Exploring statistical and population aspects of network complexity

The characterization and the definition of the complexity of objects is an important but very difficult problem that attracted much interest in many different fields. In this paper we introduce a new measure, called network diversity score (NDS), which allows us to quantify structural properties of networks. We demonstrate numerically that our diversity score is capable of distinguishing ordered, random and complex networks from each other and, hence, allowing us to categorize networks with respect to their structural complexity. We study 16 additional network complexity measures and find that none of these measures has similar good categorization capabilities. In contrast to many other measures suggested so far aiming for a characterization of the structural complexity of networks, our score is different for a variety of reasons. First, our score is multiplicatively composed of four individual scores, each assessing different structural properties of a network. That means our composite score reflects the structural diversity of a network. Second, our score is defined for a population of networks instead of individual networks. We will show that this removes an unwanted ambiguity, inherently present in measures that are based on single networks. In order to apply our measure practically, we provide a statistical estimator for the diversity score, which is based on a finite number of samples.     

A decade of seasonal dynamics and co-occurrences within freshwater bacterioplankton communities from eutrophic Lake Mendota,WI, USA

With an unprecedented decade-long time series from a temperate eutrophic lake, we analyzed bacterial and environmental co-occurrence networks to gain insight into seasonal dynamics at the community level. We found that (1) bacterial co-occurrence networks were non-random, (2) season explained the network complexity and (3) co-occurrence network complexity was negatively correlated with the underlying community diversity across different seasons. Network complexity was not related to the variance of associated environmental factors. Temperature and productivity may drive changes in diversity across seasons in temperate aquatic systems, much as they control diversity across latitude. While the implications of bacterioplankton network structure on ecosystem function are still largely unknown, network analysis, in conjunction with traditional multivariate techniques, continues to increase our understanding of bacterioplankton temporal dynamics.     

Nested liana-tree network in three distinct neotropical vegetation formations

Despite the increasing number of studies on lianas, few of them have focused on liana and host-tree (phorophyte) interactions from a network perspective. Most studies found some network structure in other systems, such as plant facilitation and host-epiphyte. However, a recent study found no structure in a small network of liana–phorophyte interactions. Our aim was to investigate the hypothesis that rich, highly diverse systems yield large interaction networks with some structure. If so, networks of liana–phorophyte interactions in highly diverse systems will have one or more of the following structures: compartmentalized, nested or compound. We sampled three highly diverse vegetation formations: a tropical rainforest, a tropical seasonally dry forest, and a woodland savanna, all in southeastern Brazil. We used simulated annealing to test compartmentalization and found no compartment in any of the three networks analyzed. By means of a modified classical temperature index, we found a nested structure in all three sites sampled. We inferred that these nested structures might result from phorophyte characteristics and sequential colonization by different liana species and might promote increased diversity in tropical tree formations. We propose that, according to the system complexity and the different variables associated with site and liana–phorophyte characteristics, a network may have a structure, which arises in more complex systems. Since we have investigated highly diverse systems with large networks, nestedness could be clearly detected in our study.     

Lessons from the macroinvertebrates: species‐genetic diversity correlations highlight important dissimilar relationships

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Biological network design strategies: discovery through dynamic optimization

An important challenge in systems biology is the inherent complexity of biological network models, which complicates the task of relating network structure to function and of understanding the conceptual design principles by which a given network operates. Here we investigate an approach to analyze the relationship between a network structure and its function using the framework of optimization. A common feature found in a variety of biochemical networks involves the opposition of a pair of enzymatic chemical modification reactions such as phosphorylation-dephosphorylation or methylation-demethylation. The modification pair frequently adjusts biochemical properties of its target, such as activating and deactivating function. We applied optimization methodology to study a reversible modification network unit commonly found in signal transduction systems, and we explored the use of this methodology to discover design principles. The results demonstrate that different sets of rate constants used to parameterize the same network topology represent different compromises made in the resulting network operating characteristics. Moreover, the same topology can be used to encode different strategies for achieving performance goals. The ability to adopt multiple strategies may lead to significantly improved performance across a range of conditions through rate modulation or evolutionary processes. The optimization framework explored here is a practical approach to support the discovery of design principles in biological networks.     

Alternative splicing and protein structure evolution

Alternative splicing is thought to be one of the major sources for functional diversity in higher eukaryotes. Interestingly, when mapping splicing events onto protein structures, about half of the events affect structured and even highly conserved regions i.e. are non-trivial on the structure level. This has led to the controversial hypothesis that such splice variants result in nonsense-mediated mRNA decay or non-functional, unstructured proteins, which do not contribute to the functional diversity of an organism. Here we show in a comprehensive study on alternative splicing that proteins appear to be much more tolerant to structural deletions, insertions and replacements than previously thought. We find literature evidence that such non-trivial splicing isoforms exhibit different functional properties compared to their native counterparts and allow for interesting regulatory patterns on the protein network level. We provide examples that splicing events may represent transitions between different folds in the protein sequence–structure space and explain these links by a common genetic mechanism. Taken together, those findings hint to a more prominent role of splicing in protein structure evolution and to a different view of phenotypic plasticity of protein structures.     

Looking for Robust Properties in the Growth of an Academic Network: The Case of the Uruguayan Biological Research Community

Graph-theoretical methods have recently been used to analyze certain properties of natural and social networks. In this work, we have investigated the early stages in the growth of a Uruguayan academic network, the Biology Area of the Programme for the Development of Basic Science (PEDECIBA). This transparent social network is a territory for the exploration of the reliability of clustering methods that can potentially be used when we are confronted with opaque natural systems that provide us with a limited spectrum of observables (happens in research on the relations between brain, thought and language). From our social net, we constructed two different graph representations based on the relationships among researchers revealed by their co-participation in Master’s thesis committees. We studied these networks at different times and found that they achieve connectedness early in their evolution and exhibit the small-world property (i.e. high clustering with short path lengths). The data seem compatible with power law distributions of connectivity, clustering coefficients and betweenness centrality. Evidence of preferential attachment of new nodes and of new links between old nodes was also found in both representations. These results suggest that there are topological properties observed throughout the growth of the network that do not depend on the representations we have chosen but reflect intrinsic properties of the academic collective under study. Researchers in PEDECIBA are classified according to their specialties. We analysed the community structure detected by a standard algorithm in both representations. We found that much of the pre-specified structure is recovered and part of the mismatches can be attributed to convergent interests between scientists from different sub-disciplines. This result shows the potentiality of some clustering methods for the analysis of partially known natural systems.     

Quantifying the relevance of different mediators in the human immune cell network

MOTIVATION: Immune cells coordinate their efforts for the correct and efficient functioning of the immune system (IS). Each cell type plays a distinct role and communicates with other cell types through mediators such as cytokines, chemokines and hormones, among others, that are crucial for the functioning of the IS and its fine tuning. Nevertheless, a quantitative analysis of the topological properties of an immunological network involving this complex interchange of mediators among immune cells is still lacking. RESULTS: Here we present a method for quantifying the relevance of different mediators in the immune network, which exploits a definition of centrality based on the concept of efficient communication. The analysis, applied to the human IS, indicates that its mediators differ significantly in their network relevance. We found that cytokines involved in innate immunity and inflammation and some hormones rank highest in the network, revealing that the most prominent mediators of the IS are molecules involved in these ancestral types of defence mechanisms which are highly integrated with the adaptive immune response, and at the interplay among the nervous, the endocrine and the immune systems. CONTACT: claudio.franceschi@unibo.it.     

Genetic and environmental variation in the visual properties of bluefin killifish, Lucania goodei

Animals use their sensory systems to detect information about the external environment in order to find mates, locate food and habitat and avoid predators. Yet, there is little understanding of the relative amounts of genetic and/or environmental variation in sensory system properties. In this paper, we demonstrate genetic and environmental variation in opsin expression in a population of bluefin killifish. We measured expression of five opsins (which correlates with relative frequency of corresponding cones) using quantitative, real-time polymerase chain reaction for offspring from a breeding study where offspring were raised under different lighting conditions. Sire (i.e. genetic) effects were present for opsin found in yellow photopigment. Dam effects were present for opsins that create violet, blue and red photopigment. Lighting conditions affected expression of all opsins except SWS2A and mimicked the pattern found among populations. These results highlight the fact that sensory systems are both plastic and yet readily evolvable traits.     

Assessing the contribution of breeds to genetic diversity in conservation schemes

The quantitative assessment of genetic diversity within and between populations is important for decision making in genetic conservation plans. In this paper we define the genetic diversity of a set of populations, S, as the maximum genetic variance that can be obtained in a random mating population that is bred from the set of populations S. First we calculated the relative contribution of populations to a core set of populations in which the overlap of genetic diversity was minimised. This implies that the mean kinship in the core set should be minimal. The above definition of diversity differs from Weitzman diversity in that it attempts to conserve the founder population (and thus minimises the loss of alleles), whereas Weitzman diversity favours the conservation of many inbred lines. The former is preferred in species where inbred lines suffer from inbreeding depression. The application of the method is illustrated by an example involving 45 Dutch poultry breeds. The calculations used were easy to implement and not computer intensive. The method gave a ranking of breeds according to their contributions to genetic diversity. Losses in genetic diversity ranged from 2.1% to 4.5% for different subsets relative to the entire set of breeds, while the loss of founder genome equivalents ranged from 22.9% to 39.3%.     

Short‐term variations in gene flow related to cyclic density fluctuations in the common vole

In highly fluctuating populations with complex social systems, genetic patterns are likely to vary in space and time due to demographic and behavioural processes. Cyclic rodents are extreme examples of demographically instable populations that often exhibit strong social organization. In such populations, kin structure and spacing behaviour may vary with density fluctuations and impact both the composition and spatial structure of genetic diversity. In this study, we analysed the multiannual genetic structure of a cyclic rodent, Microtus arvalis, using a sample of 875 individuals trapped over three complete cycles (from 1999 to 2007) and genotyped at 10 microsatellite loci. We tested the predictions that genetic diversity and gene flow intensity vary with density fluctuations. We found evidences for both spatial scale‐dependant variations in genetic diversity and higher gene flow during high density. Moreover, investigation of sex‐specific relatedness patterns revealed that, although dispersal is biased toward males in this species, distances moved by both sexes were lengthened during high density. Altogether, these results suggest that an increase in migration with density allows to restore the local loss of genetic diversity occurring during low density. We then postulate that this change in migration results from local competition, which enhances female colonization of empty spaces and male dispersal among colonies.     

On the increase in complexity in evolution II. The relativity of complexity and the principle of minimum increase

The complexity of a system cannot be measured in absolute terms, but only relative to a specified observer. As a result, the operational definition of complexity will be different for different sorts of systems and problems. In many cases it will not be as a real valued function.For the study of evolution we adopt as our definition the information content of the instructions required to build the system. Because we take into account factors such as the properties of the epigenetic system and the laws of physics and chemistry, this is not the same as the information content of the genome as measured in terms of codon or amino acid frequencies. Using this definition, we derive a principle of minimum increase in complexity, and we apply it to explain two well-known phenomena in evolution, Williston's law and parallelism. It is unlikely that any definite trends in the evolution of complex systems can be accounted for by simple optimization principles such as the law of natural selection, as these contain no arrow of time.     

Island phytophagy: explaining the remarkable diversity of plant-feeding insects

Plant-feeding insects have undergone unparalleled diversification among different plant taxa, yet explanations for variation in their diversity lack a quantitative, predictive framework. Island biogeographic theory has been applied to spatially discrete habitats but not to habitats, such as host plants, separated by genetic distance. We show that relationships between the diversity of gall-inducing flies and their host plants meet several fundamental predictions from island biogeographic theory. First, plant-taxon genetic distinctiveness, an integrator for long-term evolutionary history of plant lineages, is a significant predictor of variance in the diversity of gall-inducing flies among host-plant taxa. Second, range size and structural complexity also explain significant proportions of the variance in diversity of gall-inducing flies among different host-plant taxa. Third, as with other island systems, plant-lineage age does not predict species diversity. Island biogeographic theory, applied to habitats defined by genetic distance, provides a novel, comprehensive framework for analysing and explaining the diversity of plant-feeding insects and other host-specific taxa.     

Algebraic connectivity may explain the evolution of gene regulatory networks

Gene expression is a result of the interplay between the structure, type, kinetics, and specificity of gene regulatory interactions, whose diversity gives rise to the variety of life forms. As the dynamic behavior of gene regulatory networks depends on their structure, here we attempt to determine structural reasons which, despite the similarities in global network properties, may explain the large differences in organismal complexity. We demonstrate that the algebraic connectivity, the smallest non-trivial eigenvalue of the Laplacian, of the directed gene regulatory networks decreases with the increase of organismal complexity, and may therefore explain the difference between the variety of analyzed regulatory networks. In addition, our results point out that, for the species considered in this study, evolution favours decreasing concentration of strategically positioned feed forward loops, so that the network as a whole can increase the specificity towards changing environments. Moreover, contrary to the existing results, we show that the average degree, the length of the longest cascade, and the average cascade length of gene regulatory networks cannot recover the evolutionary relationships between organisms. Whereas the dynamical properties of special subnetworks are relatively well understood, there is still limited knowledge about the evolutionary reasons for the already identified design principles pertaining to these special subnetworks, underlying the global quantitative features of gene regulatory networks of different organisms. The behavior of the algebraic connectivity, which we show valid on gene regulatory networks extracted from curated databases, can serve as an additional evolutionary principle of organism-specific regulatory networks.     

Diversified Control Paths: A Significant Way Disease Genes Perturb the Human Regulatory Network

Background

The complexity of biological systems motivates us to use the underlying networks to provide deep understanding of disease etiology and the human diseases are viewed as perturbations of dynamic properties of networks. Control theory that deals with dynamic systems has been successfully used to capture systems-level knowledge in large amount of quantitative biological interactions. But from the perspective of system control, the ways by which multiple genetic factors jointly perturb a disease phenotype still remain.

Results

In this work, we combine tools from control theory and network science to address the diversified control paths in complex networks. Then the ways by which the disease genes perturb biological systems are identified and quantified by the control paths in a human regulatory network. Furthermore, as an application, prioritization of candidate genes is presented by use of control path analysis and gene ontology annotation for definition of similarities. We use leave-one-out cross-validation to evaluate the ability of finding the gene-disease relationship. Results have shown compatible performance with previous sophisticated works, especially in directed systems.

Conclusions

Our results inspire a deeper understanding of molecular mechanisms that drive pathological processes. Diversified control paths offer a basis for integrated intervention techniques which will ultimately lead to the development of novel therapeutic strategies.     

Genetic variation across species' geographical ranges: the central-marginal hypothesis and beyond

There is growing interest in quantifying genetic population structure across the geographical ranges of species to understand why species might exhibit stable range limits and to assess the conservation value of peripheral populations. However, many assertions regarding peripheral populations rest on the long-standing but poorly tested supposition that peripheral populations exhibit low genetic diversity and greater genetic differentiation as a consequence of smaller effective population size and greater geographical isolation relative to geographically central populations. We reviewed 134 studies representing 115 species that tested for declines in within-population genetic diversity and/or increases in among-population differentiation towards range margins using nuclear molecular genetic markers. On average, 64.2% of studies detected the expected decline in diversity, 70.2% of those that tested for it showed increased differentiation and there was a positive association between these trends. In most cases, however, the difference in genetic diversity between central and peripheral population was not large. Although these results were consistent across plants and animals, strong taxonomic and biogeographical biases in the available studies call for a cautious generalization of these results. Despite the large number of studies testing these simple predictions, very few attempted to test possible mechanisms causing reduced peripheral diversity or increased differentiation. Almost no study incorporated a phylogeographical framework to evaluate historical influences on contemporary genetic patterns. Finally, there has been little effort to test whether these geographical trends in putatively neutral variation at marker loci are reflected by quantitative genetic trait variation, which is likely to influence the adaptive potential of populations across the geographical range.     

Weighted gene coexpression network analysis strategies applied to mouse weight

Systems-oriented genetic approaches that incorporate gene expression and genotype data are valuable in the quest for genetic regulatory loci underlying complex traits. Gene coexpression network analysis lends itself to identification of entire groups of differentially regulated genes—a highly relevant endeavor in finding the underpinnings of complex traits that are, by definition, polygenic in nature. Here we describe one such approach based on liver gene expression and genotype data from an F₂ mouse intercross utilizing weighted gene coexpression network analysis (WGCNA) of gene expression data to identify physiologically relevant modules. We describe two strategies: single-network analysis and differential network analysis. Single-network analysis reveals the presence of a physiologically interesting module that can be found in two distinct mouse crosses. Module quantitative trait loci (mQTLs) that perturb this module were discovered. In addition, we report a list of genetic drivers for this module. Differential network analysis reveals differences in connectivity and module structure between two networks based on the liver expression data of lean and obese mice. Functional annotation of these genes suggests a biological pathway involving epidermal growth factor (EGF). Our results demonstrate the utility of WGCNA in identifying genetic drivers and in finding genetic pathways represented by gene modules. These examples provide evidence that integration of network properties may well help chart the path across the gene–trait chasm. Electronic supplementary material The online version of this article (doi: ) contains supplementary material, which is available to authorized users. Tova F. Fuller, Anatole Ghazalpour contributed equally to this work.