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Interacting biological systems at all organizational levels display emergent behavior. Modeling these systems is made challenging by the number and variety of biological components and interactions (from molecules in gene regulatory networks to species in ecological networks) and the often-incomplete state of system knowledge (e.g., the unknown values of kinetic parameters for biochemical reactions). Boolean networks have emerged as a powerful tool for modeling these systems. We provide a methodological overview of Boolean network models of biological systems. After a brief introduction, we describe the process of building, analyzing, and validating a Boolean model. We then present the use of the model to make predictions about the system's response to perturbations and about how to control (or at least influence) its behavior. We emphasize the interplay between structural and dynamical properties of Boolean networks and illustrate them in three case studies from disparate levels of biological organization., Comment: Review, to appear in the Cambridge Elements series

Because the attractors of biological networks reflect stable behaviors (e.g., cell phenotypes), identifying control interventions that can drive a system towards its attractors (attractor control) is of particular relevance when controlling biological systems. Driving a network's feedback vertex set (FVS) by node-state override into a state consistent with a target attractor is proven to force every system trajectory to the target attractor, but in biological networks, the FVS is typically larger than can be realistically manipulated. External control of a subset of a biological network's FVS was proposed as a strategy to drive the network to its attractors utilizing fewer interventions; however, the effectiveness of this strategy was only demonstrated on a small set of Boolean models of biological networks. Here, we extend this analysis to ensembles of biologically-inspired Boolean networks. On these models, we use three structural metrics -- PRINCE propagation, modified PRINCE propagation, and CheiRank -- to rank FVS subsets by their predicted attractor control strength. We validate the accuracy of these rankings using three dynamical measures: To Control, Away Control, and logical domain of influence. We also calculate the propagation metrics on effective graphs, which incorporate each Boolean model's functional information into edge weights. While this additional information increases the predicting power of structural metrics, we find that the increase with respect to the unweighted network is limited. The propagation metrics in conjunction with the FVS can be used to identify realizable driver node sets by emulating the dynamics that are prevalent in biological networks. This approach only uses the network's structure, and the driver sets are shown to be robust to the specific dynamical model., Comment: 17 pages, 9 figures

Traditional Chinese medicine (TCM) relies on natural medical products to treat symptoms and diseases. While clinical data have demonstrated the effectiveness of selected TCM-based treatments, the mechanistic root of how TCM herbs treat diseases remains largely unknown. More importantly, current approaches focus on single herbs or prescriptions, missing the high-level general principles of TCM. To uncover the mechanistic nature of TCM on a system level, in this work we establish a generic network medicine framework for TCM from the human protein interactome. Applying our framework reveals a network pattern between symptoms (diseases) and herbs in TCM. We first observe that genes associated with a symptom are not distributed randomly in the interactome, but cluster into localized modules; furthermore, a short network distance between two symptom modules is indicative of the symptoms' co-occurrence and similarity. Next, we show that the network proximity of a herb's targets to a symptom module is predictive of the herb's effectiveness in treating the symptom. We validate our framework with real-world hospital patient data by showing that (1) shorter network distance between symptoms of inpatients correlates with higher relative risk (co-occurrence), and (2) herb-symptom network proximity is indicative of patients' symptom recovery rate after herbal treatment. Finally, we identified novel herb-symptom pairs in which the herb's effectiveness in treating the symptom is predicted by network and confirmed in hospital data, but previously unknown to the TCM community. These predictions highlight our framework's potential in creating herb discovery or repurposing opportunities. In conclusion, network medicine offers a powerful novel platform to understand the mechanism of traditional medicine and to predict novel herbal treatment against diseases., Comment: 25 pages, 4 figures plus 1 table

In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories -- centrality measures, propagation measures, and cycle-based measures. Using each measure every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system towards or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without knowledge of the network's dynamics., Comment: 22 pages for main text 4 pages for supplementary materials, 10 figures. Supplementary tables S5 and S6 are ancillary files

We present new applications of parity inversion and time-reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor identification algorithm implemented for Boolean networks under stochastic dynamics. Its speed enables resolving a longstanding open question of how attractor count in critical random Boolean networks scales with network size, and whether the scaling matches biological observations. Via 80-fold improvement in probed network size ($N=16,384$), we find the surprisingly low scaling exponent of $0.12\pm 0.05$ -- approximately one tenth the analytical upper bound. We demonstrate a general principle: a system's relationship to its time-reversal and state-space inversion constrains its repertoire of emergent behaviors., Comment: The IPython Notebook referenced in supplemental information, along with other reproduction materials, is available at https://github.com/jcrozum/StableMotifs/tree/master/Random%20Boolean%20Network%20Application

The first mitigation response to the Covid-19 pandemic was to limit person-to-person interaction as much as possible. This was implemented by the temporary closing of many workplaces and people were required to follow social distancing. Networks are a great way to represent interactions among people and the temporary severing of these interactions. Here, we present a network model of human-human interactions that could be mediators of disease spread. The nodes of this network are individuals and different types of edges denote family cliques, workplace interactions, interactions arising from essential needs, and social interactions. Each individual can be in one of four states: susceptible, infected, immune, and dead. The network and the disease parameters are informed by the existing literature on Covid-19. Using this model, we simulate the spread of an infectious disease in the presence of various mitigation scenarios. For example, lockdown is implemented by deleting edges that denote non-essential interactions. We validate the simulation results with the real data by matching the basic and effective reproduction numbers during different phases of the spread. We also simulate different possibilities of the slow lifting of the lockdown by varying the transmission rate as facilities are slowly opened but people follow prevention measures like wearing masks etc. We make predictions on the probability and intensity of a second wave of infection in each of these scenarios., Comment: 16 pages, 10 figures

We present the epithelial-to-mesenchymal transition (EMT) from two perspectives: experimental/technological and theoretical. We review the state of the current understanding of the regulatory networks that underlie EMT in three physiological contexts: embryonic development, wound healing, and metastasis. We describe the existing experimental systems and manipulations used to better understand the molecular participants and factors that influence EMT and metastasis. We review the mathematical models of the regulatory networks involved in EMT, with a particular emphasis on the network motifs (such as coupled feedback loops) that can generate intermediate hybrid states between the epithelial and mesenchymal states. Ultimately, the understanding gained about these networks should be translated into methods to control phenotypic outcomes, especially in the context of cancer therapeutic strategies. We present emerging theories of how to drive the dynamics of a network toward a desired dynamical attractor (e.g. an epithelial cell state) and emerging synthetic biology technologies to monitor and control the state of cells., Comment: 35 pages, 7 figures

The emerging field at the intersection of quantitative biology, network modeling, and control theory has enjoyed significant progress in recent years. This Special Issue brings together a selection of papers on complementary approaches to observe, identify, and control biological and biologically inspired networks. These approaches advance the state of the art in the field by addressing challenges common to many such networks, including high dimensionality, strong nonlinearity, uncertainty, and limited opportunities for observation and intervention. Because these challenges are not unique to biological systems, it is expected that many of the results presented in these contributions will also find applications in other domains, including physical, social, and technological networks.

Complex diseases can be modeled as damage to intracellular networks that results in abnormal cell behaviors. Network-based dynamic models such as Boolean models have been employed to model a variety of biological systems including those corresponding to disease. Previous work designed compensatory interactions to stabilize an attractor of a Boolean network after single node damage. We generalize this method to a multinode damage scenario and to the simultaneous stabilization of multiple steady state attractors. We classify the emergent situations, with a special focus on combinatorial effects, and characterize each class through simulation. We explore how the structural and functional properties of the network affect its resilience and its possible repair scenarios. We demonstrate the method's applicability to two intracellular network models relevant to cancer. This work has implications in designing prevention strategies for complex disease., Comment: 16 pages, 8 figures

What can we learn about controlling a system solely from its underlying network structure? Here we adapt a recently developed framework for control of networks governed by a broad class of nonlinear dynamics that includes the major dynamic models of biological, technological, and social processes. This feedback-based framework provides realizable node overrides that steer a system towards any of its natural long term dynamic behaviors, regardless of the specific functional forms and system parameters. We use this framework on several real networks, identify the topological characteristics that underlie the predicted node overrides, and compare its predictions to those of structural controllability in control theory. Finally, we demonstrate this framework's applicability in dynamic models of gene regulatory networks and identify nodes whose override is necessary for control in the general case, but not in specific model instances., Comment: Includes main text and supporting information

Identifying control strategies for biological networks is paramount for practical applications that involve reprogramming a cell's fate, such as disease therapeutics and stem cell reprogramming. Here we develop a novel network control framework that integrates the structural and functional information available for intracellular networks to predict control targets. Formulated in a logical dynamic scheme, our approach drives any initial state to the target state with 100% effectiveness and needs to be applied only transiently for the network to reach and stay in the desired state. We illustrate our method's potential to find intervention targets for cancer treatment and cell differentiation by applying it to a leukemia signaling network and to the network controlling the differentiation of helper T cells. We find that the predicted control targets are effective in a broad dynamic framework. Moreover, several of the predicted interventions are supported by experiments., Comment: 61 pages (main text, 15 pages; supporting information, 46 pages) and 12 figures (main text, 6 figures; supporting information, 6 figures). In review

Network measures that reflect the most salient properties of complex large-scale networks are in high demand in the network research community. In this paper we adapt a combinatorial measure of negative curvature (also called hyperbolicity) to parameterized finite networks, and show that a variety of biological and social networks are hyperbolic. This hyperbolicity property has strong implications on the higher-order connectivity and other topological properties of these networks. Specifically, we derive and prove bounds on the distance among shortest or approximately shortest paths in hyperbolic networks. We describe two implications of these bounds to cross-talk in biological networks, and to the existence of central, influential neighborhoods in both biological and social networks., Comment: Physical Review E, 2014

Drug target identification is of significant commercial interest to pharmaceutical companies, and there is a vast amount of research done related to the topic of therapeutic target identification. Interdisciplinary research in this area involves both the biological network community and the graph algorithms community. Key steps of a typical therapeutic target identification problem include synthesizing or inferring the complex network of interactions relevant to the disease, connecting this network to the disease-specific behavior, and predicting which components are key mediators of the behavior. All of these steps involve graph theoretical or graph algorithmic aspects. In this perspective, we provide modelling and algorithmic perspectives for therapeutic target identification and highlight a number of algorithmic advances, which have gotten relatively little attention so far, with the hope of strengthening the ties between these two research communities.

Discrete dynamic models are a powerful tool for the understanding and modeling of large biological networks. Although a lot of progress has been made in developing analysis tools for these models, there is still a need to find approaches that can directly relate the network structure to its dynamics. Of special interest is identifying the stable patterns of activity, i.e., the attractors of the system. This is a problem for large networks, because the state space of the system increases exponentially with network size. In this work we present a novel network reduction approach that is based on finding network motifs that stabilize in a fixed state. Notably, we use a topological criterion to identify these motifs. Specifically, we find certain types of strongly connected components in a suitably expanded representation of the network. To test our method we apply it to a dynamic network model for a type of cytotoxic T cell cancer and to an ensemble of random Boolean networks of size up to 200. Our results show that our method goes beyond reducing the network and in most cases can actually predict the dynamical repertoire of the nodes (fixed states or oscillations) in the attractors of the system., Comment: 22 pages, 11 figures. The following article appeared in Chaos and may be found at http://link.aip.org/link/?CHA/23/025111 . It is part of the focus issue "Quantitative Approaches to Genetic Networks"

Feature article on how networks that govern communication, growth, herd behavior, and other key processes in nature and society are becoming increasingly amenable to modeling, forecast, and control., Comment: Review of current research on network dynamics

It is well-known that biological and social interaction networks have a varying degree of redundancy, though a consensus of the precise cause of this is so far lacking. In this paper, we introduce a topological redundancy measure for labeled directed networks that is formal, computationally efficient and applicable to a variety of directed networks such as cellular signaling, metabolic and social interaction networks. We demonstrate the computational efficiency of our measure by computing its value and statistical significance on a number of biological and social networks with up to several thousands of nodes and edges. Our results suggest a number of interesting observations: (1) social networks are more redundant that their biological counterparts, (2) transcriptional networks are less redundant than signaling networks, (3) the topological redundancy of the C. elegans metabolic network is largely due to its inclusion of currency metabolites, and (4) the redundancy of signaling networks is highly (negatively) correlated with the monotonicity of their dynamics., Comment: 18 pages

In this study, we explored the utility of a descriptive and predictive bionetwork model for phospholipase C-coupled calcium signaling pathways, built with non-kinetic experimental information. Boolean models generated from these data yield oscillatory activity patterns for both the endoplasmic reticulum resident inositol-1,4,5-trisphosphate receptor (IP3R) and the plasma-membrane resident canonical transient receptor potential channel 3 (TRPC3). These results are specific as randomization of the Boolean operators ablates oscillatory pattern formation. Furthermore, knock-out simulations of the IP3R, TRPC3, and multiple other proteins recapitulate experimentally derived results. The potential of this approach can be observed by its ability to predict previously undescribed cellular phenotypes using in vitro experimental data. Indeed our cellular analysis of the developmental and calcium-regulatory protein, DANGER1a, confirms the counter-intuitive predictions from our Boolean models in two highly relevant cellular models. Based on these results, we theorize that with sufficient legacy knowledge and/or computational biology predictions, Boolean networks provide a robust method for predictive-modeling of any biological system., Comment: 19 pages, 5 Figures, 2 Tables

Community detection and analysis is an important methodology for understanding the organization of various real-world networks and has applications in problems as diverse as consensus formation in social communities or the identification of functional modules in biochemical networks. Currently used algorithms that identify the community structures in large-scale real-world networks require a priori information such as the number and sizes of communities or are computationally expensive. In this paper we investigate a simple label propagation algorithm that uses the network structure alone as its guide and requires neither optimization of a pre-defined objective function nor prior information about the communities. In our algorithm every node is initialized with a unique label and at every step each node adopts the label that most of its neighbors currently have. In this iterative process densely connected groups of nodes form a consensus on a unique label to form communities. We validate the algorithm by applying it to networks whose community structures are known. We also demonstrate that the algorithm takes an almost linear time and hence it is computationally less expensive than what was possible so far., Comment: 12 pages, 9 figures

Plants both lose water and take in carbon dioxide through microscopic stomatal pores, each of which is regulated by a surrounding pair of guard cells. During drought, the plant hormone abscisic acid (ABA) inhibits stomatal opening and promotes stomatal closure, thereby promoting water conservation. Here we synthesize experimental results into a consistent guard cell signal transduction network for ABA-induced stomatal closure, and develop a dynamic model of this process. Our model captures the regulation of more than forty identified network components, and accords well with previous experimental results at both the pathway and whole cell physiological level. Our analysis reveals the novel predictions that the disruption of membrane depolarizability, anion efflux, actin cytoskeleton reorganization, cytosolic pH increase, the phosphatidic acid pathway or of K+ efflux through slowly activating K+ channels at the plasma membrane lead to the strongest reduction in ABA responsiveness. Initial experimental analysis assessing ABA-induced stomatal closure in the presence of cytosolic pH clamp imposed by the weak acid butyrate is consistent with model prediction. Our method can be readily applied to other biological signaling networks to identify key regulatory components in systems where quantitative information is limited., Comment: 17 pages, 8 figures

Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the topologies of different systems. Attempts to explain these similarities have led to the ongoing development and refinement of network models and graph-theoretical analysis techniques with which to characterize and understand complexity. In this tutorial, we demonstrate through illustrative examples, how network measures and models have contributed to the elucidation of the organization of complex systems., Comment: v(1) 38 pages, 7 figures, to appear in the International Journal of Bifurcation and Chaos v(2) Line spacing changed; now 23 pages, 7 figures, to appear in the Special Issue "Complex Networks' Structure and Dynamics'' of the International Journal of Bifurcation and Chaos (Volume 17, Issue 7, July 2007) edited by S. Boccaletti and V. Latora

As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes., Comment: 29 pages, 8 figures, accepted in IEE Proc. Systems Biology

Pathogen-mediated competition, through which an invasive species carrying and transmitting a pathogen can be a superior competitor to a more vulnerable resident species, is one of the principle driving forces influencing biodiversity in nature. Using an experimental system of bacteriophage-mediated competition in bacterial populations and a deterministic model, we have shown in [Joo et al 2005] that the competitive advantage conferred by the phage depends only on the relative phage pathology and is independent of the initial phage concentration and other phage and host parameters such as the infection-causing contact rate, the spontaneous and infection-induced lysis rates, and the phage burst size. Here we investigate the effects of stochastic fluctuations on bacterial invasion facilitated by bacteriophage, and examine the validity of the deterministic approach. We use both numerical and analytical methods of stochastic processes to identify the source of noise and assess its magnitude. We show that the conclusions obtained from the deterministic model are robust against stochastic fluctuations, yet deviations become prominently large when the phage are more pathological to the invading bacterial strain., Comment: 39 pages, 7 figures

We study trade-offs presented by local search algorithms in complex networks which are heterogeneous in edge weights and node degree. We show that search based on a network measure, local betweenness centrality (LBC), utilizes the heterogeneity of both node degrees and edge weights to perform the best in scale-free weighted networks. The search based on LBC is universal and performs well in a large class of complex networks., Comment: 14 pages, 5 figures, 4 tables, minor changes, added a reference

We present a novel framework for inferring regulatory and sequence-level information from gene co-expression networks. The key idea of our methodology is the systematic integration of network inference and network topological analysis approaches for uncovering biological insights. We determine the gene co-expression network of Bacillus subtilis using Affymetrix GeneChip time series data and show how the inferred network topology can be linked to sequence-level information hard-wired in the organism's genome. We propose a systematic way for determining the correlation threshold at which two genes are assessed to be co-expressed by using the clustering coefficient and we expand the scope of the gene co-expression network by proposing the slope ratio metric as a means for incorporating directionality on the edges. We show through specific examples for B. subtilis that by incorporating expression level information in addition to the temporal expression patterns, we can uncover sequence-level biological insights. In particular, we are able to identify a number of cases where (i) the co-expressed genes are part of a single transcriptional unit or operon and (ii) the inferred directionality arises due to the presence of intra-operon transcription termination sites., Comment: 7 pages, 8 figures, accepted in Bioinformatics

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. How do we quantitatively describe a network of hundreds or thousands of interacting components? Does the observed topology of cellular networks give us clues about their evolution? How does cellular networks' organization influence their function and dynamical responses? This article will review the recent advances in addressing these questions., Comment: Review article, 21 pages, 10 figures

Apparent competition between species is believed to be one of the principle driving forces that structure ecological communities, although the precise mecha nisms have yet to be characterized. Here we develop a model system that isolates phage-mediated interactions by neutralizing resource competition using two genetically identical Bordetella bronchiseptica strains that differ only in that one is the carrier of a phage and the other is susceptible to the phage. We observe and quantify the competitive advantage of the bacterial strain bearing the prophage in both invading and in resisting invasion by bacteria susceptible to the phage, and use our measurements to develop a mathematical model of phage-mediated competition. The model predicts, and experimental evidence confirms, that the competitive advantage conferred by the phage depends only on the relative phage pathology and is independent of other phage and host parameters. This work combines experimental and mathematical approaches to the study of phage-driven competition, and provides an experimentally tested framework for evaluation of the effects of pathogens/parasites on interspecific competition., Comment: 10pages, 8 figures

Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre- and posttranslational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations., Comment: 29 pages, 5 figures, accepted to the Journal of Theoretical Biology

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can lead to a 25% loss of transmission efficiency by triggering an overload cascade in the network. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that 40% of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than 1% node loss., Comment: 6 pages, 6 figures

High-throughput protein interaction detection methods are strongly affected by false positive and false negative results. Focused experiments are needed to complement the large-scale methods by validating previously detected interactions but it is often difficult to decide which proteins to probe as interaction partners. Developing reliable computational methods assisting this decision process is a pressing need in bioinformatics. We show that we can use the conserved properties of the protein network to identify and validate interaction candidates. We apply a number of machine learning algorithms to the protein connectivity information and achieve a surprisingly good overall performance in predicting interacting proteins. Using a 'leave-one-out' approach we find average success rates between 20-50% for predicting the correct interaction partner of a protein. We demonstrate that the success of these methods is based on the presence of conserved interaction motifs within the network. A reference implementation and a table with candidate interacting partners for each yeast protein are available at http://www.protsuggest.org

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure have so far precluded us from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior., Comment: 10 pages, 4 figures, to appear in PRE

Evolution has provided many organisms with sophisticated sensory systems that enable them to respond to signals in their environment. The response frequently involves alteration in the pattern of movement, such as the chemokinesis of the bacterium Escherichia coli, which swims by rotating its flagella. When rotated counterclockwise (CCW) the flagella coalesce into a propulsive bundle, producing a relatively straight ``run'', and when rotated clockwise (CW) they fly apart, resulting in a ``tumble'' which reorients the cell with little translocation. A stochastic process generates the runs and tumbles, and in a chemoeffector gradient runs that carry the cell in a favorable direction are extended. The overall structure of the signal transduction pathways is well-characterized in E. coli, but important details are still not understood. Only recently has a source of gain in the signal transduction network been identified experimentally, and here we present a mathematical model based on dynamic assembly of receptor teams that can explain this observation., Comment: Accepted for publication in the Biophysical Journal

Expression of the Drosophila segment polarity genes is initiated by a prepattern of pair-rule gene products and maintained by a network of regulatory interactions throughout several stages of embryonic development. Analysis of a model of gene interactions based on differential equations showed that wild-type expression patterns of these genes can be obtained for a wide range of kinetic parameters, which suggests that the steady states are determined by the topology of the network and the type of regulatory interactions between components, not the detailed form of the rate laws. To investigate this, we propose and analyze a Boolean model of this network which is based on a binary ON/OFF representation of transcription and protein levels, and in which the interactions are formulated as logical functions. In this model the spatial and temporal patterns of gene expression are determined by the topology of the network and whether components are present or absent, rather than the absolute levels of the mRNAs and proteins and the functional details of their interactions. The model is able to reproduce the wild type gene expression patterns, as well as the ectopic expression patterns observed in over-expression experiments and various mutants. Furthermore, we compute explicitly all steady states of the network and identify the basin of attraction of each steady state. The model gives important insights into the functioning of the segment polarity gene network, such as the crucial role of the wingless and sloppy paired genes, and the network's ability to correct errors in the prepattern., Comment: 24 pages, 9 figures

Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks., Comment: 54 pages, submitted to Reviews of Modern Physics

Many complex systems, such as communication networks, display a surprising degree of robustness: while key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. In this paper we demonstrate that error tolerance is not shared by all redundant systems, but it is displayed only by a class of inhomogeneously wired networks, called scale-free networks. We find that scale-free networks, describing a number of systems, such as the World Wide Web, Internet, social networks or a cell, display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected by even unrealistically high failure rates. However, error tolerance comes at a high price: these networks are extremely vulnerable to attacks, i.e. to the selection and removal of a few nodes that play the most important role in assuring the network's connectivity., Comment: 14 pages, 4 figures, Latex

Networks grow and evolve by local events, such as the addition of new nodes and links, or rewiring of links from one node to another. We show that depending on the frequency of these processes two topologically different networks can emerge, the connectivity distribution following either a generalized power-law or an exponential. We propose a continuum theory that predicts these two regimes as well as the scaling function and the exponents, in good agreement with the numerical results. Finally, we use the obtained predictions to fit the connectivity distribution of the network describing the professional links between movie actors., Comment: 13 pages, 3 figures

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scale-free distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems., Comment: 11 pages, 2 figures

Random networks with complex topology are common in Nature, describing systems as diverse as the world wide web or social and business networks. Recently, it has been demonstrated that most large networks for which topological information is available display scale-free features. Here we study the scaling properties of the recently introduced scale-free model, that can account for the observed power-law distribution of the connectivities. We develop a mean-field method to predict the growth dynamics of the individual vertices, and use this to calculate analytically the connectivity distribution and the scaling exponents. The mean-field method can be used to address the properties of two variants of the scale-free model, that do not display power-law scaling., Comment: 19 pages, 6 figures

Despite its increasing role in communication, the world wide web remains the least controlled medium: any individual or institution can create websites with unrestricted number of documents and links. While great efforts are made to map and characterize the Internet's infrastructure, little is known about the topology of the web. Here we take a first step to fill this gap: we use local connectivity measurements to construct a topological model of the world wide web, allowing us to explore and characterize its large scale properties., Comment: 5 pages, 1 figure, updated with most recent results on the size of the www

While there have been important theoretical advances in understanding the universality classes of interfaces moving in porous media, the developed tools cannot be directly applied to experiments. Here we introduce a method that can identify the universality class from snapshots of the interface profile. We test the method on discrete models whose universality class is well known, and use it to identify the universality class of interfaces obtained in experiments on fluid flow in porous media., Comment: 4 pages, 5 figures