Logical analysis of systems comprising feedback loops

A system-level model reveals that transcriptional stochasticity is required for hematopoietic stem cell differentiation

HSCs differentiation has been difficult to study experimentally due to the high number of components and interactions involved, as well as the impact of diverse physiological conditions. From a 200-node network, that was grounded on experimental data, we derived a 21-node regulatory network by collapsing linear pathways and retaining the functional feedback loops. This regulatory network core integrates key nodes and interactions underlying HSCs differentiation, including transcription factors, metabolic, and redox signaling pathways. We used Boolean, continuous, and stochastic dynamic models to simulate the hypoxic conditions of the HSCs niche, as well as the patterns and temporal sequences of HSCs transitions and differentiation. Our findings indicate that HSCs differentiation is a plastic process in which cell fates can transdifferentiate among themselves. Additionally, we found that cell heterogeneity is fundamental for HSCs differentiation. Lastly, we found that oxygen activates ROS production, inhibiting quiescence and promoting growth and differentiation pathways of HSCs.

BDEtools: A MATLAB Package for Boolean Delay Equation Modeling

Boolean Delay Equations (BDEs) can simulate surprisingly complex behavior, despite their relative simplicity. In addition to steady-state dynamics, BDEs can also generate periodic and quasiperiodic oscillations, m:n frequency locking, and even chaos. Further, the enumerability of Boolean update functions and their compact parametrization means that BDEs can be leveraged to generate low-level descriptions of biological networks, from which more detailed formulations (e.g., differential equation models) can be constructed. However, although several studies have demonstrated the utility of BDE modeling in computational biology, a current barrier to the wider adoption of the BDE approach is the absence of freely available simulation software. In this work, we present BDEtools-an open-source MATLAB package for numerically solving BDE models. After giving a brief introduction to BDE modeling, we describe the package's solver algorithms, together with several utility functions that can be used to provide solver inputs and to process solver outputs. We also demonstrate the functionality of BDEtools by illustrating its application to an established model of a gene regulatory network that controls circadian rhythms. BDEtools makes it straightforward for researchers to quickly build reliable BDE models of biological networks. We hope that its ease of use and free availability will encourage more researchers to explore BDE formulations of their systems of interest. Through the continued use of BDEs by the computational biology community, we will, no doubt, discover their potential applicability to a broader class of biological networks.

Decoding the Role of Epigenetics in Breast Cancer Using Formal Modeling and Machine-Learning Methods

Breast carcinogenesis is known to be instigated by genetic and epigenetic modifications impacting multiple cellular signaling cascades, thus making its prevention and treatments a challenging endeavor. However, epigenetic modification, particularly DNA methylation-mediated silencing of key TSGs, is a hallmark of cancer progression. One such tumor suppressor gene (TSG) RUNX3 (Runt-related transcription factor 3) has been a new insight in breast cancer known to be suppressed due to local promoter hypermethylation mediated by DNA methyltransferase 1 (DNMT1). However, the precise mechanism of epigenetic-influenced silencing of the RUNX3 signaling resulting in cancer invasion and metastasis remains inadequately characterized. In this study, a biological regulatory network (BRN) has been designed to model the dynamics of the DNMT1–RUNX3 network augmented by other regulators such as p21, c-myc, and p53. For this purpose, the René Thomas qualitative modeling was applied to compute the unknown parameters and the subsequent trajectories signified important behaviors of the DNMT1–RUNX3 network (i.e., recovery cycle, homeostasis, and bifurcation state). As a result, the biological system was observed to invade cancer metastasis due to persistent activation of oncogene c-myc accompanied by consistent downregulation of TSG RUNX3. Conversely, homeostasis was achieved in the absence of c-myc and activated TSG RUNX3. Furthermore, DNMT1 was endorsed as a potential epigenetic drug target to be subjected to the implementation of machine-learning techniques for the classification of the active and inactive DNMT1 modulators. The best-performing ML model successfully classified the active and least-active DNMT1 inhibitors exhibiting 97% classification accuracy. Collectively, this study reveals the underlined epigenetic events responsible for RUNX3-implicated breast cancer metastasis along with the classification of DNMT1 modulators that can potentially drive the perception of epigenetic-based tumor therapy.

Automated inference of Boolean models from molecular interaction maps using CaSQ

Motivation

Molecular interaction maps have emerged as a meaningful way of representing biological mechanisms in a comprehensive and systematic manner. However, their static nature provides limited insights to the emerging behaviour of the described biological system under different conditions. Computational modelling provides the means to study dynamic properties through in silico simulations and perturbations. We aim to bridge the gap between static and dynamic representations of biological systems with CaSQ, a software tool that infers Boolean rules based on the topology and semantics of molecular interaction maps built with CellDesigner.

Results

We developed CaSQ by defining conversion rules and logical formulas for inferred Boolean models according to the topology and the annotations of the starting molecular interaction maps. We used CaSQ to produce executable files of existing molecular maps that differ in size, complexity and the use of Systems Biology Graphical Notation (SBGN) standards. We also compared, where possible, the manually built logical models corresponding to a molecular map to the ones inferred by CaSQ. The tool is able to process large and complex maps built with CellDesigner (either following SBGN standards or not) and produce Boolean models in a standard output format, Systems Biology Marked Up Language-qualitative (SBML-qual), that can be further analyzed using popular modelling tools. References, annotations and layout of the CellDesigner molecular map are retained in the obtained model, facilitating interoperability and model reusability.

Availability and implementation

The present tool is available online: https://lifeware.inria.fr/∼soliman/post/casq/ and distributed as a Python package under the GNU GPLv3 license. The code can be accessed here: https://gitlab.inria.fr/soliman/casq.

Supplementary information

Supplementary data are available at Bioinformatics online.

Incorporating Time Delays in Process Hitting Framework for Dynamical Modeling of Large Biological Regulatory Networks

Modeling and simulation of molecular systems helps in understanding the behavioral mechanism of biological regulation. Time delays in production and degradation of expressions are important parameters in biological regulation. Constraints on time delays provide insight into the dynamical behavior of a Biological Regulatory Network (BRN). A recently introduced Process Hitting (PH) Framework has been found efficient in static analysis of large BRNs, however, it lacks the inference of time delays and thus determination of their constraints associated with the evolution of the expression levels of biological entities of BRN is not possible. In this paper we propose a Hybrid Process Hitting scheme for introducing time delays in Process Hitting Framework for dynamical modeling and analysis of Large Biological Regulatory Networks. It provides valuable insights into the time delays corresponding to the changes in the expression levels of biological entities thus possibly helping in identification of therapeutic targets. The proposed framework is applied to a well-known BRNs of Bacteriophage λ and ERBB Receptor-regulated G1/S transition involved in the breast cancer to demonstrate the viability of our approach. Using the proposed approach, we are able to perform goal-oriented reduction of the BRN and also determine the constraints on time delays characterizing the evolution (dynamics) of the reduced BRN.

Parameter estimation of qualitative biological regulatory networks on high performance computing hardware

BACKGROUND

Biological Regulatory Networks (BRNs) are responsible for developmental and maintenance related functions in organisms. These functions are implemented by the dynamics of BRNs and are sensitive to regulations enforced by specific activators and inhibitors. The logical modeling formalism by René Thomas incorporates this sensitivity with a set of logical parameters modulated by available regulators, varying with time. With the increase in complexity of BRNs in terms of number of entities and their interactions, the task of parameters estimation becomes computationally expensive with existing sequential SMBioNET tool. We extend the existing sequential implementation of SMBioNET by using a data decomposition approach using a Java messaging library called MPJ Express. The approach divides the parameters space into different regions and each region is then explored in parallel on High Performance Computing (HPC) hardware.

RESULTS

The performance of the parallel approach is evaluated on BRNs of different sizes, and experimental results on multicore and cluster computers showed almost linear speed-up. This parallel code can be executed on a wide range of concurrent hardware including laptops equipped with multicore processors, and specialized distributed memory computer systems. To demonstrate the application of parallel implementation, we selected a case study of Hexosamine Biosynthetic Pathway (HBP) in cancer progression to identify potential therapeutic targets against cancer. A set of logical parameters were computed for HBP model that directs the biological system to a state of recovery. Furthermore, the parameters also suggest a potential therapeutic intervention that restores homeostasis. Additionally, the performance of parallel application was also evaluated on a network (comprising of 23 entities) of Fibroblast Growth Factor Signalling in Drosophila melanogaster.

CONCLUSIONS

Qualitative modeling framework is widely used for investigating dynamics of biological regulatory networks. However, computation of model parameters in qualitative modeling is computationally intensive. In this work, we presented results of our Java based parallel implementation that provides almost linear speed-up on both multicore and cluster platforms. The parallel implementation is available at https://psmbionet.github.io .

Revisiting a skeleton model for the mammalian cell cycle: From bistability to Cdk oscillations and cellular heterogeneity

A network of cyclin-dependent kinases (Cdks) regulated by multiple negative and positive feedback loops controls progression in the mammalian cell cycle. We previously proposed a detailed computational model for this network, which consists of four coupled Cdk modules. Both this detailed model and a reduced, skeleton version show that the Cdk network is capable of temporal self-organization in the form of sustained Cdk oscillations, which correspond to the orderly progression along the different cell cycle phases G1, S (DNA replication), G2 and M (mitosis). We use the skeleton model to revisit the role of positive feedback (PF) loops on the dynamics of the mammalian cell cycle by showing that the multiplicity of PF loops extends the range of bistability in the isolated Cdk modules controlling the G1/S and G2/M transitions. Resorting to stochastic simulations we show that, through their effect on the range of bistability, multiple PF loops enhance the robustness of Cdk oscillations with respect to molecular noise. The model predicts that a rise in the total level of Cdk1 also enlarges the domain of bistability in the isolated Cdk modules as well as the range of oscillations in the full Cdk network. Surprisingly, stochastic simulations indicate that Cdk1 overexpression reduces the robustness of Cdk oscillations towards molecular noise; this result is due to the increased distance between the two branches of the bistable switch at higher levels of Cdk1. At intermediate levels of growth factor stochastic simulations show that cells may randomly switch between cell cycle arrest and cell proliferation, as a consequence of fluctuations. In the presence of Cdk1 overexpression, these transitions occur even at low levels of growth factor. Extending stochastic simulations from single cells to cell populations suggests that stochastic switches between cell cycle arrest and proliferation may provide a source of heterogeneity in a cell population, as observed in cancer cells characterized by Cdk1 overexpression.

Hybrid models of genetic networks: Mathematical challenges and biological relevance

We review results concerning dynamics in a class of hybrid ordinary differential equations which incorporates logical control to yield piecewise linear equations. These equations relate qualitative features of the structure of networks to qualitative properties of the dynamics. Because of their simple structure, they have been studied using techniques from discrete mathematics and nonlinear dynamics. Initially developed as a qualitataive description of gene regulatory networks, many generalizations of the basic approach have been developed. In particular, we show how this qualitative approach may be adapted to switching biochemical systems without degradation, illustrated by an example of a motif in which two branches of a pathway may be regulated differently when the thresholds for the two pathways are separated.

Structure-based control of complex networks with nonlinear dynamics

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 toward 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.

Formal Modeling of mTOR Associated Biological Regulatory Network Reveals Novel Therapeutic Strategy for the Treatment of Cancer

Cellular homeostasis is a continuous phenomenon that if compromised can lead to several disorders including cancer. There is a need to understand the dynamics of cellular proliferation to get deeper insights into the prevalence of cancer. Mechanistic Target of Rapamycin (mTOR) is implicated as the central regulator of the metabolic pathway involved in growth whereas its two distinct complexes mTORC1 and mTORC2 perform particular functions in cellular propagation. To date, mTORC1 is a well defined therapeutic target to inhibit uncontrolled cell division, while the role of mTORC2 is not well characterized. Therefore, the current study is designed to understand the signaling dynamics of mTOR and its partner proteins such as PI3K, PTEN, mTORC2, PKB (Akt), mTORC1, and FOXO. For this purpose, a qualitative model of mTOR-associated Biological Regulatory Network (BRN) is constructed to predict its regulatory behaviors which may not be predictable otherwise. The depleted expression of PTEN and FOXO along with the overexpression of PI3K, mTORC2, mTORC1 and Akt is predicted as a stable steady state which is in accordance with their observed expression levels in the progression of various cancers. The qualitative model also predicts the homeostasis of all the entities in the form of qualitative cycles. The significant qualitative (discrete) cycle is identified by analyzing betweenness centralities of the qualitative (discrete) states. This cycle is further refined as a linear hybrid automaton model with the production (activation) and degradation (inhibition) time delays in order to analyze the real-time constraints for its existence. The analysis of the hybrid model provides a formal proof that during homeostasis the inhibition time delay of Akt is less than the inhibition time delay of mTORC2. In conclusion, our observations characterize that in homeostasis Akt is degraded with a faster rate than mTORC2 which suggests that the inhibition of Akt along with the activation of mTORC2 may be a better therapeutic strategy for the treatment of cancer.

Mathematical models of cell phenotype regulation and reprogramming: Make cancer cells sensitive again!

A cell's phenotype is the observable actualization of complex interactions between its genome, epigenome, and local environment. While traditional views in cancer have held that cellular and tumor phenotypes are largely functions of genomic instability, increasing attention has recently been given to epigenetic and microenvironmental influences. Such non-genetic factors allow cancer cells to experience intrinsic diversity and plasticity, and at the tumor level can result in phenotypic heterogeneity and treatment evasion. In 2006, Takahashi and Yamanaka exploited the epigenome's plasticity by "reprogramming" differentiated cells into a pluripotent state by inducing expression of a cocktail of four transcription factors. Recent advances in cancer biology have shown not only that cellular reprogramming is possible for malignant cells, but it may provide a foundation for future therapies. Nevertheless, cell reprogramming experiments are frequently plagued by low efficiency, activation of aberrant transcriptional programs, instability, and often rely on expertise gathered from systems which may not translate directly to cancer. Here, we review a theoretical framework tracing back to Waddington's epigenetic landscape which may be used to derive quantitative and qualitative understanding of cellular reprogramming. Implications for tumor heterogeneity, evolution and adaptation are discussed in the context of designing new treatments to re-sensitize recalcitrant tumors. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer?, edited by Dr. Robert A. Gatenby.

Modeling Asymmetric Cell Division in Caulobacter crescentus Using a Boolean Logic Approach

Caulobacter crescentus is a model organism for the study of asymmetric division and cell type differentiation, as its cell division cycle generates a pair of daughter cells that differ from one another in their morphology and behavior. One of these cells (called stalked) develops a structure that allows it to attach to solid surfaces and is the only one capable of dividing, while the other (called swarmer) develops a flagellum that allows it to move in liquid media and divides only after differentiating into a stalked cell type. Although many genes, proteins, and other molecules involved in the asymmetric division exhibited by C. crescentus have been discovered and characterized for several decades, it remains as a challenging task to understand how cell properties arise from the high number of interactions between these molecular components. This chapter describes a modeling approach based on the Boolean logic framework that provides a means for the integration of knowledge and study of the emergence of asymmetric division. The text illustrates how the simulation of simple logic models gives valuable insight into the dynamic behavior of the regulatory and signaling networks driving the emergence of the phenotypes exhibited by C. crescentus. These models provide useful tools for the characterization and analysis of other complex biological networks.

Formal modeling and analysis of the hexosamine biosynthetic pathway: role of O-linked N-acetylglucosamine transferase in oncogenesis and cancer progression

The alteration of glucose metabolism, through increased uptake of glucose and glutamine addiction, is essential to cancer cell growth and invasion. Increased flux of glucose through the Hexosamine Biosynthetic Pathway (HBP) drives increased cellular O-GlcNAcylation (hyper-O-GlcNAcylation) and contributes to cancer progression by regulating key oncogenes. However, the association between hyper-O-GlcNAcylation and activation of these oncogenes remains poorly characterized. Here, we implement a qualitative modeling framework to analyze the role of the Biological Regulatory Network in HBP activation and its potential effects on key oncogenes. Experimental observations are encoded in a temporal language format and model checking is applied to infer the model parameters and qualitative model construction. Using this model, we discover step-wise genetic alterations that promote cancer development and invasion due to an increase in glycolytic flux, and reveal critical trajectories involved in cancer progression. We compute delay constraints to reveal important associations between the production and degradation rates of proteins. O-linked N-acetylglucosamine transferase (OGT), an enzyme used for addition of O-GlcNAc during O-GlcNAcylation, is identified as a key regulator to promote oncogenesis in a feedback mechanism through the stabilization of c-Myc. Silencing of the OGT and c-Myc loop decreases glycolytic flux and leads to programmed cell death. Results of network analyses also identify a significant cycle that highlights the role of p53-Mdm2 circuit oscillations in cancer recovery and homeostasis. Together, our findings suggest that the OGT and c-Myc feedback loop is critical in tumor progression, and targeting these mediators may provide a mechanism-based therapeutic approach to regulate hyper-O-GlcNAcylation in human cancer.

A Generalized Gene-Regulatory Network Model of Stem Cell Differentiation for Predicting Lineage Specifiers

Identification of cell-fate determinants for directing stem cell differentiation remains a challenge. Moreover, little is known about how cell-fate determinants are regulated in functionally important subnetworks in large gene-regulatory networks (i.e., GRN motifs). Here we propose a model of stem cell differentiation in which cell-fate determinants work synergistically to determine different cellular identities, and reside in a class of GRN motifs known as feedback loops. Based on this model, we develop a computational method that can systematically predict cell-fate determinants and their GRN motifs. The method was able to recapitulate experimentally validated cell-fate determinants, and validation of two predicted cell-fate determinants confirmed that overexpression of ESR1 and RUNX2 in mouse neural stem cells induces neuronal and astrocyte differentiation, respectively. Thus, the presented GRN-based model of stem cell differentiation and computational method can guide differentiation experiments in stem cell research and regenerative medicine.

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A network-based method for predicting lineage specifiers and key network motifs

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A computational guidance to stem cell differentiation experiments

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Overexpression of ESR1 in mNSCs induces neuronal differentiation

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Overexpression of RUNX2 in mNSCs induces astrocyte differentiation

Logical Modeling and Dynamical Analysis of Cellular Networks

The logical (or logic) formalism is increasingly used to model regulatory and signaling networks. Complementing these applications, several groups contributed various methods and tools to support the definition and analysis of logical models. After an introduction to the logical modeling framework and to several of its variants, we review here a number of recent methodological advances to ease the analysis of large and intricate networks. In particular, we survey approaches to determine model attractors and their reachability properties, to assess the dynamical impact of variations of external signals, and to consistently reduce large models. To illustrate these developments, we further consider several published logical models for two important biological processes, namely the differentiation of T helper cells and the control of mammalian cell cycle.

Dynamical modeling of the cell cycle and cell fate emergence in Caulobacter crescentus

The division of Caulobacter crescentus, a model organism for studying cell cycle and differentiation in bacteria, generates two cell types: swarmer and stalked. To complete its cycle, C. crescentus must first differentiate from the swarmer to the stalked phenotype. An important regulator involved in this process is CtrA, which operates in a gene regulatory network and coordinates many of the interactions associated to the generation of cellular asymmetry. Gaining insight into how such a differentiation phenomenon arises and how network components interact to bring about cellular behavior and function demands mathematical models and simulations. In this work, we present a dynamical model based on a generalization of the Boolean abstraction of gene expression for a minimal network controlling the cell cycle and asymmetric cell division in C. crescentus. This network was constructed from data obtained from an exhaustive search in the literature. The results of the simulations based on our model show a cyclic attractor whose configurations can be made to correspond with the current knowledge of the activity of the regulators participating in the gene network during the cell cycle. Additionally, we found two point attractors that can be interpreted in terms of the network configurations directing the two cell types. The entire network is shown to be operating close to the critical regime, which means that it is robust enough to perturbations on dynamics of the network, but adaptable to environmental changes.

On the modelling and analysis of the regulatory network of dengue virus pathogenesis and clearance

Dengue virus can ignite both protective and pathogenic responses in human. The pathogenesis is related with modified functioning of our immune system during infection. Pattern recognition receptors like Toll like receptor 3 is vital for the induction of innate immunity in case of Dengue infection. Toll like receptor 3 induces TRIF mediated activation of Type 1 interferons and Fc receptor mediated induction of cytokines. Interferons have been related with clearance of Dengue virus but it has adopted modified regulatory mechanisms to counter this effect. SOCS protein is also induced due to the interferon and cytokine mediated signalling which can subsequently play its part in the regulation of interferon and cytokine production. Our hypothesis in this study relates the pathogenesis of Dengue virus with the SOCS mediated inhibition of our innate immunity. We used the qualitative formalism of René Thomas to model the biological regulatory network of Toll like receptor 3 mediated signalling pathway in an association with pathogenesis of dengue. Logical parameters for the qualitative modelling were inferred using a model checking approach implemented in SMBioNet. A linear hybrid model, parametric linear hybrid automaton, was constructed to incorporate the activation and inhibition time delays in the qualitative model. The qualitative model captured all the possible expression dynamics of the proteins in the form of paths, some of which were observed as abstract cycles (representing homoeostasis) and diverging paths towards stable states. The analysis of the qualitative model highlighted the importance of SOCS protein in elevating propagation of dengue virus through inhibition of type 1 interferons. Detailed qualitative analysis of regulatory network endorses our hypothesis that elevated levels of cytokine subsequently induce SOCS expression which in turn results into the continuous down-regulation of Toll like receptor 3 and interferon. This may result into the Dengue pathogenesis during the stage of immunosuppression. Further analysis with HyTech (HYbrid TECHnology) tool provided us with the real-time constraints (delay constraints) of the proteins involved in the cyclic paths of the regulatory network backing the evidence provided by the qualitative analysis. The HyTech results also suggest that the role of SOCS is vital in homoeostasis.

Formal modelling of toll like receptor 4 and JAK/STAT signalling pathways: insight into the roles of SOCS-1, interferon-β and proinflammatory cytokines in sepsis

Sepsis is one of the major causes of human morbidity and results in a considerable number of deaths each year. Lipopolysaccharide-induced sepsis has been associated with TLR4 signalling pathway which in collaboration with the JAK/STAT signalling regulate endotoxemia and inflammation. However, during sepsis our immune system cannot maintain a balance of cytokine levels and results in multiple organ damage and eventual death. Different opinions have been made in previous studies about the expression patterns and the role of proinflammatory cytokines in sepsis that attracted our attention towards qualitative properties of TLR4 and JAK/STAT signalling pathways using computer-aided studies. René Thomas' formalism was used to model septic and non-septic dynamics of TLR4 and JAK/STAT signalling. Comparisons among dynamics were made by intervening or removing the specific interactions among entities. Among our predictions, recurrent induction of proinflammatory cytokines with subsequent downregulation was found as the basic characteristic of septic model. This characteristic was found in agreement with previous experimental studies, which implicate that inflammation is followed by immunomodulation in septic patients. Moreover, intervention in downregulation of proinflammatory cytokines by SOCS-1 was found desirable to boost the immune responses. On the other hand, interventions either in TLR4 or transcriptional elements such as NFκB and STAT were found effective in the downregulation of immune responses. Whereas, IFN-β and SOCS-1 mediated downregulation at different levels of signalling were found to be associated with variations in the levels of proinflammatory cytokines. However, these predictions need to be further validated using wet laboratory experimental studies to further explore the roles of inhibitors such as SOCS-1 and IFN-β, which may alter the levels of proinflammatory cytokines at different stages of sepsis.

Bacterial sugar utilization gives rise to distinct single-cell behaviours

Inducible utilization pathways reflect widespread microbial strategies to uptake and consume sugars from the environment. Despite their broad importance and extensive characterization, little is known how these pathways naturally respond to their inducing sugar in individual cells. Here, we performed single-cell analyses to probe the behaviour of representative pathways in the model bacterium Escherichia coli. We observed diverse single-cell behaviours, including uniform responses (d-lactose, d-galactose, N-acetylglucosamine, N-acetylneuraminic acid), 'all-or-none' responses (d-xylose, l-rhamnose) and complex combinations thereof (l-arabinose, d-gluconate). Mathematical modelling and probing of genetically modified pathways revealed that the simple framework underlying these pathways - inducible transport and inducible catabolism - could give rise to most of these behaviours. Sugar catabolism was also an important feature, as disruption of catabolism eliminated tunable induction as well as enhanced memory of previous conditions. For instance, disruption of catabolism in pathways that respond to endogenously synthesized sugars led to full pathway induction even in the absence of exogenous sugar. Our findings demonstrate the remarkable flexibility of this simple biological framework, with direct implications for environmental adaptation and the engineering of synthetic utilization pathways as titratable expression systems and for metabolic engineering.

Introduction to focus issue: quantitative approaches to genetic networks

All cells of living organisms contain similar genetic instructions encoded in the organism's DNA. In any particular cell, the control of the expression of each different gene is regulated, in part, by binding of molecular complexes to specific regions of the DNA. The molecular complexes are composed of protein molecules, called transcription factors, combined with various other molecules such as hormones and drugs. Since transcription factors are coded by genes, cellular function is partially determined by genetic networks. Recent research is making large strides to understand both the structure and the function of these networks. Further, the emerging discipline of synthetic biology is engineering novel gene circuits with specific dynamic properties to advance both basic science and potential practical applications. Although there is not yet a universally accepted mathematical framework for studying the properties of genetic networks, the strong analogies between the activation and inhibition of gene expression and electric circuits suggest frameworks based on logical switching circuits. This focus issue provides a selection of papers reflecting current research directions in the quantitative analysis of genetic networks. The work extends from molecular models for the binding of proteins, to realistic detailed models of cellular metabolism. Between these extremes are simplified models in which genetic dynamics are modeled using classical methods of systems engineering, Boolean switching networks, differential equations that are continuous analogues of Boolean switching networks, and differential equations in which control is based on power law functions. The mathematical techniques are applied to study: (i) naturally occurring gene networks in living organisms including: cyanobacteria, Mycoplasma genitalium, fruit flies, immune cells in mammals; (ii) synthetic gene circuits in Escherichia coli and yeast; and (iii) electronic circuits modeling genetic networks using field-programmable gate arrays. Mathematical analyses will be essential for understanding naturally occurring genetic networks in diverse organisms and for providing a foundation for the improved development of synthetic genetic networks.

The interplay of intrinsic and extrinsic bounded noises in biomolecular networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: (i) the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, (ii) a model of enzymatic futile cycle and (iii) a genetic toggle switch. In (ii) and (iii) we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.

A simple theoretical framework for understanding heterogeneous differentiation of CD4+ T cells

BACKGROUND

CD4+ T cells have several subsets of functional phenotypes, which play critical yet diverse roles in the immune system. Pathogen-driven differentiation of these subsets of cells is often heterogeneous in terms of the induced phenotypic diversity. In vitro recapitulation of heterogeneous differentiation under homogeneous experimental conditions indicates some highly regulated mechanisms by which multiple phenotypes of CD4+ T cells can be generated from a single population of naïve CD4+ T cells. Therefore, conceptual understanding of induced heterogeneous differentiation will shed light on the mechanisms controlling the response of populations of CD4+ T cells under physiological conditions.

RESULTS

We present a simple theoretical framework to show how heterogeneous differentiation in a two-master-regulator paradigm can be governed by a signaling network motif common to all subsets of CD4+ T cells. With this motif, a population of naïve CD4+ T cells can integrate the signals from their environment to generate a functionally diverse population with robust commitment of individual cells. Notably, two positive feedback loops in this network motif govern three bistable switches, which in turn, give rise to three types of heterogeneous differentiated states, depending upon particular combinations of input signals. We provide three prototype models illustrating how to use this framework to explain experimental observations and make specific testable predictions.

CONCLUSIONS

The process in which several types of T helper cells are generated simultaneously to mount complex immune responses upon pathogenic challenges can be highly regulated, and a simple signaling network motif can be responsible for generating all possible types of heterogeneous populations with respect to a pair of master regulators controlling CD4+ T cell differentiation. The framework provides a mathematical basis for understanding the decision-making mechanisms of CD4+ T cells, and it can be helpful for interpreting experimental results. Mathematical models based on the framework make specific testable predictions that may improve our understanding of this differentiation system.

Formal modeling and analysis of the MAL-associated biological regulatory network: insight into cerebral malaria

The discrete modeling formalism of René Thomas is a well known approach for the modeling and analysis of Biological Regulatory Networks (BRNs). This formalism uses a set of parameters which reflect the dynamics of the BRN under study. These parameters are initially unknown but may be deduced from the appropriately chosen observed dynamics of a BRN. The discrete model can be further enriched by using the model checking tool HyTech along with delay parameters. This paves the way to accurately analyse a BRN and to make predictions about critical trajectories which lead to a normal or diseased response. In this paper, we apply the formal discrete and hybrid (discrete and continuous) modeling approaches to characterize behavior of the BRN associated with MyD88-adapter-like (MAL)--a key protein involved with innate immune response to infections. In order to demonstrate the practical effectiveness of our current work, different trajectories and corresponding conditions that may lead to the development of cerebral malaria (CM) are identified. Our results suggest that the system converges towards hyperinflammation if Bruton's tyrosine kinase (BTK) remains constitutively active along with pre-existing high cytokine levels which may play an important role in CM pathogenesis.

Qualitative modelling and formal verification of the FLR1 gene mancozeb response in Saccharomyces cerevisiae

BACKGROUND

Qualitative models allow understanding the relation between the structure and the dynamics of gene regulatory networks. The dynamical properties of these models can be automatically analysed by means of formal verification methods, like model checking. This facilitates the model-validation process and the test of new hypotheses to reconcile model predictions with the experimental data.

RESULTS

The authors report in this study the qualitative modelling and simulation of the transcriptional regulatory network controlling the response of the model eukaryote Saccharomyces cerevisiae to the agricultural fungicide mancozeb. The model allowed the analysis of the regulation level and activity of the components of the gene mancozeb-induced network controlling the transcriptional activation of the FLR1 gene, which is proposed to confer multidrug resistance through its putative role as a drug eflux pump. Formal verification analysis of the network allowed us to confront model predictions with the experimental data and to assess the model robustness to parameter ordering and gene deletion.

CONCLUSIONS

This analysis enabled us to better understand the mechanisms regulating the FLR1 gene mancozeb response and confirmed the need of a new transcription factor for the full transcriptional activation of YAP1. The result is a computable model of the FLR1 gene response to mancozeb, permitting a quick and cost-effective test of hypotheses prior to experimental validation.

Modular random Boolean networks

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors, and are closer to criticality when chaotic dynamics would be expected, than classical RBNs.

On the intrinsic inevitability of cancer: from foetal to fatal attraction

The cracks in the paradigm of oncogenic mutations and somatic evolution as driving force of tumorigenesis, lucidly exposed by the dynamic heterogeneity of "cancer stem cells" or the diffuse results of cancer genome sequencing projects, indicate the need for a more encompassing theory of cancer that reaches beyond the current proximate explanations based on individual genetic pathways. One such integrative concept, derived from first principles of the dynamics of gene regulatory networks, is that cancerous cell states are attractor states, just like normal cell types are. Here we extend the concept of cancer attractors to illuminate a more profound property of cancer initiation: its inherent inevitability in the light of metazoan evolution. Using Waddington's Epigenetic Landscape as a conceptual aid, for which we present a mathematical and evolutionary foundation, we propose that cancer is intrinsically linked to ontogenesis and phylogenesis. This explanatory rather than enumerating review uses a formal argumentation structure that is atypical in modern experimental biology but may hopefully offer a new coherent perspective to reconcile many conflicts between new findings and the old thinking in the categories of linear oncogenic pathways.

Interactions among oscillatory pathways in NF-kappa B signaling

BACKGROUND

Sustained stimulation with tumour necrosis factor alpha (TNF-alpha) induces substantial oscillations--observed at both the single cell and population levels--in the nuclear factor kappa B (NF-kappa B) system. Although the mechanism has not yet been elucidated fully, a core system has been identified consisting of a negative feedback loop involving NF-kappa B (RelA:p50 hetero-dimer) and its inhibitor I-kappa B-alpha. Many authors have suggested that this core oscillator should couple to other oscillatory pathways.

RESULTS

First we analyse single-cell data from experiments in which the NF-kappa B system is forced by short trains of strong pulses of TNF-alpha. Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency. Using a recent model of the NF-kappa B system due to Caroline Horton, we carried out extensive numerical simulations to analyze the response frequencies induced by trains of pulses of TNF-alpha stimulation having a wide range of frequencies and amplitudes. These studies suggest that for sufficiently weak stimulation, various nonlinear resonances should be observable. To explore further the possibility of probing alternative feedback mechanisms, we also coupled the model to sinusoidal signals with a wide range of strengths and frequencies. Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics.

CONCLUSIONS

Our numerical results suggest that the entrainment phenomena observed in pulse-stimulated experiments is a consequence of the high intensity of the stimulation. Computational studies based on current models suggest that resonant interactions between periodic pulsatile forcing and the system's natural frequencies may become evident for sufficiently weak stimulation. Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

From biological pathways to regulatory networks

This paper presents a general theoretical framework for generating Boolean networks whose state transitions realize a set of given biological pathways or minor variations thereof. This ill-posed inverse problem, which is of crucial importance across practically all areas of biology, is solved by using Karnaugh maps which are classical tools for digital system design. It is shown that the incorporation of prior knowledge, presented in the form of biological pathways, can bring about a dramatic reduction in the cardinality of the network search space. Constraining the connectivity of the network, the number and relative importance of the attractors, and concordance with observed time-course data are additional factors that can be used to further reduce the cardinality of the search space. The networks produced by the approaches developed here should facilitate the understanding of multivariate biological phenomena and the subsequent design of intervention approaches that are more likely to be successful in practice. As an example, the results of this paper are applied to the widely studied p53 pathway and it is shown that the resulting network exhibits dynamic behavior consistent with experimental observations from the published literature.

The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation

Differentiation from a multipotent stem or progenitor state to a mature cell is an essentially irreversible process. The associated changes in gene expression patterns exhibit time-directionality. This "arrow of time" in the collective change of gene expression across multiple stable gene expression patterns (attractors) is not explained by the regulated activation, the suppression of individual genes which are bidirectional molecular processes, or by the standard dynamical models of the underlying gene circuit which only account for local stability of attractors. To capture the global dynamics of this nonequilibrium system and gain insight in the time-asymmetry of state transitions, we computed the quasipotential landscape of the stochastic dynamics of a canonical gene circuit that governs branching cell fate commitment. The potential landscape reveals the global dynamics and permits the calculation of potential barriers between cell phenotypes imposed by the circuit architecture. The generic asymmetry of barrier heights indicates that the transition from the uncommitted multipotent state to differentiated states is inherently unidirectional. The model agrees with observations and predicts the extreme conditions for reprogramming cells back to the undifferentiated state.

Signal transduction in the human thyrocyte and its perversion in thyroid tumors

The study of normal signal transduction pathways regulating the proliferation and differentiation of a cell type allows to predict and to understand the perversions of these pathways which lead to tumorigenesis. In the case of the human thyroid cell, three cascades are mostly involved in tumorigenesis: The pathways and genetic events affecting them are described. Caveats in the use of models and the interpretation of results are formulated and the still pending questions are outlined.

Cancer stem cells: a reality, a myth, a fuzzy concept or a misnomer? An analysis

The concept of cancer stem cells (CSC) embodies two aspects: the stem cell as the initial target of the oncogenic process and the existence of two populations of cells in cancers: the CSC and derived cells. The second is discussed in this review. CSC are defined as cells having three properties: a selectively endowed tumorigenic capacity, an ability to recreate the full repertoire of cancer cells of the parent tumor and the expression of a distinctive repertoire of surface biomarkers. In operational terms, the CSC are among all cancer cells those able to initiate a xenotransplant. Other explicit or implicit assumptions exist, including the concept of CSC as a single unique infrequent population of cells. To avoid such assumptions, we propose to use the operational term tumor-propagating cells (TPC); indeed, the cells that initiate transplants did not initiate the cancer. The experimental evidence supporting the explicit definition is analyzed. Cancers indeed contain a fraction of cells mainly responsible for the tumor development. However, there is evidence that these cells do not represent one homogenous population. Moreover, there is no evidence that the derived cells result from an asymmetric, qualitative and irreversible process. A more general model is proposed of which the CSC model could be one extreme case. We propose that the TPC are multiple evolutionary selected cancer cells with the most competitive properties [maintained by (epi-)genetic mechanisms], at least partially reversible, quantitative rather than qualitative and resulting from a stochastic rather than deterministic process.

The powerful law of the power law and other myths in network biology

For almost 10 years, topological analysis of different large-scale biological networks (metabolic reactions, protein interactions, transcriptional regulation) has been highlighting some recurrent properties: power law distribution of degree, scale-freeness, small world, which have been proposed to confer functional advantages such as robustness to environmental changes and tolerance to random mutations. Stochastic generative models inspired different scenarios to explain the growth of interaction networks during evolution. The power law and the associated properties appeared so ubiquitous in complex networks that they were qualified as "universal laws". However, these properties are no longer observed when the data are subjected to statistical tests: in most cases, the data do not fit the expected theoretical models, and the cases of good fitting merely result from sampling artefacts or improper data representation. The field of network biology seems to be founded on a series of myths, i.e. widely believed but false ideas. The weaknesses of these foundations should however not be considered as a failure for the entire domain. Network analysis provides a powerful frame for understanding the function and evolution of biological processes, provided it is brought to an appropriate level of description, by focussing on smaller functional modules and establishing the link between their topological properties and their dynamical behaviour.

Reprogramming cell fates: reconciling rarity with robustness

The stunning possibility of "reprogramming" differentiated somatic cells to express a pluripotent stem cell phenotype (iPS, induced pluripotent stem cell) and the "ground state" character of pluripotency reveal fundamental features of cell fate regulation that lie beyond existing paradigms. The rarity of reprogramming events appears to contradict the robustness with which the unfathomably complex phenotype of stem cells can reliably be generated. This apparent paradox, however, is naturally explained by the rugged "epigenetic landscape" with valleys representing "preprogrammed" attractor states that emerge from the dynamical constraints of the gene regulatory network. This article provides a pedagogical primer to the fundamental principles of gene regulatory networks as integrated dynamic systems and reviews recent insights in gene expression noise and fate determination, thereby offering a formal framework that may help us to understand why cell fate reprogramming events are inherently rare and yet so robust.

Stochasticity and the molecular mechanisms of induced pluripotency

The generation of induced pluripotent stem cells from adult somatic cells by ectopic expression of key transcription factors holds significant medical promise. However, current techniques for inducing pluripotency rely on viral infection and are therefore not, at present, viable within a clinical setting. Thus, there is now a need to better understand the molecular basis of stem cell pluripotency and lineage specification in order to investigate alternative methods to induce pluripotency for clinical application. However, the complexity of the underlying molecular circuitry makes this a conceptually difficult task. In order to address these issues, we considered a computational model of transcriptional control of cell fate specification. The model comprises two mutually interacting sub-circuits: a central pluripotency circuit consisting of interactions between stem-cell specific transcription factors OCT4, SOX2 and NANOG coupled to a differentiation circuit consisting of interactions between lineage-specifying master genes.The molecular switches which arise from feedback loops within these circuits give rise to a well-defined sequence of successive gene restrictions corresponding to a controlled differentiation cascade in response to environmental stimuli. Furthermore, we found that this differentiation cascade is strongly unidirectional: once silenced, core transcription factors cannot easily be reactivated. In the context of induced pluripotency, this indicates that differentiated cells are robustly resistant to reprogramming to a more primitive state. However, our model suggests that under certain circumstances, amplification of low-level fluctuations in transcriptional status (transcriptional "noise") may be sufficient to trigger reactivation of the core pluripotency switch and reprogramming to a pluripotent state. This interpretation offers an explanation of a number of experimental observations concerning the molecular mechanisms of cellular reprogramming by defined factors and suggests a role for stochasticity in reprogramming of somatic cells to pluripotency.

Oscillations in the immune system

Oscillations are surprisingly common in the immune system, both in its healthy state and in disease. The most famous example is that of periodic fevers caused by the malaria parasite. A number of hereditary disorders, which also cause periodic fevers, have also been known for a long time. Various reports of oscillations in cytokine concentrations following antigen challenge have been published over at least the past three decades. Oscillations can also occur at the intracellular level. Calcium oscillations following T-cell activation are familiar to all immunologists, and metabolic and reactive oxygen species oscillations in neutrophils have been well documented. More recently, oscillations in nuclear factor kappaB activity following stimulation by tumor necrosis factor alpha have received considerable publicity. However, despite all of these examples, oscillations in the immune system still tend to be considered mainly as pathological aberrations, and their causes and significance remained largely unknown. This is partly because of a lack of awareness within the immunological community of the appropriate theoretical frameworks for describing and analyzing such behavior. We provide an introduction to these frameworks and give a survey of the currently known oscillations that occur within the immune system.

Algorithmic and complexity results for decompositions of biological networks into monotone subsystems

A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson [Goemans, M., Williamson, D., 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42 (6), 1115-1145]. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model, and it was found to perform close to optimally.

MicroRNAs acting in a double-negative feedback loop to control a neuronal cell fate decision

The elucidation of the architecture of gene regulatory networks that control cell-type specific gene expression programs represents a major challenge in developmental biology. We describe here a cell fate decision between two alternative neuronal fates and the architecture of a gene regulatory network that controls this cell fate decision. The two Caenorhabditis elegans taste receptor neurons "ASE left" (ASEL) and "ASE right" (ASER) share many bilaterally symmetric features, but each cell expresses a distinct set of chemoreceptors that endow the gustatory system with the capacity to sense and discriminate specific environmental inputs. We show that these left/right asymmetric fates develop from a precursor state in which both ASE neurons express equivalent features. This hybrid precursor state is unstable and transitions into the stable ASEL or ASER terminal end state. Although this transition is spatially stereotyped in wild-type animals, mutant analysis reveals that each cell has the potential to transition into either the ASEL or ASER stable end state. The stability and irreversibility of the terminal differentiated state is ensured by the interactions of two microRNAs (miRNAs) and their transcription factor targets in a double-negative feedback loop. Simple feedback loops are found as common motifs in many gene regulatory networks, but the loop described here is unusually complex and involves miRNAs. The interaction of miRNAs in double-negative feedback loops may not only be a means for miRNAs to regulate their own expression but may also represent a general paradigm for how terminal cell fates are selected and stabilized.

A novel transcriptional autoregulatory loop enhances expression of the Pantoea stewartii subsp. stewartii Hrp type III secretion system

The hrp type III secretion regulon of Pantoea stewartii is regulated by a cascade involving the HrpX/HrpY two-component system, the HrpS enhancer-binding protein and the HrpL alternate sigma factor. hrpXY is both constitutive and autoregulated; HrpY controls hrpS; and HrpS activates hrpL. These regulatory genes are arranged in the order hrpL, hrpXY and hrpS and constitute three operons. This study describes a novel autoregulatory loop involving HrpS. Genetic experiments using a chromosomal hrpS-lacZ fusion demonstrated that ectopic expression of HrpS increases hrpS transcription and that this effect is blocked by polar mutations in hrpXY and hrpL and by a nonpolar mutation in hrpY. RT-PCR and Northern blot analysis revealed a hrpL-hrpXY polycistronic mRNA. These results suggest that HrpS-mediated autoregulation is due to activation of hrpS by increased levels of HrpY resulting from read-through transcription of hrpXY from the hrpL promoter. This novel autoregulatory loop may serve to rapidly induce hrp genes during infection and to compensate for negative regulatory mechanisms that keep the regulon off in the insect vector.

Application of formal methods to biological regulatory networks: extending Thomas’ asynchronous logical approach with temporal logic

Based on the discrete definition of biological regulatory networks developed by René Thomas, we provide a computer science formal approach to treat temporal properties of biological regulatory networks, expressed in computational tree logic. It is then possible to build all the models satisfying a set of given temporal properties. Our approach is illustrated with the mucus production in Pseudomonas aeruginosa. This application of formal methods from computer science to biological regulatory networks should open the way to many other fruitful applications.

A logical analysis of the process of T cell activation: different consequences depending on the state of CD28 engagement

The adaptive immune system is a complex organized action of several immune cell types like, T cells, B cells, dendritic cells, mast cells, and their ability to recognize self and foreign molecular information. Based on logical analysis, a model has been developed that describes TCR-ligand association coupled to intracellular signaling events that result in a proliferation signal. The model demonstrates that after TCR-ligand binding, the activation of tyrosine kinases in one of the paths leads to oscillations between the subsequent states of activation and deactivation of Ca(2+) initiation. In our studies the effect of costimulation on the primary signal has also been explored. Analysis reveals that costimulation increases by more than 2.5 fold the number of paths rendering a cell proliferation signal compared to the outcome when costimulation is blocked. Traversal of 97% of these paths attains a costimulation threshold of activation. We also examined a hypothesis that couples the primary signal and costimulation by modeling costimulation to act as an inhibitor on the Inhibitor proteins. Using this hypothesis our analysis showed a 25% increase in the number of paths leading to cell proliferation in comparison to when costimulation is blocked. Our model also reveals that this hypothesis actually decrease by approximately 50% the number of paths attaining cell proliferation compared to the number of available paths leading to cell proliferation when costimulation does not act as an inhibitor on Inhibitor proteins. This suggests that costimulation influences cell proliferation by providing a greater diversity of paths that converge to this state. However, costimulation should be thought independent of its regulatory interaction with the inhibitor proteins.