diseases involving mitochondrial enzymes. Moreover, the lipid-extended network still shows power-law features. Power-law distributions are intrinsic to metabolic network organization and evolution. Hubs in the lipid-extended mitochondrial network strongly suggest that the "RNA world" and the "lipid world" hypothesis are both correct.     

Biology,Methodology or Chance? The Degree Distributions of Bipartite Ecological Networks

The distribution of the number of links per species, or degree distribution, is widely used as a summary of the topology of complex networks. Degree distributions have been studied in a range of ecological networks, including both mutualistic bipartite networks of plants and pollinators or seed dispersers and antagonistic bipartite networks of plants and their consumers. The shape of a degree distribution, for example whether it follows an exponential or power-law form, is typically taken to be indicative of the processes structuring the network. The skewed degree distributions of bipartite mutualistic and antagonistic networks are usually assumed to show that ecological or co-evolutionary processes constrain the relative numbers of specialists and generalists in the network. I show that a simple null model based on the principle of maximum entropy cannot be rejected as a model for the degree distributions in most of the 115 bipartite ecological networks tested here. The model requires knowledge of the number of nodes and links in the network, but needs no other ecological information. The model cannot be rejected for 159 (69%) of the 230 degree distributions of the 115 networks tested. It performed equally well on the plant and animal degree distributions, and cannot be rejected for 81 (70%) of the 115 plant distributions and 78 (68%) of the animal distributions. There are consistent differences between the degree distributions of mutualistic and antagonistic networks, suggesting that different processes are constraining these two classes of networks. Fit to the MaxEnt null model is consistently poor among the largest mutualistic networks. Potential ecological and methodological explanations for deviations from the model suggest that spatial and temporal heterogeneity are important drivers of the structure of these large networks.     

Scaling and structure of dicotyledonous leaf venation networks

There have been numerous attempts to derive general models for the structure and function of resource delivery networks in biology. Such theories typically predict the quantitative structure of vascular networks across scales. For example, fractal branching models of plant structure predict that the network dimensions within plant stems or leaves should be scale-free. However, very few empirical examples of such networks are available with which to evaluate such hypotheses. Here, we apply recently developed leaf network extraction software to a global leaf dataset. We find that leaf networks are neither entirely scale-free nor governed entirely by a characteristic scale. Indeed, we find many network properties, such as vein length distributions, which are governed by characteristic scales, and other network properties, notably vein diameter distributions, which are typified by power-law behaviour. Our findings suggest that theories of network structure will remain incomplete until they address the multiple constraints on network architecture.     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Duplication models for biological networks.

Are biological networks different from other large complex networks? Both large biological and nonbiological networks exhibit power-law graphs (number of nodes with degree k, N(k) approximately k(-beta)), yet the exponents, beta, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks) and is fundamentally different from the mechanisms thought to dominate the growth of most nonbiological networks (such as the Internet). The preferential choice models used for nonbiological networks like web graphs can only produce power-law graphs with exponents greater than 2. We use combinatorial probabilistic methods to examine the evolution of graphs by node duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.     

A versatile graphene-based fluorescence "on/off" switch for multiplex detection of various targets

We have designed a versatile molecular beacon (MB)-like probe for the multiplex sensing of targets such as sequence-specific DNA, protein, metal ions and small molecule compounds based on the self-assembled ssDNA-graphene oxide (ssDNA-GO) architecture. The probe employs fluorescence "on/off" switching strategy in a single step in homogeneous solution. Compared to traditional molecular beacons, the proposed design is simple to prepare and manipulate and has little background interference, but still gives superior sensitivity and rapid response. More importantly, this ssDNA-GO architecture can serve as a universal beacon platform by simply changing the types of ssDNA sequences for the different targets. In this work, the ssDNA-GO architecture probe has been successfully applied in the multiplex detection of sequence-specific DNA, thrombin, Ag(+), Hg(2+) and cysteine, and the limit of detection was 1 nM, 5 nM, 20 nM, 5.7 nM and 60 nM, respectively. The results demonstrate that the ssDNA-GO architecture can be an excellent and versatile platform for sensing multiplex analytes, easily replacing the universal molecular beacon.     

Random Phenotypic Variation of Yeast (Saccharomyces cerevisiae) Single-Gene Knockouts Fits a Double Pareto-Lognormal Distribution

Background

Distributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.

Objective and Methods

If our assumptions are true, the DPLN distribution should provide a better fit to random phenotypic variation in a large series of single-gene knockout lines than other skewed or symmetrical distributions. We fit a large published data set of single-gene knockout lines in Saccharomyces cerevisiae to seven different probability distributions: DPLN, right Pareto-lognormal (RPLN), left Pareto-lognormal (LPLN), normal, lognormal, exponential, and Pareto. The best model was judged by the Akaike Information Criterion (AIC).

Results

Phenotypic variation among gene knockouts in S. cerevisiae fits a double Pareto-lognormal (DPLN) distribution better than any of the alternative distributions, including the right Pareto-lognormal and lognormal distributions.

Conclusions and Significance

A DPLN distribution is consistent with the hypothesis that developmental stability is mediated, in part, by distributed robustness, the resilience of gene regulatory, metabolic, and protein-protein interaction networks. Alternatively, multiplicative cell growth, and the mixing of lognormal distributions having different variances, may generate a DPLN distribution.     

The birth-death-mutation process: a new paradigm for fat tailed distributions

Fat tailed statistics and power-laws are ubiquitous in many complex systems. Usually the appearance of of a few anomalously successful individuals (bio-species, investors, websites) is interpreted as reflecting some inherent "quality" (fitness, talent, giftedness) as in Darwin's theory of natural selection. Here we adopt the opposite, "neutral", outlook, suggesting that the main factor explaining success is merely luck. The statistics emerging from the neutral birth-death-mutation (BDM) process is shown to fit marvelously many empirical distributions. While previous neutral theories have focused on the power-law tail, our theory economically and accurately explains the entire distribution. We thus suggest the BDM distribution as a standard neutral model: effects of fitness and selection are to be identified by substantial deviations from it.     

Emergence of Scale-Free Close-Knit Friendship Structure in Online Social Networks

Although the structural properties of online social networks have attracted much attention, the properties of the close-knit friendship structures remain an important question. Here, we mainly focus on how these mesoscale structures are affected by the local and global structural properties. Analyzing the data of four large-scale online social networks reveals several common structural properties. It is found that not only the local structures given by the indegree, outdegree, and reciprocal degree distributions follow a similar scaling behavior, the mesoscale structures represented by the distributions of close-knit friendship structures also exhibit a similar scaling law. The degree correlation is very weak over a wide range of the degrees. We propose a simple directed network model that captures the observed properties. The model incorporates two mechanisms: reciprocation and preferential attachment. Through rate equation analysis of our model, the local-scale and mesoscale structural properties are derived. In the local-scale, the same scaling behavior of indegree and outdegree distributions stems from indegree and outdegree of nodes both growing as the same function of the introduction time, and the reciprocal degree distribution also shows the same power-law due to the linear relationship between the reciprocal degree and in/outdegree of nodes. In the mesoscale, the distributions of four closed triples representing close-knit friendship structures are found to exhibit identical power-laws, a behavior attributed to the negligible degree correlations. Intriguingly, all the power-law exponents of the distributions in the local-scale and mesoscale depend only on one global parameter, the mean in/outdegree, while both the mean in/outdegree and the reciprocity together determine the ratio of the reciprocal degree of a node to its in/outdegree. Structural properties of numerical simulated networks are analyzed and compared with each of the four real networks. This work helps understand the interplay between structures on different scales in online social networks.     

Neurobiologically realistic determinants of self-organized criticality in networks of spiking neurons

Self-organized criticality refers to the spontaneous emergence of self-similar dynamics in complex systems poised between order and randomness. The presence of self-organized critical dynamics in the brain is theoretically appealing and is supported by recent neurophysiological studies. Despite this, the neurobiological determinants of these dynamics have not been previously sought. Here, we systematically examined the influence of such determinants in hierarchically modular networks of leaky integrate-and-fire neurons with spike-timing-dependent synaptic plasticity and axonal conduction delays. We characterized emergent dynamics in our networks by distributions of active neuronal ensemble modules (neuronal avalanches) and rigorously assessed these distributions for power-law scaling. We found that spike-timing-dependent synaptic plasticity enabled a rapid phase transition from random subcritical dynamics to ordered supercritical dynamics. Importantly, modular connectivity and low wiring cost broadened this transition, and enabled a regime indicative of self-organized criticality. The regime only occurred when modular connectivity, low wiring cost and synaptic plasticity were simultaneously present, and the regime was most evident when between-module connection density scaled as a power-law. The regime was robust to variations in other neurobiologically relevant parameters and favored systems with low external drive and strong internal interactions. Increases in system size and connectivity facilitated internal interactions, permitting reductions in external drive and facilitating convergence of postsynaptic-response magnitude and synaptic-plasticity learning rate parameter values towards neurobiologically realistic levels. We hence infer a novel association between self-organized critical neuronal dynamics and several neurobiologically realistic features of structural connectivity. The central role of these features in our model may reflect their importance for neuronal information processing.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Examining the potential effects of species aggregation on the network structure of food webs

One of the key measures that have been used to describe the topological properties of complex networks is the “degree distribution”, which is a measure that describes the frequency distribution of number of links per node. Food webs are complex ecological networks that describe the trophic relationships among species in a community, and the topological properties of empirical food webs, including degree distributions, have been examined previously. Previously, the “niche model” has been shown to accurately predict degree distributions of empirical food webs, however, the niche model-generated food webs were referenced against empirical food webs that had their species grouped together based on their taxonomic and/or trophic relationships (aggregated food webs). Here, we explore the effects of species aggregation on the ability of the niche model to predict the total- (sum of prey and predator links per node), in- (number of predator links per node), and out- (number of prey links per node) degree distributions of empirical food webs by examining two food webs that can be aggregated at different levels of resolution. The results showed that (1) the cumulative total- and out-degree distributions were consistent with the niche model predictions when the species were aggregated, (2) when the species were disaggregated (i.e., higher resolution), there were mixed conclusions with regards to the niche model's ability to predict total- and out-degree distributions, (3) the model's ability to predict the in-degree distributions of the two food webs was generally inadequate. Although it has been argued that universal functional form based on the niche model could describe the degree distribution patterns of empirical food webs, we believe there are some limitations to the model's ability to accurately predict the structural properties of food webs.     

Resolution of electron microscope autoradiography. IV. Application to analysis of autoradiographs

The previous publications of this series described the expected grain distributions around model radioactive structures in EM autoradiographs as a function of the specimen resolution. This family of expected distributions was called the "universal curves". In the present study, experiments on 14C-sources were compared, significant differences were found depending on the energy of the isotope. These differences were primarily in the tails of the distributions, and are therefore important in correcting for cross-scatter when analyzing electron microscope autoradiographs. Using the universal curves unique for 125I, 3H, and 14C, we designed three sets of transparent overlays, or "masks", one set for each of these isotopes. The masks can be used by an investigator in a manner similar to that suggested by Blackett and Parry to generate grain distributions in autoradiographs on the basis of any desired hypothesis regarding the levels of radioactivity in different structures. A subsequent comparison between these generated distributions and those obtained from the observed grains in these autoradiographs leads to a determination of the most likely levels of radioactivity in the tissue. A computer (described in an Appendix by Land and Salpeter) can be used to find the "best fit" levels of radioactivity in complex cases. The accuracy of the masks was checked on generated line sources for each of the three isotopes.     

Two Fundamental Mechanisms Govern the Stiffening of Cross-linked Networks

Biopolymer networks, such as those constituting the cytoskeleton of a cell or biological tissue, exhibit a nonlinear strain-stiffening behavior when subjected to large deformations. Interestingly, rheological experiments on various in vitro biopolymer networks have shown similar strain-stiffening trends regardless of the differences in their microstructure or constituents, suggesting a universal stiffening mechanism. In this article, we use computer simulations of a random network comprised of cross-linked biopolymer-like fibers to substantiate the notion that this universality lies in the existence of two fundamental stiffening mechanisms. After showing that the large strain response is accompanied by the development of a stress path, i.e., a percolating path of axially stressed fibers and cross-links, we demonstrate that the strain stiffening can be caused by two distinctly different mechanisms: 1) the pulling out of stress-path undulations; and 2) reorientation of the stress path. The former mechanism is bending-dominated and can be recognized by a power-law dependence with exponent 3/2 of the shear modulus on stress, whereas the latter mechanism is stretching-dominated and characterized by a power-law exponent 1/2. We demonstrate how material properties of the constituents, as well as the network microstructure, can affect the transition between the two stiffening mechanisms and, as such, control the dominant power-law scaling behavior.     

High-confidence discovery of genetic network regulators in expression quantitative trait loci data

Expression QTL (eQTL) studies involve the collection of microarray gene expression data and genetic marker data from segregating individuals in a population to search for genetic determinants of differential gene expression. Previous studies have found large numbers of trans-regulated genes (regulated by unlinked genetic loci) that link to a single locus or eQTL "hotspot," and it would be desirable to find the mechanism of coregulation for these gene groups. However, many difficulties exist with current network reconstruction algorithms such as low power and high computational cost. A common observation for biological networks is that they have a scale-free or power-law architecture. In such an architecture, highly influential nodes exist that have many connections to other nodes. If we assume that this type of architecture applies to genetic networks, then we can simplify the problem of genetic network reconstruction by focusing on discovery of the key regulatory genes at the top of the network. We introduce the concept of "shielding" in which a specific gene expression variable (the shielder) renders a set of other gene expression variables (the shielded genes) independent of the eQTL. We iteratively build networks from the eQTL to the shielder down using tests of conditional independence. We have proposed a novel test for controlling the shielder false-positive rate at a predetermined level by requiring a threshold number of shielded genes per shielder. Using simulation, we have demonstrated that we can control the shielder false-positive rate as well as obtain high shielder and edge specificity. In addition, we have shown our method to be robust to violation of the latent variable assumption, an important feature in the practical application of our method. We have applied our method to a yeast expression QTL data set in which microarray and marker data were collected from the progeny of a backcross of two species of Saccharomyces cerevisiae (Brem et al. 2002). Seven genetic networks have been discovered, and bioinformatic analysis of the discovered regulators and corresponding regulated genes has generated plausible hypotheses for mechanisms of regulation that can be tested in future experiments.     

Bow ties, metabolism and disease

Highly organized, universal structures underlying biological and technological networks mediate effective trade-offs among efficiency, robustness and evolvability, with predictable fragilities that can be used to understand disease pathogenesis. The aims of this article are to describe the features of one common organizational architecture in biology, the bow tie. Large-scale organizational frameworks such as the bow tie are necessary starting points for higher-resolution modeling of complex biologic processes     

Universality splitting in distribution of number of miRNA co-targets

In a recent work (Basu et al., in EPL 105:28007, 2014) it was pointed out that the link-weight distribution of microRNA co-target network of a wide class of species are universal up to scaling. The number cell types, widely accepted as a measure of complexity, turns out to be proportional to these scale-factor. In this article we discuss additional universal features of these networks and show that, this universality splits if one considers distribution of number of common targets of three or more number of microRNAs. These distributions for different species can be collapsed onto two distinct set of universal functions, revealing the fact that the species which appeared in early evolution have different complexity measure compared to those appeared late.     

Build-up mechanisms determining the topology of mutualistic networks

The frequency distribution of the number of interactions per species (i.e., degree distribution) within plant-animal mutualistic assemblages often decays as a power-law with an exponential truncation. Such a truncation suggests that there are ecological factors limiting the frequency of supergeneralist species. However, it is not clear whether these patterns can emerge from intrinsic features of the interacting assemblages, such as differences between plant and animal species richness (richness ratio). Here, we show that high richness ratios often characterize plant-animal mutualisms. Then, we demonstrate that exponential truncations are expected in bipartite networks generated by a simple model that incorporates build-up mechanisms that lead to a high richness ratio. Our results provide a simple interpretation for the truncations commonly observed in the degree distributions of mutualistic networks that complements previous ones based on biological effects.