Extreme self-organization in networks constructed from gene expression data

Multiplex network disintegration strategy inference based on deep network representation learning

Multiplex networks have attracted more and more attention because they can model the coupling of network nodes between layers more accurately. The interaction of nodes between layers makes the attack effect on multiplex networks not simply a linear superposition of the attack effect on single-layer networks, and the disintegration of multiplex networks has become a research hotspot and difficult. Traditional multiplex network disintegration methods generally adopt approximate and heuristic strategies. However, these two methods have a number of drawbacks and fail to meet our requirements in terms of effectiveness and timeliness. In this paper, we develop a novel deep learning framework, called MINER (Multiplex network disintegration strategy Inference based on deep NEtwork Representation learning), which transforms the disintegration strategy inference of multiplex networks into the encoding and decoding process based on deep network representation learning. In the encoding process, the attention mechanism encodes the coupling relationship of corresponding nodes between layers, and reinforcement learning is adopted to evaluate the disintegration action in the decoding process. Experiments indicate that the trained MINER model can be directly transferred and applied to the disintegration of multiplex networks with different scales. We extend it to scenarios that consider node attack cost constraints and also achieve excellent performance. This framework provides a new way to understand and employ multiplex networks.

Minimal physicalism as a scale-free substrate for cognition and consciousness

Theories of consciousness and cognition that assume a neural substrate automatically regard phylogenetically basal, nonneural systems as nonconscious and noncognitive. Here, we advance a scale-free characterization of consciousness and cognition that regards basal systems, including synthetic constructs, as not only informative about the structure and function of experience in more complex systems but also as offering distinct advantages for experimental manipulation. Our "minimal physicalist" approach makes no assumptions beyond those of quantum information theory, and hence is applicable from the molecular scale upwards. We show that standard concepts including integrated information, state broadcasting via small-world networks, and hierarchical Bayesian inference emerge naturally in this setting, and that common phenomena including stigmergic memory, perceptual coarse-graining, and attention switching follow directly from the thermodynamic requirements of classical computation. We show that the self-representation that lies at the heart of human autonoetic awareness can be traced as far back as, and serves the same basic functions as, the stress response in bacteria and other basal systems.

Morphological Coordination: A Common Ancestral Function Unifying Neural and Non-Neural Signaling

Nervous systems are traditionally thought of as providing sensing and behavioral coordination functions at the level of the whole organism. What is the evolutionary origin of the mechanisms enabling the nervous systems’ information processing ability? Here, we review evidence from evolutionary, developmental, and regenerative biology suggesting a deeper, ancestral function of both pre-neural and neural cell-cell communication systems: the long-distance coordination of cell division and differentiation required to create and maintain body-axis symmetries. This conceptualization of the function of nervous system activity sheds new light on the evolutionary transition from the morphologically rudimentary, non-neural Porifera and Placazoa to the complex morphologies of Ctenophores, Cnidarians, and Bilaterians. It further allows a sharp formulation of the distinction between long-distance axis-symmetry coordination based on external coordinates, e.g., by whole-organism scale trophisms as employed by plants and sessile animals, and coordination based on body-centered coordinates as employed by motile animals. Thus we suggest that the systems that control animal behavior evolved from ancient mechanisms adapting preexisting ionic and neurotransmitter mechanisms to regulate individual cell behaviors during morphogenesis. An appreciation of the ancient, non-neural origins of bioelectrically mediated computation suggests new approaches to the study of embryological development, including embryological dysregulation, cancer, regenerative medicine, and synthetic bioengineering.

Somatic multicellularity as a satisficing solution to the prediction-error minimization problem

Adaptive success in the biosphere requires the dynamic ability to adjust physiological, transcriptional, and behavioral responses to environmental conditions. From chemical networks to organisms to whole communities, biological entities at all levels of organization seek to optimize their predictive power. Here, we argue that this fundamental drive provides a novel perspective on the origin of multicellularity. One way for unicellular organisms to minimize surprise with respect to external inputs is to be surrounded by reproductively-disabled, i.e. somatic copies of themselves - highly predictable agents which in effect reduce uncertainty in their microenvironments. We show that the transition to multicellularity can be modeled as a phase transition driven by environmental threats. We present modeling results showing how multicellular bodies can arise if non-reproductive somatic cells protect their reproductive parents from environmental lethality. We discuss how a somatic body can be interpreted as a Markov blanket around one or more reproductive cells, and how the transition to somatic multicellularity can be represented as a transition from exposure of reproductive cells to a high-uncertainty environment to their protection from environmental uncertainty by this Markov blanket. This is, effectively, a transition by the Markov blanket from transparency to opacity for the variational free energy of the environment. We suggest that the ability to arrest the cell cycle of daughter cells and redirect their resource utilization from division to environmental threat amelioration is the key innovation of obligate multicellular eukaryotes, that the nervous system evolved to exercise this control over long distances, and that cancer is an escape by somatic cells from the control of reproductive cells. Our quantitative model illustrates the evolutionary dynamics of this system, provides a novel hypothesis for the origin of multicellular animal bodies, and suggests a fundamental link between the architectures of complex organisms and information processing in proto-cognitive cellular agents.

Causal Queries from Observational Data in Biological Systems via Bayesian Networks: An Empirical Study in Small Networks

Biological networks are a very convenient modeling and visualization tool to discover knowledge from modern high-throughput genomics and post-genomics data sets. Indeed, biological entities are not isolated but are components of complex multilevel systems. We go one step further and advocate for the consideration of causal representations of the interactions in living systems. We present the causal formalism and bring it out in the context of biological networks, when the data is observational. We also discuss its ability to decipher the causal information flow as observed in gene expression. We also illustrate our exploration by experiments on small simulated networks as well as on a real biological data set.

Sciences of Observation

Multiple sciences have converged, in the past two decades, on a hitherto mostly unremarked question: what is observation? Here, I examine this evolution, focusing on three sciences: physics, especially quantum information theory, developmental biology, especially its molecular and “evo-devo” branches, and cognitive science, especially perceptual psychology and robotics. I trace the history of this question to the late 19th century, and through the conceptual revolutions of the 20th century. I show how the increasing interdisciplinary focus on the process of extracting information from an environment provides an opportunity for conceptual unification, and sketch an outline of what such a unification might look like.

Epistatic Networks Jointly Influence Phenotypes Related to Metabolic Disease and Gene Expression in Diversity Outbred Mice

In this study, Tyler et al. analyzed the complex genetic architecture of metabolic disease-related traits using the Diversity Outbred mouse population

Genetic studies of multidimensional phenotypes can potentially link genetic variation, gene expression, and physiological data to create multi-scale models of complex traits. The challenge of reducing these data to specific hypotheses has become increasingly acute with the advent of genome-scale data resources. Multi-parent populations derived from model organisms provide a resource for developing methods to understand this complexity. In this study, we simultaneously modeled body composition, serum biomarkers, and liver transcript abundances from 474 Diversity Outbred mice. This population contained both sexes and two dietary cohorts. Transcript data were reduced to functional gene modules with weighted gene coexpression network analysis (WGCNA), which were used as summary phenotypes representing enriched biological processes. These module phenotypes were jointly analyzed with body composition and serum biomarkers in a combined analysis of pleiotropy and epistasis (CAPE), which inferred networks of epistatic interactions between quantitative trait loci that affect one or more traits. This network frequently mapped interactions between alleles of different ancestries, providing evidence of both genetic synergy and redundancy between haplotypes. Furthermore, a number of loci interacted with sex and diet to yield sex-specific genetic effects and alleles that potentially protect individuals from the effects of a high-fat diet. Although the epistatic interactions explained small amounts of trait variance, the combination of directional interactions, allelic specificity, and high genomic resolution provided context to generate hypotheses for the roles of specific genes in complex traits. Our approach moves beyond the cataloging of single loci to infer genetic networks that map genetic etiology by simultaneously modeling all phenotypes.

Functional connectivity in islets of Langerhans from mouse pancreas tissue slices

We propose a network representation of electrically coupled beta cells in islets of Langerhans. Beta cells are functionally connected on the basis of correlations between calcium dynamics of individual cells, obtained by means of confocal laser-scanning calcium imaging in islets from acute mouse pancreas tissue slices. Obtained functional networks are analyzed in the light of known structural and physiological properties of islets. Focusing on the temporal evolution of the network under stimulation with glucose, we show that the dynamics are more correlated under stimulation than under non-stimulated conditions and that the highest overall correlation, largely independent of Euclidean distances between cells, is observed in the activation and deactivation phases when cells are driven by the external stimulus. Moreover, we find that the range of interactions in networks during activity shows a clear dependence on the Euclidean distance, lending support to previous observations that beta cells are synchronized via calcium waves spreading throughout islets. Most interestingly, the functional connectivity patterns between beta cells exhibit small-world properties, suggesting that beta cells do not form a homogeneous geometric network but are connected in a functionally more efficient way. Presented results provide support for the existing knowledge of beta cell physiology from a network perspective and shed important new light on the functional organization of beta cell syncitia whose structural topology is probably not as trivial as believed so far.

Statistical inference and reverse engineering of gene regulatory networks from observational expression data

In this paper, we present a systematic and conceptual overview of methods for inferring gene regulatory networks from observational gene expression data. Further, we discuss two classic approaches to infer causal structures and compare them with contemporary methods by providing a conceptual categorization thereof. We complement the above by surveying global and local evaluation measures for assessing the performance of inference algorithms.

The role of indirect connections in gene networks in predicting function

MOTIVATION

Gene networks have been used widely in gene function prediction algorithms, many based on complex extensions of the 'guilt by association' principle. We sought to provide a unified explanation for the performance of gene function prediction algorithms in exploiting network structure and thereby simplify future analysis.

RESULTS

We use co-expression networks to show that most exploited network structure simply reconstructs the original correlation matrices from which the co-expression network was obtained. We show the same principle works in predicting gene function in protein interaction networks and that these methods perform comparably to much more sophisticated gene function prediction algorithms.

AVAILABILITY AND IMPLEMENTATION

Data and algorithm implementation are fully described and available at http://www.chibi.ubc.ca/extended. Programs are provided in Matlab m-code.

CONTACT

paul@chibi.ubc.ca

KIAA0101 interacts with BRCA1 and regulates centrosome number

To find genes and proteins that collaborate with BRCA1 or BRCA2 in the pathogenesis of breast cancer, we used an informatics approach and found a candidate BRCA interactor, KIAA0101, to function like BRCA1 in exerting a powerful control over centrosome number. The effect of KIAA0101 on centrosomes is likely direct, as its depletion does not affect the cell cycle, KIAA0101 localizes to regions coincident with the centrosomes, and KIAA0101 binds to BRCA1. We analyzed whether KIAA0101 protein is overexpressed in breast cancer tumor samples in tissue microarrays, and we found that overexpression of KIAA0101 correlated with positive Ki67 staining, a biomarker associated with increased disease severity. Furthermore, overexpression of the KIAA0101 gene in breast tumors was found to be associated with significantly decreased survival time. This study identifies KIAA0101 as a protein important for breast tumorigenesis, and as this factor has been reported as a UV repair factor, it may link the UV damage response to centrosome control.

The impact of multifunctional genes on "guilt by association" analysis

Many previous studies have shown that by using variants of "guilt-by-association", gene function predictions can be made with very high statistical confidence. In these studies, it is assumed that the "associations" in the data (e.g., protein interaction partners) of a gene are necessary in establishing "guilt". In this paper we show that multifunctionality, rather than association, is a primary driver of gene function prediction. We first show that knowledge of the degree of multifunctionality alone can produce astonishingly strong performance when used as a predictor of gene function. We then demonstrate how multifunctionality is encoded in gene interaction data (such as protein interactions and coexpression networks) and how this can feed forward into gene function prediction algorithms. We find that high-quality gene function predictions can be made using data that possesses no information on which gene interacts with which. By examining a wide range of networks from mouse, human and yeast, as well as multiple prediction methods and evaluation metrics, we provide evidence that this problem is pervasive and does not reflect the failings of any particular algorithm or data type. We propose computational controls that can be used to provide more meaningful control when estimating gene function prediction performance. We suggest that this source of bias due to multifunctionality is important to control for, with widespread implications for the interpretation of genomics studies.

Benchmarking regulatory network reconstruction with GRENDEL

MOTIVATION

Over the past decade, the prospect of inferring networks of gene regulation from high-throughput experimental data has received a great deal of attention. In contrast to the massive effort that has gone into automated deconvolution of biological networks, relatively little effort has been invested in benchmarking the proposed algorithms. The rate at which new network inference methods are being proposed far outpaces our ability to objectively evaluate and compare them. This is largely due to a lack of fully understood biological networks to use as gold standards.

RESULTS

We have developed the most realistic system to date that generates synthetic regulatory networks for benchmarking reconstruction algorithms. The improved biological realism of our benchmark leads to conclusions about the relative accuracies of reconstruction algorithms that are significantly different from those obtained with A-BIOCHEM, an established in silico benchmark.

AVAILABILITY

The synthetic benchmark utility and the specific benchmark networks that were used in our analyses are available at http://mblab.wustl.edu/software/grendel/.

Natural selection governs local, but not global, evolutionary gene coexpression networks in Caenorhabditis elegans

Background

Large-scale evaluation of gene expression variation among Caenorhabditis elegans lines that have diverged from a common ancestor allows for the analysis of a novel class of biological networks – evolutionary gene coexpression networks. Comparative analysis of these evolutionary networks has the potential to uncover the effects of natural selection in shaping coexpression network topologies since C. elegans mutation accumulation (MA) lines evolve essentially free from the effects of natural selection, whereas natural isolate (NI) populations are subject to selective constraints.

Results

We compared evolutionary gene coexpression networks for C. elegans MA lines versus NI populations to evaluate the role that natural selection plays in shaping the evolution of network topologies. MA and NI evolutionary gene coexpression networks were found to have very similar global topological properties as measured by a number of network topological parameters. Observed MA and NI networks show node degree distributions and average values for node degree, clustering coefficient, path length, eccentricity and betweeness that are statistically indistinguishable from one another yet highly distinct from randomly simulated networks. On the other hand, at the local level the MA and NI coexpression networks are highly divergent; pairs of genes coexpressed in the MA versus NI lines are almost entirely different as are the connectivity and clustering properties of individual genes.

Conclusion

It appears that selective forces shape how local patterns of coexpression change over time but do not control the global topology of C. elegans evolutionary gene coexpression networks. These results have implications for the evolutionary significance of global network topologies, which are known to be conserved across disparate complex systems.

Topology and predictability of El Niño and La Niña networks

We construct the networks of the surface temperature field for El Niño and for La Niña years and investigate their structure. We find that the El Niño network possesses significantly fewer links and lower clustering coefficient and characteristic path length than the La Niña network, which indicates that the former network is less communicative and less stable than the latter. We conjecture that because of this, predictability of temperature should decrease during El Niño years. Here we verify that indeed during El Niño years predictability is lower compared to La Niña years.

Rank-based edge reconstruction for scale-free genetic regulatory networks

Background

The reconstruction of genetic regulatory networks from microarray gene expression data has been a challenging task in bioinformatics. Various approaches to this problem have been proposed, however, they do not take into account the topological characteristics of the targeted networks while reconstructing them.

Results

In this study, an algorithm that explores the scale-free topology of networks was proposed based on the modification of a rank-based algorithm for network reconstruction. The new algorithm was evaluated with the use of both simulated and microarray gene expression data. The results demonstrated that the proposed algorithm outperforms the original rank-based algorithm. In addition, in comparison with the Bayesian Network approach, the results show that the proposed algorithm gives much better recovery of the underlying network when sample size is much smaller relative to the number of genes.

Conclusion

The proposed algorithm is expected to be useful in the reconstruction of biological networks whose degree distributions follow the scale-free topology.

Sample scale-free gene regulatory network using gene ontology

Currently there are various approaches to the reconstruction of gene regulatory networks from different sources of data. However, none of these methods incorporates explicitly scale-free property, one of the most important features of the targeted network, into their algorithms. In this paper, several network sampling strategies are explored on a set assembled from previous published gene interactions in yeast, expecting to reconstruct regulatory networks that are scale-free.

Visualization, modelling and prediction in soil microbiology

The introduction of new approaches for characterizing microbial communities and imaging soil environments has benefited soil microbiology by providing new ways of detecting and locating microorganisms. Consequently, soil microbiology is poised to progress from simply cataloguing microbial complexity to becoming a systems science. A systems approach will enable the structures of microbial communities to be characterized and will inform how microbial communities affect soil function. Systems approaches require accurate analyses of the spatio-temporal properties of the different microenvironments present in soil. In this Review we advocate the need for the convergence of the experimental and theoretical approaches that are used to characterize and model the development of microbial communities in soils.

Subdimension-based similarity measure for DNA microarray data clustering

Microarray data analysis is useful for understanding biological processes. A number of clustering algorithms have been used to achieve this task. However, the performance of these methods can be significantly degraded due to the presence of nonsignificant conditions. In this paper, we propose a robust clustering algorithm based on a similarity measure. The key concept of the proposed similarity measure is to measure the similarity between two data points by their subdimensions. For example, assume that x1, x2, and x3 are ten-dimensional data vectors. The data point x3 is said to be closer to x1 than x2 if more than half of the dimensions of x1 and x3 are closer to x1 than x2. Thus, if two patterns are very similar except for a small amount of features, this measure will preserve the similarity. We have performed eight experiments to test the robustness of the proposed method, including three synthetic data sets, three real world data sets, and two microarray data sets. We also have compared the proposed method with four different clustering algorithms. Experimental results show that the proposed method yields better results than existing clustering algorithms.

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.

A medical bioinformatics approach for metabolic disorders: Biomedical data prediction, modeling, and systematic analysis

During the past century, studies of metabolic disorders have focused research efforts to improve clinical diagnosis and management, to illuminate metabolic mechanisms, and to find effective treatments. The availability of human genome sequences and transcriptomic, proteomic, and metabolomic data provides us with a challenging opportunity to develop computational approaches for systematic analysis of metabolic disorders. In this paper, we present a strategy of bioinformatics analysis to exploit the current data available both on genomic and metabolic levels and integrate these at novel levels of understanding of metabolic disorders. PathAligner is applied to predict biomedical data based on a given disorder. A case study on urea cycle disorders is demonstrated. A Petri net model is constructed to estimate the regulation both on genomic and metabolic levels. We also analyze the transcription factors, signaling pathways and associated disorders to interpret the occurrence and regulation of the urea cycle.

AVAILABILITY

PathAligner's metabolic disorder analyzer is available at http://bibiserv.techfak.uni-bielefeld.de/pathaligner/pathaligner_MDA.html. Supplementary materials are available at http://www.techfak.uni-bielefeld.de/~mchen/metabolic_disorders.

Microarray missing data imputation based on a set theoretic framework and biological knowledge

Gene expressions measured using microarrays usually suffer from the missing value problem. However, in many data analysis methods, a complete data matrix is required. Although existing missing value imputation algorithms have shown good performance to deal with missing values, they also have their limitations. For example, some algorithms have good performance only when strong local correlation exists in data while some provide the best estimate when data is dominated by global structure. In addition, these algorithms do not take into account any biological constraint in their imputation. In this paper, we propose a set theoretic framework based on projection onto convex sets (POCS) for missing data imputation. POCS allows us to incorporate different types of a priori knowledge about missing values into the estimation process. The main idea of POCS is to formulate every piece of prior knowledge into a corresponding convex set and then use a convergence-guaranteed iterative procedure to obtain a solution in the intersection of all these sets. In this work, we design several convex sets, taking into consideration the biological characteristic of the data: the first set mainly exploit the local correlation structure among genes in microarray data, while the second set captures the global correlation structure among arrays. The third set (actually a series of sets) exploits the biological phenomenon of synchronization loss in microarray experiments. In cyclic systems, synchronization loss is a common phenomenon and we construct a series of sets based on this phenomenon for our POCS imputation algorithm. Experiments show that our algorithm can achieve a significant reduction of error compared to the KNNimpute, SVDimpute and LSimpute methods.

Copying nodes versus editing links: the source of the difference between genetic regulatory networks and the WWW

We study two kinds of networks: genetic regulatory networks and the World Wide Web. We systematically test microscopic mechanisms to find the set of such mechanisms that optimally explain each networks' specific properties. In the first case we formulate a model including mainly random unbiased gene duplications and mutations. In the second case, the basic moves are website generation and rapid surf-induced link creation (/destruction). The different types of mechanisms reproduce the appropriate observed network properties. We use those to show that different kinds of networks have strongly system-dependent macroscopic experimental features. The diverging properties result from dissimilar node and link basic dynamics. The main non-uniform properties include the clustering coefficient, small-scale motifs frequency, time correlations, centrality and the connectivity of outgoing links. Some other features are generic such as the large-scale connectivity distribution of incoming links (scale-free) and the network diameter (small-worlds). The common properties are just the general hallmark of autocatalysis (self-enhancing processes), while the specific properties hinge on the specific elementary mechanisms.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics Online.

Functional network analysis reveals extended gliomagenesis pathway maps and three novel MYC-interacting genes in human gliomas

Gene expression profiling has proven useful in subclassification and outcome prognostication for human glial brain tumors. The analysis of biological significance of the hundreds or thousands of alterations in gene expression found in genomic profiling remains a major challenge. Moreover, it is increasingly evident that genes do not act as individual units but collaborate in overlapping networks, the deregulation of which is a hallmark of cancer. Thus, we have here applied refined network knowledge to the analysis of key functions and pathways associated with gliomagenesis in a set of 50 human gliomas of various histogenesis, using cDNA microarrays, inferential and descriptive statistics, and dynamic mapping of gene expression data into a functional annotation database. Highest-significance networks were assembled around the myc oncogene in gliomagenesis and around the integrin signaling pathway in the glioblastoma subtype, which is paradigmatic for its strong migratory and invasive behavior. Three novel MYC-interacting genes (UBE2C, EMP1, and FBXW7) with cancer-related functions were identified as network constituents differentially expressed in gliomas, as was CD151 as a new component of a network that mediates glioblastoma cell invasion. Complementary, unsupervised relevance network analysis showed a conserved self-organization of modules of interconnected genes with functions in cell cycle regulation in human gliomas. This approach has extended existing knowledge about the organizational pattern of gene expression in human gliomas and identified potential novel targets for future therapeutic development.

Self-organized scale-free networks

Growth and preferential attachments have been coined as the two fundamental mechanisms responsible for the scale-free feature in complex networks, as characterized by an algebraic degree distribution. There are situations, particularly in biological networks, where growth is absent or not important, yet some of these networks still exhibit the scale-free feature with a small degree exponent. Here we propose two classes of models to account for this phenomenon. We show analytically and numerically that, in the first model, a spectrum of algebraic degree distributions with a small exponent can be generated. The second model incorporates weights for nodes, and it is able to generate robust scale-free degree distribution with larger algebraic exponents. Our results imply that it is natural for a complex network to self-organize itself into a scale-free state without growth.

DNA-microarray analysis of brain cancer: molecular classification for therapy

Primary brain tumours are among the most lethal of all cancers, largely as a result of their lack of responsiveness to current therapy. Numerous new therapies hold great promise for the treatment of patients with brain cancer, but the main challenge is to determine which treatment is most likely to benefit an individual patient. DNA-microarray-based technologies, which allow simultaneous analysis of expression of thousands of genes, have already begun to uncover previously unrecognized patient subsets that differ in their survival. Here, we review the progress made so far in using DNA microarrays to optimize brain cancer therapy.

Conservation and coevolution in the scale-free human gene coexpression network

The role of natural selection in biology is well appreciated. Recently, however, a critical role for physical principles of network self-organization in biological systems has been revealed. Here, we employ a systems level view of genome-scale sequence and expression data to examine the interplay between these two sources of order, natural selection and physical self-organization, in the evolution of human gene regulation. The topology of a human gene coexpression network, derived from tissue-specific expression profiles, shows scale-free properties that imply evolutionary self-organization via preferential node attachment. Genes with numerous coexpressed partners (the hubs of the coexpression network) evolve more slowly on average than genes with fewer coexpressed partners, and genes that are coexpressed show similar rates of evolution. Thus, the strength of selective constraints on gene sequences is affected by the topology of the gene coexpression network. This connection is strong for the coding regions and 3' untranslated regions (UTRs), but the 5' UTRs appear to evolve under a different regime. Surprisingly, we found no connection between the rate of gene sequence divergence and the extent of gene expression profile divergence between human and mouse. This suggests that distinct modes of natural selection might govern sequence versus expression divergence, and we propose a model, based on rapid, adaptation-driven divergence and convergent evolution of gene expression patterns, for how natural selection could influence gene expression divergence.

Functional modules by relating protein interaction networks and gene expression

Genes and proteins are organized on the basis of their particular mutual relations or according to their interactions in cellular and genetic networks. These include metabolic or signaling pathways and protein interaction, regulatory or co-expression networks. Integrating the information from the different types of networks may lead to the notion of a functional network and functional modules. To find these modules, we propose a new technique which is based on collective, multi-body correlations in a genetic network. We calculated the correlation strength of a group of genes (e.g. in the co-expression network) which were identified as members of a module in a different network (e.g. in the protein interaction network) and estimated the probability that this correlation strength was found by chance. Groups of genes with a significant correlation strength in different networks have a high probability that they perform the same function. Here, we propose evaluating the multi-body correlations by applying the superparamagnetic approach. We compare our method to the presently applied mean Pearson correlations and show that our method is more sensitive in revealing functional relationships.

Connectivity distribution of spatial networks

We study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff. We derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes. We show that for regular spatial densities, the corresponding spatial network has a connectivity distribution decreasing faster than an exponential. In contrast, we also show that scale-free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution (integral of higher powers of the distribution) diverges. We illustrate our results on a simple example for which we present simulation results. Finally, we speculate on the role played by the limiting case P(k) proportional, variant k(-1) which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data.

Potts ferromagnets on coexpressed gene networks: identifying maximally stable partitions

Clustering gene expression data by exploiting phase transitions in granular ferromagnets requires transforming the data to a granular substrate. We present a method using the recently introduced homogeneity order parameter Lambda [H. Agrawal, Phys. Rev. Lett. 89, 268702 (2002)]] for optimizing the parameter controlling the "granularity" and thus the stability of partitions. The model substrates obtained for gene expression data have a highly granular structure. We explore properties of phase transition in high q ferromagnetic Potts models on these substrates and show that the maximum of the width of superparamagnetic domain, corresponding to maximally stable partitions, coincides with the minimum of Lambda.