Robustness and fragility of Boolean models for genetic regulatory networks

A theory for the arrangement of sensory organs in Drosophila

We study the arrangements of recurved bristles on the anterior wing margin of wild-type and mutant Drosophila. The epidermal or neural fate of a proneural cell depends on the concentrations of proteins of the achaete-scute complex. At puparium formation, concentrations of proteins are nearly identical in all cells of the anterior wing and each cell has the potential for neural fate. In wild-type flies, the action of regulatory networks drives the initial state to one where a bristle grows out of every fifth cell. Recent experiments have shown that the frequency of recurved bristles can be made to change by adjusting the mean concentrations of the zinc-finger transcription factor Senseless and the micro-RNA miR-9a. Specifically, mutant flies with reduced levels of miR-9a exhibit ectopic bristles, and those with lower levels of both miR-9a and Senseless show regular organization of recurved bristles, but with a lower periodicity of 4. We argue that these characteristics can be explained assuming an underlying Turing-type bifurcation whereby a periodic pattern spontaneously emerges from a uniform background. However, bristle patterns occur in a discrete array of cells, and are not mediated by diffusion. We argue that intracellular actions of transmembrane proteins such as Delta and Notch can play a role of diffusion in destabilizing the homogeneous state. In contrast to diffusion, intercellular actions can be activating or inhibiting; further, there can be lateral cross-species interactions. We introduce a phenomenological model to study bristle arrangements and make several model-independent predictions that can be tested in experiments. In our theory, miRNA-9a is one of the components of the underlying network and has no special regulatory role. The loss of periodicity in its absence is due to the transfer of the system to a bistable state.

Inference of gene regulatory networks using boolean-network inference methods

The modeling of genetic networks especially from microarray and related data has become an important aspect of the biosciences. This review takes a fresh look at a specific family of models used for constructing genetic networks, the so-called Boolean networks. The review outlines the various different types of Boolean network developed to date, from the original Random Boolean Network to the current Probabilistic Boolean Network. In addition, some of the different inference methods available to infer these genetic networks are also examined. Where possible, particular attention is paid to input requirements as well as the efficiency, advantages and drawbacks of each method. Though the Boolean network model is one of many models available for network inference today, it is well established and remains a topic of considerable interest in the field of genetic network inference. Hybrids of Boolean networks with other approaches may well be the way forward in inferring the most informative networks.

A Boolean model of the Pseudomonas syringae hrp regulon predicts a tightly regulated system

The Type III secretion system (TTSS) is a protein secretion machinery used by certain gram-negative bacterial pathogens of plants and animals to deliver effector molecules to the host and is at the core of the ability to cause disease. Extensive molecular and biochemical study has revealed the components and their interactions within this system but reductive approaches do not consider the dynamical properties of the system as a whole. In order to gain a better understanding of these dynamical behaviours and to create a basis for the refinement of the experimentally derived knowledge we created a Boolean model of the regulatory interactions within the hrp regulon of Pseudomonas syringae pathovar tomato strain DC3000 Pseudomonas syringae. We compared simulations of the model with experimental data and found them to be largely in accordance, though the hrpV node shows some differences in state changes to that expected. Our simulations also revealed interesting dynamical properties not previously predicted. The model predicts that the hrp regulon is a biologically stable two-state system, with each of the stable states being strongly attractive, a feature indicative of selection for a tightly regulated and responsive system. The model predicts that the state of the GacS/GacA node confers control, a prediction that is consistent with experimental observations that the protein has a role as master regulator. Simulated gene "knock out" experiments with the model predict that HrpL is a central information processing point within the network.

Timing robustness in the budding and fission yeast cell cycles

Robustness of biological models has emerged as an important principle in systems biology. Many past analyses of Boolean models update all pending changes in signals simultaneously (i.e., synchronously), making it impossible to consider robustness to variations in timing that result from noise and different environmental conditions. We checked previously published mathematical models of the cell cycles of budding and fission yeast for robustness to timing variations by constructing Boolean models and analyzing them using model-checking software for the property of speed independence. Surprisingly, the models are nearly, but not totally, speed-independent. In some cases, examination of timing problems discovered in the analysis exposes apparent inaccuracies in the model. Biologically justified revisions to the model eliminate the timing problems. Furthermore, in silico random mutations in the regulatory interactions of a speed-independent Boolean model are shown to be unlikely to preserve speed independence, even in models that are otherwise functional, providing evidence for selection pressure to maintain timing robustness. Multiple cell cycle models exhibit strong robustness to timing variation, apparently due to evolutionary pressure. Thus, timing robustness can be a basis for generating testable hypotheses and can focus attention on aspects of a model that may need refinement.

Discrete logic modelling as a means to link protein signalling networks with functional analysis of mammalian signal transduction

Large-scale protein signalling networks are useful for exploring complex biochemical pathways but do not reveal how pathways respond to specific stimuli. Such specificity is critical for understanding disease and designing drugs. Here we describe a computational approach—implemented in the free CNO software—for turning signalling networks into logical models and calibrating the models against experimental data. When a literature-derived network of 82 proteins covering the immediate-early responses of human cells to seven cytokines was modelled, we found that training against experimental data dramatically increased predictive power, despite the crudeness of Boolean approximations, while significantly reducing the number of interactions. Thus, many interactions in literature-derived networks do not appear to be functional in the liver cells from which we collected our data. At the same time, CNO identified several new interactions that improved the match of model to data. Although missing from the starting network, these interactions have literature support. Our approach, therefore, represents a means to generate predictive, cell-type-specific models of mammalian signalling from generic protein signalling networks.

Robustness in regulatory interaction networks. A generic approach with applications at different levels: physiologic, metabolic and genetic

Regulatory interaction networks are often studied on their dynamical side (existence of attractors, study of their stability). We focus here also on their robustness, that is their ability to offer the same spatiotemporal patterns and to resist to external perturbations such as losses of nodes or edges in the networks interactions architecture, changes in their environmental boundary conditions as well as changes in the update schedule (or updating mode) of the states of their elements (e.g., if these elements are genes, their synchronous coexpression mode versus their sequential expression). We define the generic notions of boundary, core, and critical vertex or edge of the underlying interaction graph of the regulatory network, whose disappearance causes dramatic changes in the number and nature of attractors (e.g., passage from a bistable behaviour to a unique periodic regime) or in the range of their basins of stability. The dynamic transition of states will be presented in the framework of threshold Boolean automata rules. A panorama of applications at different levels will be given: brain and plant morphogenesis, bulbar cardio-respiratory regulation, glycolytic/oxidative metabolic coupling, and eventually cell cycle and feather morphogenesis genetic control.

Modelling the Drosophila embryo

I provide a historical overview on the use of mathematical models to gain insight into pattern formation during early development of the fruit fly Drosophila melanogaster. It is my intention to illustrate how the aims and methodology of modelling have changed from the early beginnings of a theoretical developmental biology in the 1960s to modern-day systems biology. I show that even early modelling attempts addressed interesting and relevant questions, which were not tractable by experimental approaches. Unfortunately, their validation was severely hampered by a lack of specificity and appropriate experimental evidence. There is a simple lesson to be learned from this: we cannot deduce general rules for pattern formation from first principles or spurious reproduction of developmental phenomena. Instead, we must infer such rules (if any) from detailed and accurate studies of specific developmental systems. To achieve this, mathematical modelling must be closely integrated with experimental approaches. I report on progress that has been made in this direction in the past few years and illustrate the kind of novel insights that can be gained from such combined approaches. These insights demonstrate the great potential (and some pitfalls) of an integrative, systems-level investigation of pattern formation.

Transforming Boolean models to continuous models: methodology and application to T-cell receptor signaling

Background

The understanding of regulatory and signaling networks has long been a core objective in Systems Biology. Knowledge about these networks is mainly of qualitative nature, which allows the construction of Boolean models, where the state of a component is either 'off' or 'on'. While often able to capture the essential behavior of a network, these models can never reproduce detailed time courses of concentration levels.

Nowadays however, experiments yield more and more quantitative data. An obvious question therefore is how qualitative models can be used to explain and predict the outcome of these experiments.

Results

In this contribution we present a canonical way of transforming Boolean into continuous models, where the use of multivariate polynomial interpolation allows transformation of logic operations into a system of ordinary differential equations (ODE). The method is standardized and can readily be applied to large networks. Other, more limited approaches to this task are briefly reviewed and compared. Moreover, we discuss and generalize existing theoretical results on the relation between Boolean and continuous models. As a test case a logical model is transformed into an extensive continuous ODE model describing the activation of T-cells. We discuss how parameters for this model can be determined such that quantitative experimental results are explained and predicted, including time-courses for multiple ligand concentrations and binding affinities of different ligands. This shows that from the continuous model we may obtain biological insights not evident from the discrete one.

Conclusion

The presented approach will facilitate the interaction between modeling and experiments. Moreover, it provides a straightforward way to apply quantitative analysis methods to qualitatively described systems.

Logical modelling of cell cycle control in eukaryotes: a comparative study

Dynamical modelling is at the core of the systems biology paradigm. However, the development of comprehensive quantitative models is complicated by the daunting complexity of regulatory networks controlling crucial biological processes such as cell division, the paucity of currently available quantitative data, as well as the limited reproducibility of large-scale experiments. In this context, qualitative modelling approaches offer a useful alternative or complementary framework to build and analyse simplified, but still rigorous dynamical models. This point is illustrated here by analysing recent logical models of the molecular network controlling mitosis in different organisms, from yeasts to mammals. After a short introduction covering cell cycle and logical modelling, we compare the assumptions and properties underlying these different models. Next, leaning on their transposition into a common logical framework, we compare their functional structure in terms of regulatory circuits. Finally, we discuss assets and prospects of qualitative approaches for the modelling of the cell cycle.

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs.

Geometry and topology of parameter space: investigating measures of robustness in regulatory networks

The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.

The logic of EGFR/ErbB signaling: theoretical properties and analysis of high-throughput data

The epidermal growth factor receptor (EGFR) signaling pathway is probably the best-studied receptor system in mammalian cells, and it also has become a popular example for employing mathematical modeling to cellular signaling networks. Dynamic models have the highest explanatory and predictive potential; however, the lack of kinetic information restricts current models of EGFR signaling to smaller sub-networks. This work aims to provide a large-scale qualitative model that comprises the main and also the side routes of EGFR/ErbB signaling and that still enables one to derive important functional properties and predictions. Using a recently introduced logical modeling framework, we first examined general topological properties and the qualitative stimulus-response behavior of the network. With species equivalence classes, we introduce a new technique for logical networks that reveals sets of nodes strongly coupled in their behavior. We also analyzed a model variant which explicitly accounts for uncertainties regarding the logical combination of signals in the model. The predictive power of this model is still high, indicating highly redundant sub-structures in the network. Finally, one key advance of this work is the introduction of new techniques for assessing high-throughput data with logical models (and their underlying interaction graph). By employing these techniques for phospho-proteomic data from primary hepatocytes and the HepG2 cell line, we demonstrate that our approach enables one to uncover inconsistencies between experimental results and our current qualitative knowledge and to generate new hypotheses and conclusions. Our results strongly suggest that the Rac/Cdc42 induced p38 and JNK cascades are independent of PI3K in both primary hepatocytes and HepG2. Furthermore, we detected that the activation of JNK in response to neuregulin follows a PI3K-dependent signaling pathway.

Logical modelling of regulatory networks with GINsim 2.3

Many important problems in cell biology require the consideration of dense nonlinear interactions between functional modules. The requirement of computer simulation for the understanding of cellular processes is now widely accepted, and a variety of modelling frameworks have been designed to meet this need. Here, we present a novel public release of the Gene Interaction Network simulation suite (GINsim), a software designed for the qualitative modelling and analysis of regulatory networks. The main functionalities of GINsim are illustrated through the analysis of a logical model for the core network controlling the fission yeast cell cycle. The last public release of GINsim (version 2.3), as well as development versions, can be downloaded from the dedicated website (http://gin.univ-mrs.fr/GINsim/), which further includes a model library, along with detailed tutorial and user manual.

Fuzzy logic analysis of kinase pathway crosstalk in TNF/EGF/insulin-induced signaling

When modeling cell signaling networks, a balance must be struck between mechanistic detail and ease of interpretation. In this paper we apply a fuzzy logic framework to the analysis of a large, systematic dataset describing the dynamics of cell signaling downstream of TNF, EGF, and insulin receptors in human colon carcinoma cells. Simulations based on fuzzy logic recapitulate most features of the data and generate several predictions involving pathway crosstalk and regulation. We uncover a relationship between MK2 and ERK pathways that might account for the previously identified pro-survival influence of MK2. We also find unexpected inhibition of IKK following EGF treatment, possibly due to down-regulation of autocrine signaling. More generally, fuzzy logic models are flexible, able to incorporate qualitative and noisy data, and powerful enough to produce quantitative predictions and new biological insights about the operation of signaling networks.

Cells use networks of interacting proteins to interpret intra-cellular state and extra-cellular cues and to execute cell-fate decisions. Even when individual proteins are well understood at a molecular level, the dynamics and behavior of networks as a whole are harder to understand. However, deciphering the operation of such networks is key to understanding disease processes and therapeutic opportunities. As a means to study signaling networks, we have modified and applied a fuzzy logic approach originally developed for industrial control. We use fuzzy logic to model the responses of colon cancer cells in culture to combinations of pro-survival and pro-death cytokines, making it possible to interpret quantitative data in the context of abstract information drawn from the literature. Our work establishes that fuzzy logic can be used to understand complex signaling pathways with respect to multi-factorial activity-based protein data and prior knowledge.

Logical analysis of the budding yeast cell cycle

The budding yeast Saccharomyces cerevisiae is a model organism that is commonly used to investigate control of the eukaryotic cell cycle. Moreover, because of the extensive experimental data on wild type and mutant phenotypes, it is also particularly suitable for mathematical modelling and analysis. Here, I present a new Boolean model of the budding yeast cell cycle. This model is consistent with a wide range of wild type and mutant phenotypes and shows remarkable robustness against perturbations, both to reaction times and the states of component genes/proteins. Because of its simple logical nature, the model is suitable for sub-network analysis, which can be used to identify a four node core regulatory circuit underlying cell cycle regulation. Sub-network analysis can also be used to identify key sub-dynamics that are essential for viable cell cycle control, as well as identifying the sub-dynamics that are most variable between different mutants.

Shape, size, and robustness: feasible regions in the parameter space of biochemical networks

The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

Developing models with a large number of parameters for describing the dynamics of a biochemical network is a common exercise today. The dependence of predictions of such a network model on the choice of parameters is important to understand for two reasons. For the purpose of fitting biological data and making predictions, we need to know which combinations of parameters are strongly constrained by observations and also which combinations seriously affect a particular prediction. In addition, we expect naturally evolved networks to be somewhat robust to parameter changes. If the functioning of the network requires fine-tuning in many parameters, then mutations causing changes in regulatory interactions could quickly make the network dysfunctional. For predictions involving gene products being ON or OFF, we found a method that facilitates the study parameter dependence. As an example, we analyzed several competing models of the segment polarity network in Drosophila. We explicitly describe the region in the parameter space where the wild-type expression pattern of key genes becomes feasible for each model. We also study how random walks in the parameter space exit from the feasible region of a network model, allowing us to compare the relative robustness of the alternative models.

Structural properties of gene regulatory networks: definitions and connections

The study of gene regulatory networks is a significant problem in systems biology. Of particular interest is the problem of determining the unknown or hidden higher level regulatory signals by using gene expression data from DNA microarray experiments. Several studies in this area have demonstrated the critical aspect of the network structure in tackling the network modelling problem. Structural analysis of systems has proved useful in a number of contexts, viz., observability, controllability, fault diagnosis, sparse matrix computations etc. In this contribution, we formally define structural properties that are relevant to Gene Regulatory Networks. We explore the structural implications of certain quantitative methods and explain completely the connections between the identifiability conditions and structural criteria of observability and distinguishability. We illustrate these concepts in case studies using representative biologically motivated network examples. The present work bridges the quantitative modelling methods with those based on the structural analysis.

Deterministic and stochastic models of genetic regulatory networks

Traditionally molecular biology research has tended to reduce biological pathways to composite units studied as isolated parts of the cellular system. With the advent of high throughput methodologies that can capture thousands of data points, and powerful computational approaches, the reality of studying cellular processes at a systems level is upon us. As these approaches yield massive datasets, systems level analyses have drawn upon other fields such as engineering and mathematics, adapting computational and statistical approaches to decipher relationships between molecules. Guided by high quality datasets and analyses, one can begin the process of predictive modeling. The findings from such approaches are often surprising and beyond normal intuition. We discuss four classes of dynamical systems used to model genetic regulatory networks. The discussion is divided into continuous and discrete models, as well as deterministic and stochastic model classes. For each combination of these categories, a model is presented and discussed in the context of the yeast cell cycle, illustrating how different types of questions can be addressed by different model classes.

Discrete dynamic modeling with asynchronous update, or how to model complex systems in the absence of quantitative information

A major aim of systems biology is the study of the inter-relationships found within and between large biological data sets. Here we describe one systems biology method, in which the tools of network analysis and discrete dynamic (Boolean) modeling are used to develop predictive models of cellular signaling in cases where detailed temporal and kinetic information regarding the propagation of the signal through the system is lacking. This approach is also applicable to data sets derived from some other types of biological systems, such as transcription factor-mediated regulation of gene expression during the control of developmental fate, or host defense responses following pathogen attack, and is equally applicable to plant and non-plant systems. The method also allows prediction of how elimination of one or more individual signaling components will affect the ultimate outcome, thus allowing the researcher to model the effects of genetic knockout or pharmacological block. The method also serves as a starting point from which more quantitative models can be developed as additional information becomes available.

Discrete dynamic modeling of cellular signaling networks

Understanding signal transduction in cellular systems is a central issue in systems biology. Numerous experiments from different laboratories generate an abundance of individual components and causal interactions mediating environmental and developmental signals. However, for many signal transduction systems there is insufficient information on the overall structure and the molecular mechanisms involved in the signaling network. Moreover, lack of kinetic and temporal information makes it difficult to construct quantitative models of signal transduction pathways. Discrete dynamic modeling, combined with network analysis, provides an effective way to integrate fragmentary knowledge of regulatory interactions into a predictive mathematical model which is able to describe the time evolution of the system without the requirement for kinetic parameters. This chapter introduces the fundamental concepts of discrete dynamic modeling, particularly focusing on Boolean dynamic models. We describe this method step-by-step in the context of cellular signaling networks. Several variants of Boolean dynamic models including threshold Boolean networks and piecewise linear systems are also covered, followed by two examples of successful application of discrete dynamic modeling in cell biology.

Algebraic models of biochemical networks

With the rise of systems biology as an important paradigm in the life sciences and the availability and increasingly good quality of high-throughput molecular data, the role of mathematical models has become central in the understanding of the relationship between structure and function of organisms. This chapter focuses on a particular type of models, so-called algebraic models, which are generalizations of Boolean networks. It provides examples of such models and discusses several available methods to construct such models from high-throughput time course data. One specific such method, Polynome, is discussed in detail.

Dynamical modeling of the cholesterol regulatory pathway with Boolean networks

BACKGROUND

Qualitative dynamics of small gene regulatory networks have been studied in quite some details both with synchronous and asynchronous analysis. However, both methods have their drawbacks: synchronous analysis leads to spurious attractors and asynchronous analysis lacks computational efficiency, which is a problem to simulate large networks. We addressed this question through the analysis of a major biosynthesis pathway. Indeed the cholesterol synthesis pathway plays a pivotal role in dislypidemia and, ultimately, in cancer through intermediates such as mevalonate, farnesyl pyrophosphate and geranyl geranyl pyrophosphate, but no dynamic model of this pathway has been proposed until now.

RESULTS

We set up a computational framework to dynamically analyze large biological networks. This framework associates a classical and computationally efficient synchronous Boolean analysis with a newly introduced method based on Markov chains, which identifies spurious cycles among the results of the synchronous simulation. Based on this method, we present here the results of the analysis of the cholesterol biosynthesis pathway and its physiological regulation by the Sterol Response Element Binding Proteins (SREBPs), as well as the modeling of the action of statins, inhibitor drugs, on this pathway. The in silico experiments show the blockade of the cholesterol endogenous synthesis by statins and its regulation by SREPBs, in full agreement with the known biochemical features of the pathway.

CONCLUSION

We believe that the method described here to identify spurious cycles opens new routes to compute large and biologically relevant models, thanks to the computational efficiency of synchronous simulation. Furthermore, to the best of our knowledge, we present here the first dynamic systems biology model of the human cholesterol pathway and several of its key regulatory control elements, hoping it would provide a good basis to perform in silico experiments and confront the resulting properties with published and experimental data. The model of the cholesterol pathway and its regulation, along with Boolean formulae used for simulation are available on our web site http://Bioinformaticsu613.free.fr. Graphical results of the simulation are also shown online. The SBML model is available in the BioModels database http://www.ebi.ac.uk/biomodels/ with submission ID: MODEL0568648427.

Boolean network simulations for life scientists

Modern life sciences research increasingly relies on computational solutions, from large scale data analyses to theoretical modeling. Within the theoretical models Boolean networks occupy an increasing role as they are eminently suited at mapping biological observations and hypotheses into a mathematical formalism. The conceptual underpinnings of Boolean modeling are very accessible even without a background in quantitative sciences, yet it allows life scientists to describe and explore a wide range of surprisingly complex phenomena. In this paper we provide a clear overview of the concepts used in Boolean simulations, present a software library that can perform these simulations based on simple text inputs and give three case studies. The large scale simulations in these case studies demonstrate the Boolean paradigms and their applicability as well as the advanced features and complex use cases that our software package allows. Our software is distributed via a liberal Open Source license and is freely accessible from

Network model of survival signaling in large granular lymphocyte leukemia

T cell large granular lymphocyte (T-LGL) leukemia features a clonal expansion of antigen-primed, competent, cytotoxic T lymphocytes (CTL). To systematically understand signaling components that determine the survival of CTL in T-LGL leukemia, we constructed a T-LGL survival signaling network by integrating the signaling pathways involved in normal CTL activation and the known deregulations of survival signaling in leukemic T-LGL. This network was subsequently translated into a predictive, discrete, dynamic model. Our model suggests that the persistence of IL-15 and PDGF is sufficient to reproduce all known deregulations in leukemic T-LGL. This finding leads to the following predictions: (i) Inhibiting PDGF signaling induces apoptosis in leukemic T-LGL. (ii) Sphingosine kinase 1 and NFkappaB are essential for the long-term survival of CTL in T-LGL leukemia. (iii) NFkappaB functions downstream of PI3K and prevents apoptosis through maintaining the expression of myeloid cell leukemia sequence 1. (iv) T box expressed in T cells (T-bet) should be constitutively activated concurrently with NFkappaB activation to reproduce the leukemic T-LGL phenotype. We validated these predictions experimentally. Our study provides a model describing the signaling network involved in maintaining the long-term survival of competent CTL in humans. The model will be useful in identifying potential therapeutic targets for T-LGL leukemia and generating long-term competent CTL necessary for tumor and cancer vaccine development.

Studying the effect of cell division on expression patterns of the segment polarity genes

The segment polarity gene family, and its gene regulatory network, is at the basis of Drosophila embryonic development. The network's capacity for generating and robustly maintaining a specific gene expression pattern has been investigated through mathematical modelling. The models have provided several useful insights by suggesting essential network links, or uncovering the importance of the relative time scales of different biological processes in the formation of the segment polarity genes' expression patterns. But the developmental pattern formation process raises many other questions. Two of these questions are analysed here: the dependence of the signalling protein sloppy paired on the segment polarity genes and the effect of cell division on the segment polarity genes' expression patterns. This study suggests that cell division increases the robustness of the segment polarity network with respect to perturbations in biological processes.

Biological switches and clocks

To introduce this special issue on biological switches and clocks, we review the historical development of mathematical models of bistability and oscillations in chemical reaction networks. In the 1960s and 1970s, these models were limited to well-studied biochemical examples, such as glycolytic oscillations and cyclic AMP signalling. After the molecular genetics revolution of the 1980s, the field of molecular cell biology was thrown wide open to mathematical modellers. We review recent advances in modelling the gene-protein interaction networks that control circadian rhythms, cell cycle progression, signal processing and the design of synthetic gene networks.

The effect of negative feedback loops on the dynamics of boolean networks

Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distance-to-positive-feedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distance-to-positive-feedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.

The signaling petri net-based simulator: a non-parametric strategy for characterizing the dynamics of cell-specific signaling networks

Reconstructing cellular signaling networks and understanding how they work are major endeavors in cell biology. The scale and complexity of these networks, however, render their analysis using experimental biology approaches alone very challenging. As a result, computational methods have been developed and combined with experimental biology approaches, producing powerful tools for the analysis of these networks. These computational methods mostly fall on either end of a spectrum of model parameterization. On one end is a class of structural network analysis methods; these typically use the network connectivity alone to generate hypotheses about global properties. On the other end is a class of dynamic network analysis methods; these use, in addition to the connectivity, kinetic parameters of the biochemical reactions to predict the network's dynamic behavior. These predictions provide detailed insights into the properties that determine aspects of the network's structure and behavior. However, the difficulty of obtaining numerical values of kinetic parameters is widely recognized to limit the applicability of this latter class of methods. Several researchers have observed that the connectivity of a network alone can provide significant insights into its dynamics. Motivated by this fundamental observation, we present the signaling Petri net, a non-parametric model of cellular signaling networks, and the signaling Petri net-based simulator, a Petri net execution strategy for characterizing the dynamics of signal flow through a signaling network using token distribution and sampling. The result is a very fast method, which can analyze large-scale networks, and provide insights into the trends of molecules' activity-levels in response to an external stimulus, based solely on the network's connectivity. We have implemented the signaling Petri net-based simulator in the PathwayOracle toolkit, which is publicly available at http://bioinfo.cs.rice.edu/pathwayoracle. Using this method, we studied a MAPK1,2 and AKT signaling network downstream from EGFR in two breast tumor cell lines. We analyzed, both experimentally and computationally, the activity level of several molecules in response to a targeted manipulation of TSC2 and mTOR-Raptor. The results from our method agreed with experimental results in greater than 90% of the cases considered, and in those where they did not agree, our approach provided valuable insights into discrepancies between known network connectivities and experimental observations.

Quantitative analysis of robustness and fragility in biological networks based on feedback dynamics

MOTIVATION

It has been widely reported that biological networks are robust against perturbations such as mutations. On the contrary, it has also been known that biological networks are often fragile against unexpected mutations. There is a growing interest in these intriguing observations and the underlying design principle that causes such robust but fragile characteristics of biological networks. For relatively small networks, a feedback loop has been considered as an important motif for realizing the robustness. It is still, however, not clear how a number of coupled feedback loops actually affect the robustness of large complex biological networks. In particular, the relationship between fragility and feedback loops has not yet been investigated till now.

RESULTS

Through extensive computational experiments, we found that networks with a larger number of positive feedback loops and a smaller number of negative feedback loops are likely to be more robust against perturbations. Moreover, we found that the nodes of a robust network subject to perturbations are mostly involved with a smaller number of feedback loops compared with the other nodes not usually subject to perturbations. This topological characteristic eventually makes the robust network fragile against unexpected mutations at the nodes not previously exposed to perturbations.

Host-pathogen systems biology: logical modelling of hepatocyte growth factor and Helicobacter pylori induced c-Met signal transduction

Background

The hepatocyte growth factor (HGF) stimulates mitogenesis, motogenesis, and morphogenesis in a wide range of tissues, including epithelial cells, on binding to the receptor tyrosine kinase c-Met. Abnormal c-Met signalling contributes to tumour genesis, in particular to the development of invasive and metastatic phenotypes. The human microbial pathogen Helicobacter pylori can induce chronic gastritis, peptic ulceration and more rarely, gastric adenocarcinoma. The H. pylori effector protein cytotoxin associated gene A (CagA), which is translocated via a type IV secretion system (T4SS) into epithelial cells, intracellularly modulates the c-Met receptor and promotes cellular processes leading to cell scattering, which could contribute to the invasiveness of tumour cells. Using a logical modelling framework, the presented work aims at analysing the c-Met signal transduction network and how it is interfered by H. pylori infection, which might be of importance for tumour development.

Results

A logical model of HGF and H. pylori induced c-Met signal transduction is presented in this work. The formalism of logical interaction hypergraphs (LIH) was used to construct the network model. The molecular interactions included in the model were all assembled manually based on a careful meta-analysis of published experimental results. Our model reveals the differences and commonalities of the response of the network upon HGF and H. pylori induced c-Met signalling. As another important result, using the formalism of minimal intervention sets, phospholipase Cγ1 (PLCγ1) was identified as knockout target for repressing the activation of the extracellular signal regulated kinase 1/2 (ERK1/2), a signalling molecule directly linked to cell scattering in H. pylori infected cells. The model predicted only an effect on ERK1/2 for the H. pylori stimulus, but not for HGF treatment. This result could be confirmed experimentally in MDCK cells using a specific pharmacological inhibitor against PLCγ1. The in silico predictions for the knockout of two other network components were also verified experimentally.

Conclusion

This work represents one of the first approaches in the direction of host-pathogen systems biology aiming at deciphering signalling changes brought about by pathogenic bacteria. The suitability of our network model is demonstrated by an in silico prediction of a relevant target against pathogen infection.

Dynamic simulation of regulatory networks using SQUAD

BACKGROUND

The ambition of most molecular biologists is the understanding of the intricate network of molecular interactions that control biological systems. As scientists uncover the components and the connectivity of these networks, it becomes possible to study their dynamical behavior as a whole and discover what is the specific role of each of their components. Since the behavior of a network is by no means intuitive, it becomes necessary to use computational models to understand its behavior and to be able to make predictions about it. Unfortunately, most current computational models describe small networks due to the scarcity of kinetic data available. To overcome this problem, we previously published a methodology to convert a signaling network into a dynamical system, even in the total absence of kinetic information. In this paper we present a software implementation of such methodology.

RESULTS

We developed SQUAD, a software for the dynamic simulation of signaling networks using the standardized qualitative dynamical systems approach. SQUAD converts the network into a discrete dynamical system, and it uses a binary decision diagram algorithm to identify all the steady states of the system. Then, the software creates a continuous dynamical system and localizes its steady states which are located near the steady states of the discrete system. The software permits to make simulations on the continuous system, allowing for the modification of several parameters. Importantly, SQUAD includes a framework for perturbing networks in a manner similar to what is performed in experimental laboratory protocols, for example by activating receptors or knocking out molecular components. Using this software we have been able to successfully reproduce the behavior of the regulatory network implicated in T-helper cell differentiation.

CONCLUSION

The simulation of regulatory networks aims at predicting the behavior of a whole system when subject to stimuli, such as drugs, or determine the role of specific components within the network. The predictions can then be used to interpret and/or drive laboratory experiments. SQUAD provides a user-friendly graphical interface, accessible to both computational and experimental biologists for the fast qualitative simulation of large regulatory networks for which kinetic data is not necessarily available.

Identifying dynamical modules from genetic regulatory systems: applications to the segment polarity network

Background

It is widely accepted that genetic regulatory systems are 'modular', in that the whole system is made up of smaller 'subsystems' corresponding to specific biological functions. Most attempts to identify modules in genetic regulatory systems have relied on the topology of the underlying network. However, it is the temporal activity (dynamics) of genes and proteins that corresponds to biological functions, and hence it is dynamics that we focus on here for identifying subsystems.

Results

Using Boolean network models as an exemplar, we present a new technique to identify subsystems, based on their dynamical properties. The main part of the method depends only on the stable dynamics (attractors) of the system, thus requiring no prior knowledge of the underlying network. However, knowledge of the logical relationships between the network components can be used to describe how each subsystem is regulated. To demonstrate its applicability to genetic regulatory systems, we apply the method to a model of the Drosophila segment polarity network, providing a detailed breakdown of the system.

Conclusion

We have designed a technique for decomposing any set of discrete-state, discrete-time attractors into subsystems. Having a suitable mathematical model also allows us to describe how each subsystem is regulated and how robust each subsystem is against perturbations. However, since the subsystems are found directly from the attractors, a mathematical model or underlying network topology is not necessarily required to identify them, potentially allowing the method to be applied directly to experimental expression data.

A logical model provides insights into T cell receptor signaling

Cellular decisions are determined by complex molecular interaction networks. Large-scale signaling networks are currently being reconstructed, but the kinetic parameters and quantitative data that would allow for dynamic modeling are still scarce. Therefore, computational studies based upon the structure of these networks are of great interest. Here, a methodology relying on a logical formalism is applied to the functional analysis of the complex signaling network governing the activation of T cells via the T cell receptor, the CD4/CD8 co-receptors, and the accessory signaling receptor CD28. Our large-scale Boolean model, which comprises 94 nodes and 123 interactions and is based upon well-established qualitative knowledge from primary T cells, reveals important structural features (e.g., feedback loops and network-wide dependencies) and recapitulates the global behavior of this network for an array of published data on T cell activation in wild-type and knock-out conditions. More importantly, the model predicted unexpected signaling events after antibody-mediated perturbation of CD28 and after genetic knockout of the kinase Fyn that were subsequently experimentally validated. Finally, we show that the logical model reveals key elements and potential failure modes in network functioning and provides candidates for missing links. In summary, our large-scale logical model for T cell activation proved to be a promising in silico tool, and it inspires immunologists to ask new questions. We think that it holds valuable potential in foreseeing the effects of drugs and network modifications.

Representing perturbed dynamics in biological network models

We study the dynamics of gene activities in relatively small size biological networks (up to a few tens of nodes), e.g., the activities of cell-cycle proteins during the mitotic cell-cycle progression. Using the framework of deterministic discrete dynamical models, we characterize the dynamical modifications in response to structural perturbations in the network connectivities. In particular, we focus on how perturbations affect the set of fixed points and sizes of the basins of attraction. Our approach uses two analytical measures: the basin entropy H and the perturbation size Delta , a quantity that reflects the distance between the set of fixed points of the perturbed network and that of the unperturbed network. Applying our approach to the yeast-cell-cycle network introduced by Li [Proc. Natl. Acad. Sci. U.S.A. 101, 4781 (2004)] provides a low-dimensional and informative fingerprint of network behavior under large classes of perturbations. We identify interactions that are crucial for proper network function, and also pinpoint functionally redundant network connections. Selected perturbations exemplify the breadth of dynamical responses in this cell-cycle model.

Robustness can evolve gradually in complex regulatory gene networks with varying topology

The topology of cellular circuits (the who-interacts-with-whom) is key to understand their robustness to both mutations and noise. The reason is that many biochemical parameters driving circuit behavior vary extensively and are thus not fine-tuned. Existing work in this area asks to what extent the function of any one given circuit is robust. But is high robustness truly remarkable, or would it be expected for many circuits of similar topology? And how can high robustness come about through gradual Darwinian evolution that changes circuit topology gradually, one interaction at a time? We here ask these questions for a model of transcriptional regulation networks, in which we explore millions of different network topologies. Robustness to mutations and noise are correlated in these networks. They show a skewed distribution, with a very small number of networks being vastly more robust than the rest. All networks that attain a given gene expression state can be organized into a graph whose nodes are networks that differ in their topology. Remarkably, this graph is connected and can be easily traversed by gradual changes of network topologies. Thus, robustness is an evolvable property. This connectedness and evolvability of robust networks may be a general organizational principle of biological networks. In addition, it exists also for RNA and protein structures, and may thus be a general organizational principle of all biological systems.

Modeling systems-level regulation of host immune responses

Many pathogens are able to manipulate the signaling pathways responsible for the generation of host immune responses. Here we examine and model a respiratory infection system in which disruption of host immune functions or of bacterial factors changes the dynamics of the infection. We synthesize the network of interactions between host immune components and two closely related bacteria in the genus Bordetellae. We incorporate existing experimental information on the timing of immune regulatory events into a discrete dynamic model, and verify the model by comparing the effects of simulated disruptions to the experimental outcome of knockout mutations. Our model indicates that the infection time course of both Bordetellae can be separated into three distinct phases based on the most active immune processes. We compare and discuss the effect of the species-specific virulence factors on disrupting the immune response during their infection of naive, antibody-treated, diseased, or convalescent hosts. Our model offers predictions regarding cytokine regulation, key immune components, and clearance of secondary infections; we experimentally validate two of these predictions. This type of modeling provides new insights into the virulence, pathogenesis, and host adaptation of disease-causing microorganisms and allows systems-level analysis that is not always possible using traditional methods.

The immune response is a complex network of processes activated in a host upon infection. Pathogens seek to disrupt or evade these processes to ensure their own survival and proliferation. This article provides a systems-level analysis of the immune response against two related bacterial species in the Bordetella genus, B. bronchiseptica and B. pertussis. B. pertussis, the causative agent of whooping cough, has lost many of the virulence factors of its B. bronchiseptica–like progenitor, and is using different strategies for the modulation of the immune system. We have synthesized two separate network models for the interaction of these pathogens with their hosts. Each network is then translated into a predictive dynamic model and is validated by comparison with available experimental data. The model offers predictions regarding cytokine regulation and the effects of perturbations of the immune system, as well as the time course of infections in hosts that had previously encountered the pathogens. We experimentally validate the prediction that convalescent hosts can rapidly clear both pathogens, while antibody transfer cannot substantially reduce the duration of a B. pertussis infection. This type of modeling provides new insights into the virulence, pathogenesis, and host adaptation of disease-causing microorganisms and can be readily extended to other pathogens.

Monotone and near-monotone biochemical networks

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a "small" number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion.

The impact of function perturbations in Boolean networks

MOTIVATION

A network is said to be robust relative to a certain network characteristic if a small change in network structure does not significantly affect the characteristic. From the perspective of network stability, robustness is desirable; however, from the perspective of intervention to exert influence on network behavior, it is undesirable. For Boolean networks, there are two fundamental types of robustness. One type pertains to perturbing the state of the network and the other to perturbing the rule-based structure.

RESULTS

This article explores the impact of function perturbations in Boolean networks from two aspects: (1) analysis: predict the impact on network state transitions and attractors via analytical approaches or identify a perturbation by observing its consequences; (2) synthesis: preserve or modify the network characteristics, especially attractors, by introducing a judicious change to the functions. The results are applied to achieve intervention that structurally alters the network to achieve a more favorable steady-state distribution and to identify the function perturbation that has led to altered observed behavior. The intervention procedure is applied to a WNT5A network to reduce the risk of metastasis in melanoma, and the identification procedure is applied to a Drosophila melanogaster segmentation polarity gene network to identify regulatory function perturbation.

Predicting essential components of signal transduction networks: a dynamic model of guard cell abscisic acid signaling

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. Dozens of cellular components have been identified to function in ABA regulation of guard cell volume and thus of stomatal aperture, but a dynamic description is still not available for this complex process. 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 40 identified network components, and accords well with previous experimental results at both the pathway and whole-cell physiological level. By simulating gene disruptions and pharmacological interventions we find that the network is robust against a significant fraction of possible perturbations. 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 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. Simulations of stomatal response as derived from our model provide an efficient tool for the identification of candidate manipulations that have the best chance of conferring increased drought stress tolerance and for the prioritization of future wet bench analyses. Our method can be readily applied to other biological signaling networks to identify key regulatory components in systems where quantitative information is limited.

Superstability of the yeast cell-cycle dynamics: ensuring causality in the presence of biochemical stochasticity

Gene regulatory dynamics are governed by molecular processes and therefore exhibits an inherent stochasticity. However, for the survival of an organism it is a strict necessity that this intrinsic noise does not prevent robust functioning of the system. It is still an open question how dynamical stability is achieved in biological systems despite the omnipresent fluctuations. In this paper we investigate the cell cycle of the budding yeast Saccharomyces cerevisiae as an example of a well-studied organism. We study a genetic network model of 11 genes that coordinate the cell-cycle dynamics using a modeling framework which generalizes the concept of discrete threshold dynamics. By allowing for fluctuations in the process times, we introduce noise into the model, accounting for the effects of biochemical stochasticity. We study the dynamical attractor of the cell cycle and find a remarkable robustness against fluctuations of this kind. We identify mechanisms that ensure reliability in spite of fluctuations: 'Catcher states' and persistence of activity levels contribute significantly to the stability of the yeast cell cycle despite the inherent stochasticity.

Visual setup of logical models of signaling and regulatory networks with ProMoT

BACKGROUND

The analysis of biochemical networks using a logical (Boolean) description is an important approach in Systems Biology. Recently, new methods have been proposed to analyze large signaling and regulatory networks using this formalism. Even though there is a large number of tools to set up models describing biological networks using a biochemical (kinetic) formalism, however, they do not support logical models.

RESULTS

Herein we present a flexible framework for setting up large logical models in a visual manner with the software tool ProMoT. An easily extendible library, ProMoT's inherent modularity and object-oriented concept as well as adaptive visualization techniques provide a versatile environment. Both the graphical and the textual description of the logical model can be exported to different formats.

CONCLUSION

New features of ProMoT facilitate an efficient set-up of large Boolean models of biochemical interaction networks. The modeling environment is flexible; it can easily be adapted to specific requirements, and new extensions can be introduced. ProMoT is freely available from http://www.mpi-magdeburg.mpg.de/projects/promot/.

Few crucial links assure checkpoint efficiency in the yeast cell-cycle network

MOTIVATION

The ability of cells to complete mitosis with high fidelity relies on elaborate checkpoint mechanisms. We study S- and M-phase checkpoint responses in silico in the budding yeast with a stochastic dynamical model for the cell-cycle. We aim to provide an unbiased functional classification of network interactions that reflect the contribution of each link to checkpoint efficiency in the presence of cellular fluctuations.

RESULTS

We developed an algorithm BNetDyn to compute stochastic dynamical trajectories for an input gene network and its structural perturbations. User specified output measures like the mutual information between trigger and output nodes are then evaluated on the stationary state of the Markov process. Systematic perturbations of the yeast cell-cycle model by Li et al. classify each link according to its effect on checkpoint efficiencies and stabilities of the main cell-cycle phases. This points to the crosstalk in the cascades downstream of the SBF/MBF transcription activator complexes as determinant for checkpoint optimality; a finding that consistently reflects recent experiments. Finally our stochastic analysis emphasizes how dynamical stability in the yeast cell-cycle network crucially relies on backward inhibitory circuits next to forward induction.

AVAILABILITY

C++ source code and network models can be downloaded at http://www.vital-it.ch/Software/

Theodor Bücher Lecture. Metabolomics, modelling and machine learning in systems biology - towards an understanding of the languages of cells. Delivered on 3 July 2005 at the 30th FEBS Congress and the 9th IUBMB conference in Budapest

The newly emerging field of systems biology involves a judicious interplay between high-throughput 'wet' experimentation, computational modelling and technology development, coupled to the world of ideas and theory. This interplay involves iterative cycles, such that systems biology is not at all confined to hypothesis-dependent studies, with intelligent, principled, hypothesis-generating studies being of high importance and consequently very far from aimless fishing expeditions. I seek to illustrate each of these facets. Novel technology development in metabolomics can increase substantially the dynamic range and number of metabolites that one can detect, and these can be exploited as disease markers and in the consequent and principled generation of hypotheses that are consistent with the data and achieve this in a value-free manner. Much of classical biochemistry and signalling pathway analysis has concentrated on the analyses of changes in the concentrations of intermediates, with 'local' equations - such as that of Michaelis and Menten v=(Vmax x S)/(S+K m) - that describe individual steps being based solely on the instantaneous values of these concentrations. Recent work using single cells (that are not subject to the intellectually unsupportable averaging of the variable displayed by heterogeneous cells possessing nonlinear kinetics) has led to the recognition that some protein signalling pathways may encode their signals not (just) as concentrations (AM or amplitude-modulated in a radio analogy) but via changes in the dynamics of those concentrations (the signals are FM or frequency-modulated). This contributes in principle to a straightforward solution of the crosstalk problem, leads to a profound reassessment of how to understand the downstream effects of dynamic changes in the concentrations of elements in these pathways, and stresses the role of signal processing (and not merely the intermediates) in biological signalling. It is this signal processing that lies at the heart of understanding the languages of cells. The resolution of many of the modern and postgenomic problems of biochemistry requires the development of a myriad of new technologies (and maybe a new culture), and thus regular input from the physical sciences, engineering, mathematics and computer science. One solution, that we are adopting in the Manchester Interdisciplinary Biocentre (http://www.mib.ac.uk/) and the Manchester Centre for Integrative Systems Biology (http://www.mcisb.org/), is thus to colocate individuals with the necessary combinations of skills. Novel disciplines that require such an integrative approach continue to emerge. These include fields such as chemical genomics, synthetic biology, distributed computational environments for biological data and modelling, single cell diagnostics/bionanotechnology, and computational linguistics/text mining.