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PubMed2005

High-betweenness proteins in the yeast protein interaction network.

Structural features found in biomolecular networks that are absent in random networks produced by simple algorithms can provide insight into the function and evolution of cell regulatory networks. Here we analyze "betweenness" of network nodes, a graph theoretical centrality measure, in the yeast protein interaction network. Proteins that have high betweenness, but low connectivity (degree), were found to be abundant in the yeast proteome. This finding is not explained by algorithms proposed to explain the scale-free property of protein interaction networks, where low-connectivity proteins also have low betweenness. These data suggest the existence of some modular organization of the network, and that the high-betweenness, low-connectivity proteins may act as important links between these modules. We found that proteins with high betweenness are more likely to be essential and that evolutionary age of proteins is positively correlated with betweenness. By comparing different models of genome evolution that generate scale-free networks, we show that rewiring of interactions via mutation is an important factor in the production of such proteins. The evolutionary and functional significance of these observations are discussed.

Journal Article↗
PubMed2005

On the relationship between deterministic and probabilistic directed Graphical models: from Bayesian networks to recursive neural networks.

Machine learning methods that can handle variable-size structured data such as sequences and graphs include Bayesian networks (BNs) and Recursive Neural Networks (RNNs). In both classes of models, the data is modeled using a set of observed and hidden variables associated with the nodes of a directed acyclic graph. In BNs, the conditional relationships between parent and child variables are probabilistic, whereas in RNNs they are deterministic and parameterized by neural networks. Here, we study the formal relationship between both classes of models and show that when the source nodes variables are observed, RNNs can be viewed as limits, both in distribution and probability, of BNs with local conditional distributions that have vanishing covariance matrices and converge to delta functions. Conditions for uniform convergence are also given together with an analysis of the behavior and exactness of Belief Propagation (BP) in 'deterministic' BNs. Implications for the design of mixed architectures and the corresponding inference algorithms are briefly discussed.

Bayes Theorem↗
PubMed2006

Complexity and non-commutativity of learning operations on graphs.

We present results from numerical studies of supervised learning operations in small recurrent networks considered as graphs, leading from a given set of input conditions to predetermined outputs. Graphs that have optimized their output for particular inputs with respect to predetermined outputs are asymptotically stable and can be characterized by attractors, which form a representation space for an associative multiplicative structure of input operations. As the mapping from a series of inputs onto a series of such attractors generally depends on the sequence of inputs, this structure is generally non-commutative. Moreover, the size of the set of attractors, indicating the complexity of learning, is found to behave non-monotonically as learning proceeds. A tentative relation between this complexity and the notion of pragmatic information is indicated.

Artificial Intelligence↗
PubMed2006

On analytical approaches to epidemics on networks.

One way to describe the spread of an infection on a network is by approximating the network by a random graph. However, the usual way of constructing a random graph does not give any control over the number of triangles in the graph, while these triangles will naturally arise in many networks (e.g. in social networks). In this paper, random graphs with a given degree distribution and a given expected number of triangles are constructed. By using these random graphs we analyze the spread of two types of infection on a network: infections with a fixed infectious period and infections for which an infective individual will infect all of its susceptible neighbors or none. These two types of infection can be used to give upper and lower bounds for R(0), the probability of extinction and other measures of dynamics of infections with more general infectious periods.

Communicable Diseases↗
PubMed2003

Global snapshot of a protein interaction network-a percolation based approach.

MOTIVATION: Biologically significant information can be revealed by modeling large-scale protein interaction data using graph theory based network analysis techniques. However, the methods that are currently being used draw conclusions about the global features of the network from local connectivity data. A more systematic approach would be to define global quantities that measure (1) how strongly a protein ties with the other parts of the network and (2) how significantly an interaction contributes to the integrity of the network, and connect them with phenotype data from other sources. In this paper, we introduce such global connectivity measures and develop a stochastic algorithm based upon percolation in random graphs to compute them. RESULTS: We show that, in terms of global connectivities, the distribution of essential proteins is distinct from the background. This observation highlights a fundamental difference between the essential and the non-essential proteins in the network. We also find that the interaction data obtained from different experimental methods such as immunoprecipitation and two-hybrid techniques contribute differently to network integrities. Such difference between different experimental methods can provide insight into the systematic bias present among these techniques. SUPPLEMENTARY INFORMATION: The full list of our results can be found in the supplemental web site http://www.nas.nasa.gov/Groups/SciTech/nano/msamanta/projects/percolation/index.php

Algorithms↗
PubMed2005

Analyzing protein lists with large networks: edge-count probabilities in random graphs with given expected degrees.

We present an analytical framework to analyze lists of proteins with large undirected graphs representing their known functional relationships. We consider edge-count variables such as the number of interactions between a protein and a list, the size of a subgraph induced by a list, and the number of interactions bridging two lists. We derive approximate analytical expressions for the probability distributions of these variables in a model of a random graph with given expected degrees. Probabilities obtained with the analytical expressions are used to mine a protein interaction network for functional modules, characterize the connectedness of protein functional categories, and measure the strength of relations between modules.

Algorithms↗
PubMed2004

How biologically relevant are interaction-based modules in protein networks?

By applying a graph-based algorithm to yeast protein-interaction networks we have extracted modular structures and show that they can be validated using information from the phylogenetic conservation of the network components. We show that the module cores, the parts with the highest intramodular connectivity, are biologically relevant components of the networks. These constituents correlate only weakly with other levels of organization. We also discuss how such structures could be used for finding targets for antimicrobial drugs.

Algorithms↗
PubMed2005

Scale-free networks in cell biology.

A cell's behavior is a consequence of the complex interactions between its numerous constituents, such as DNA, RNA, proteins and small molecules. Cells use signaling pathways and regulatory mechanisms to coordinate multiple processes, allowing them to respond to and adapt to an ever-changing environment. The large number of components, the degree of interconnectivity and the complex control of cellular networks are becoming evident in the integrated genomic and proteomic analyses that are emerging. It is increasingly recognized that the understanding of properties that arise from whole-cell function require integrated, theoretical descriptions of the relationships between different cellular components. Recent theoretical advances allow us to describe cellular network structure with graph concepts and have revealed organizational features shared with numerous non-biological networks. We now have the opportunity to describe quantitatively a network of hundreds or thousands of interacting components. Moreover, the observed topologies of cellular networks give us clues about their evolution and how their organization influences their function and dynamic responses.

Animals↗
PubMed1992

Propagation of information in MetaNet graph models.

Information flow in metabolic networks has been studied with a graph model which represents the biochemical transformations occurring in the system under investigation. The "signal strength", an algebraic expression which estimates the probability that an intermediate metabolite is bound to a given enzyme, has been used to derive the "signal transmittance", the fraction of the informational signal at one intermediate that reaches another intermediate. The transmittance has been used to derive the "response ratio", the sensitivity of the rate of change of information at one metabolite consequent to a perturbation at another metabolite. Because the graphical representation corresponds to the biochemical events presumed to occur in the network, these quantities can be used to design experiments to confirm or falsify the hypotheses underlying the model and aid in understanding the regulatory properties of the system. The technique is illustrated by an example model, and its predictions are shown to be sensitive to modest structural changes in the network.

Animals↗
PubMed2002

A duplication growth model of gene expression networks.

MOTIVATION: There has been considerable interest in developing computational techniques for inferring genetic regulatory networks from whole-genome expression profiles. When expression time series data sets are available, dynamic models can, in principle, be used to infer correlative relationships between gene expression levels, which may be causal. However, because of the range of detectable expression levels and the current quality of the data, the predictive nature of such inferred, quantitative models is questionable. Network models derived from simple rate laws offer an intermediate level analysis, going beyond simple statistical analysis, but falling short of a fully quantitative description. This work shows how such network models can be constructed and describes the global properties of the networks derived from such a model. These global properties are statistically robust and provide insights into the design of the underlying network. RESULTS: Several whole-genome expression time series data sets from yeast microarray experiments were analyzed using a Markov-modeling method (Dewey and Galas, FUNC: Integr. Genomics, 1, 269-278, 2001) to infer an approximation to the underlying genetic network. We found that the global statistical properties of all the resulting networks are similar. The overall structure of these biological networks is distinctly different from that of other recently studied networks such as the Internet or social networks. These biological networks show hierarchical, hub-like structures that have some properties similar to a class of graphs known as small world graphs. Small world networks exhibit local cliquishness while exhibiting strong global connectivity. In addition to the small world properties, the biological networks show a power law or scale free distribution of connectivities. An inverse power law, N(k) approximately k(-3/2), for the number of vertices (genes) with k connections was observed for three different data sets from yeast. We propose network growth models based on gene duplication events. Simulations of these models yield networks with the same combination of global graphical properties that we inferred from the expression data.

Algorithms↗
PubMed2005

Network clustering coefficient without degree-correlation biases.

The clustering coefficient quantifies how well connected are the neighbors of a vertex in a graph. In real networks it decreases with the vertex degree, which has been taken as a signature of the network hierarchical structure. Here we show that this signature of hierarchical structure is a consequence of degree-correlation biases in the clustering coefficient definition. We introduce a definition in which the degree-correlation biases are filtered out, and provide evidence that in real networks the clustering coefficient is constant or decays logarithmically with vertex degree.

Journal Article↗
PubMed2005

Farm animal networks: unraveling the contact structure of the British sheep population.

The spatial and temporal dynamics of many farm animal diseases depend both on disease specific parameters and on the underlying contact structure between farms. Whilst many models for farm animal diseases focus on obtaining and estimating disease transmission parameters, relatively little attention has been given to modelling the underlying network of contacts. In this paper, we present an initial analysis of two relations underlying the contact network of individual sheep breeds in Great Britain. The first relation is based on geographical proximity and the second is based on attendance at agricultural shows. These relations are combined to give a risk-potential network that is based on these two levels of interaction. The structure of each network is investigated using techniques developed in graph theory and social network analysis.

Agriculture↗
PubMed2004

Essentiality and damage in metabolic networks.

Understanding the architecture of physiological functions from annotated genome sequences is a major task for postgenomic biology. From the annotated genome sequence of the microbe Escherichia coli, we propose a general quantitative definition of enzyme importance in a metabolic network. Using a graph analysis of its metabolism, we relate the extent of the topological damage generated in the metabolic network by the deletion of an enzyme to the experimentally determined viability of the organism in the absence of that enzyme. We show that the network is robust and that the extent of the damage relates to enzyme importance. We predict that a large fraction (91%) of enzymes causes little damage when removed, while a small group (9%) can cause serious damage. Experimental results confirm that this group contains the majority of essential enzymes. The results may reveal a universal property of metabolic networks.

Computer Simulation↗
PubMed2004

Local graph alignment and motif search in biological networks.

Interaction networks are of central importance in postgenomic molecular biology, with increasing amounts of data becoming available by high-throughput methods. Examples are gene regulatory networks or protein interaction maps. The main challenge in the analysis of these data is to read off biological functions from the topology of the network. Topological motifs, i.e., patterns occurring repeatedly at different positions in the network, have recently been identified as basic modules of molecular information processing. In this article, we discuss motifs derived from families of mutually similar but not necessarily identical patterns. We establish a statistical model for the occurrence of such motifs, from which we derive a scoring function for their statistical significance. Based on this scoring function, we develop a search algorithm for topological motifs called graph alignment, a procedure with some analogies to sequence alignment. The algorithm is applied to the gene regulation network of Escherichia coli.

Algorithms↗
PubMed2005

Subnets of scale-free networks are not scale-free: sampling properties of networks.

Most studies of networks have only looked at small subsets of the true network. Here, we discuss the sampling properties of a network's degree distribution under the most parsimonious sampling scheme. Only if the degree distributions of the network and randomly sampled subnets belong to the same family of probability distributions is it possible to extrapolate from subnet data to properties of the global network. We show that this condition is indeed satisfied for some important classes of networks, notably classical random graphs and exponential random graphs. For scale-free degree distributions, however, this is not the case. Thus, inferences about the scale-free nature of a network may have to be treated with some caution. The work presented here has important implications for the analysis of molecular networks as well as for graph theory and the theory of networks in general.

Journal Article↗
PubMed2026

GenOT: generative optimal transport enables spatiotemporal interpolation and generation in cross-platform spatial transcriptomics.

Spatial transcriptomics technologies have revolutionized the analysis of spatial gene expression, yet integrating spatial information and generating data across heterogeneous samples remain challenging. We present GenOT, a generative framework combining multi-scale graph self-supervised contrastive learning with optimal transport barycenter theory for efficient cross-slice and cross-platform spatiotemporal interpolation. The core innovation of GenOT lies in introducing an optimal transport barycenter-based interpolation algorithm, which mathematically models spatial distribution differences across heterogeneous samples to reconstruct spatiotemporal gene expression dynamics. Extensive evaluations demonstrate that GenOT consistently outperforms existing approaches in spatial domain identification, cross-platform interpolation, and developmental trajectory reconstruction.

Spatial Transcriptomics↗
PubMed2000

Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices.

Neuroanatomy places critical constraints on the functional connectivity of the cerebral cortex. To analyze these constraints we have examined the relationship between structural features of networks (expressed as graphs) and the patterns of functional connectivity to which they give rise when implemented as dynamical systems. We selected among structurally varying graphs using as selective criteria a number of global information-theoretical measures that characterize functional connectivity. We selected graphs separately for increases in measures of entropy (capturing statistical independence of graph elements), integration (capturing their statistical dependence) and complexity (capturing the interplay between their functional segregation and integration). We found that dynamics with high complexity were supported by graphs whose units were organized into densely linked groups that were sparsely and reciprocally interconnected. Connection matrices based on actual neuroanatomical data describing areas and pathways of the macaque visual cortex and the cat cortex showed structural characteristics that coincided best with those of such complex graphs, revealing the presence of distinct but interconnected anatomical groupings of areas. Moreover, when implemented as dynamical systems, these cortical connection matrices generated functional connectivity with high complexity, characterized by the presence of highly coherent functional clusters. We also found that selection of graphs as they responded to input or produced output led to increases in the complexity of their dynamics. We hypothesize that adaptation to rich sensory environments and motor demands requires complex dynamics and that these dynamics are supported by neuroanatomical motifs that are characteristic of the cerebral cortex.

Animals↗