Continuous time Boolean modeling for biological signaling: application of Gillespie algorithm

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.

Estimating Attractor Reachability in Asynchronous Logical Models

Logical models are well-suited to capture salient dynamical properties of regulatory networks. For networks controlling cell fate decisions, cell fates are associated with model attractors (stable states or cyclic attractors) whose identification and reachability properties are particularly relevant. While synchronous updates assume unlikely instantaneous or identical rates associated with component changes, the consideration of asynchronous updates is more realistic but, for large models, may hinder the analysis of the resulting non-deterministic concurrent dynamics. This complexity hampers the study of asymptotical behaviors, and most existing approaches suffer from efficiency bottlenecks, being generally unable to handle cyclical attractors and quantify attractor reachability. Here, we propose two algorithms providing probability estimates of attractor reachability in asynchronous dynamics. The first algorithm, named Firefront, exhaustively explores the state space from an initial state, and provides quasi-exact evaluations of the reachability probabilities of model attractors. The algorithm progresses in breadth, propagating the probabilities of each encountered state to its successors. Second, Avatar is an adapted Monte Carlo approach, better suited for models with large and intertwined transient and terminal cycles. Avatar iteratively explores the state space by randomly selecting trajectories and by using these random walks to estimate the likelihood of reaching an attractor. Unlike Monte Carlo simulations, Avatar is equipped to avoid getting trapped in transient cycles and to identify cyclic attractors. Firefront and Avatar are validated and compared to related methods, using as test cases logical models of synthetic and biological networks. Both algorithms are implemented as new functionalities of GINsim 3.0, a well-established software tool for logical modeling, providing executable GUI, Java API, and scripting facilities.

The CoLoMoTo Interactive Notebook: Accessible and Reproducible Computational Analyses for Qualitative Biological Networks

Analysing models of biological networks typically relies on workflows in which different software tools with sensitive parameters are chained together, many times with additional manual steps. The accessibility and reproducibility of such workflows is challenging, as publications often overlook analysis details, and because some of these tools may be difficult to install, and/or have a steep learning curve. The CoLoMoTo Interactive Notebook provides a unified environment to edit, execute, share, and reproduce analyses of qualitative models of biological networks. This framework combines the power of different technologies to ensure repeatability and to reduce users' learning curve of these technologies. The framework is distributed as a Docker image with the tools ready to be run without any installation step besides Docker, and is available on Linux, macOS, and Microsoft Windows. The embedded computational workflows are edited with a Jupyter web interface, enabling the inclusion of textual annotations, along with the explicit code to execute, as well as the visualization of the results. The resulting notebook files can then be shared and re-executed in the same environment. To date, the CoLoMoTo Interactive Notebook provides access to the software tools GINsim, BioLQM, Pint, MaBoSS, and Cell Collective, for the modeling and analysis of Boolean and multi-valued networks. More tools will be included in the future. We developed a Python interface for each of these tools to offer a seamless integration in the Jupyter web interface and ease the chaining of complementary analyses.

Logical modeling of lymphoid and myeloid cell specification and transdifferentiation

Blood cells are derived from a common set of hematopoietic stem cells, which differentiate into more specific progenitors of the myeloid and lymphoid lineages, ultimately leading to differentiated cells. This developmental process is controlled by a complex regulatory network involving cytokines and their receptors, transcription factors, and chromatin remodelers. Using public data and data from our own molecular genetic experiments (quantitative PCR, Western blot, EMSA) or genome-wide assays (RNA-sequencing, ChIP-sequencing), we have assembled a comprehensive regulatory network encompassing the main transcription factors and signaling components involved in myeloid and lymphoid development. Focusing on B-cell and macrophage development, we defined a qualitative dynamical model recapitulating cytokine-induced differentiation of common progenitors, the effect of various reported gene knockdowns, and the reprogramming of pre-B cells into macrophages induced by the ectopic expression of specific transcription factors. The resulting network model can be used as a template for the integration of new hematopoietic differentiation and transdifferentiation data to foster our understanding of lymphoid/myeloid cell-fate decisions.

Logical model specification aided by model-checking techniques: application to the mammalian cell cycle regulation

MOTIVATION

Understanding the temporal behaviour of biological regulatory networks requires the integration of molecular information into a formal model. However, the analysis of model dynamics faces a combinatorial explosion as the number of regulatory components and interactions increases.

RESULTS

We use model-checking techniques to verify sophisticated dynamical properties resulting from the model regulatory structure in the absence of kinetic assumption. We demonstrate the power of this approach by analysing a logical model of the molecular network controlling mammalian cell cycle. This approach enables a systematic analysis of model properties, the delineation of model limitations, and the assessment of various refinements and extensions based on recent experimental observations. The resulting logical model accounts for the main irreversible transitions between cell cycle phases, the sequential activation of cyclins, and the inhibitory role of Skp2, and further emphasizes the multifunctional role for the cell cycle inhibitor Rb.

AVAILABILITY AND IMPLEMENTATION

The original and revised mammalian cell cycle models are available in the model repository associated with the public modelling software GINsim (http://ginsim.org/node/189).

CONTACT

thieffry@ens.fr

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Logic Modeling in Quantitative Systems Pharmacology

Here we present logic modeling as an approach to understand deregulation of signal transduction in disease and to characterize a drug's mode of action. We discuss how to build a logic model from the literature and experimental data and how to analyze the resulting model to obtain insights of relevance for systems pharmacology. Our workflow uses the free tools OmniPath (network reconstruction from the literature), CellNOpt (model fit to experimental data), MaBoSS (model analysis), and Cytoscape (visualization).

Identification of bifurcation transitions in biological regulatory networks using Answer-Set Programming

Background

Numerous cellular differentiation processes can be captured using discrete qualitative models of biological regulatory networks. These models describe the temporal evolution of the state of the network subject to different competing transitions, potentially leading the system to different attractors. This paper focusses on the formal identification of states and transitions that are crucial for preserving or pre-empting the reachability of a given behaviour.

Methods

In the context of non-deterministic automata networks, we propose a static identification of so-called bifurcations, i.e., transitions after which a given goal is no longer reachable. Such transitions are naturally good candidates for controlling the occurrence of the goal, notably by modulating their propensity. Our method combines Answer-Set Programming with static analysis of reachability properties to provide an under-approximation of all the existing bifurcations.

Results

We illustrate our discrete bifurcation analysis on several models of biological systems, for which we identify transitions which impact the reachability of given long-term behaviour. In particular, we apply our implementation on a regulatory network among hundreds of biological species, supporting the scalability of our approach.

Conclusions

Our method allows a formal and scalable identification of transitions which are responsible for the lost of capability to reach a given state. It can be applied to any asynchronous automata networks, which encompass Boolean and multi-valued models. An implementation is provided as part of the Pint software, available at http://loicpauleve.name/pint.

Electronic supplementary material

The online version of this article (doi:10.1186/s13015-017-0110-3) contains supplementary material, which is available to authorized users.

A Clb/Cdk1-mediated regulation of Fkh2 synchronizes CLB expression in the budding yeast cell cycle

Precise timing of cell division is achieved by coupling waves of cyclin-dependent kinase (Cdk) activity with a transcriptional oscillator throughout cell cycle progression. Although details of transcription of cyclin genes are known, it is unclear which is the transcriptional cascade that modulates their expression in a timely fashion. Here, we demonstrate that a Clb/Cdk1-mediated regulation of the Fkh2 transcription factor synchronizes the temporal mitotic CLB expression in budding yeast. A simplified kinetic model of the cyclin/Cdk network predicts a linear cascade where a Clb/Cdk1-mediated regulation of an activator molecule drives CLB3 and CLB2 expression. Experimental validation highlights Fkh2 as modulator of CLB3 transcript levels, besides its role in regulating CLB2 expression. A Boolean model based on the minimal number of interactions needed to capture the information flow of the Clb/Cdk1 network supports the role of an activator molecule in the sequential activation, and oscillatory behavior, of mitotic Clb cyclins. This work illustrates how transcription and phosphorylation networks can be coupled by a Clb/Cdk1-mediated regulation that synchronizes them.

A dynamic coupling of cyclin-dependent kinase with transcription factors in yeast offers insights into the timely cell cycle progression. An international team lead by Matteo Barberis from University of Amsterdam in The Netherlands studied the molecular mechanisms responsible for the coordination of DNA replication with cell division. The researchers have demonstrated how the sequential order of waves of mitotic cyclins activating cyclin-dependent kinase, or Cdk, is achieved by synchronizing Cdk with transcriptional activities. They have generated a mathematical model that predicts a cyclin/Cdk-mediated regulation of an activator molecule to stimulate mitotic cyclin expression. This prediction was successfully validated experimentally, identifying Forkhead transcription factors, or Fkh, as pivotal molecules. Cyclin waves are temporally synchronized by Fkh, and a mitotic Clb/Cdk1-mediated regulation of Fkh modulates cyclin expression. The findings reveal a novel principle of design, with kinase and transcription activities interlocked to guarantee a timely cell cycle.

Advances and challenges in logical modeling of cell cycle regulation: perspective for multi-scale, integrative yeast cell models

The eukaryotic cell cycle is robustly designed, with interacting molecules organized within a definite topology that ensures temporal precision of its phase transitions. Its underlying dynamics are regulated by molecular switches, for which remarkable insights have been provided by genetic and molecular biology efforts. In a number of cases, this information has been made predictive, through computational models. These models have allowed for the identification of novel molecular mechanisms, later validated experimentally. Logical modeling represents one of the youngest approaches to address cell cycle regulation. We summarize the advances that this type of modeling has achieved to reproduce and predict cell cycle dynamics. Furthermore, we present the challenge that this type of modeling is now ready to tackle: its integration with intracellular networks, and its formalisms, to understand crosstalks underlying systems level properties, ultimate aim of multi-scale models. Specifically, we discuss and illustrate how such an integration may be realized, by integrating a minimal logical model of the cell cycle with a metabolic network.

A comprehensive approach to the molecular determinants of lifespan using a Boolean model of geroconversion

Altered molecular responses to insulin and growth factors (GF) are responsible for late‐life shortening diseases such as type‐2 diabetes mellitus (T2DM) and cancers. We have built a network of the signaling pathways that control S‐phase entry and a specific type of senescence called geroconversion. We have translated this network into a Boolean model to study possible cell phenotype outcomes under diverse molecular signaling conditions. In the context of insulin resistance, the model was able to reproduce the variations of the senescence level observed in tissues related to T2DM's main morbidity and mortality. Furthermore, by calibrating the pharmacodynamics of mTOR inhibitors, we have been able to reproduce the dose‐dependent effect of rapamycin on liver degeneration and lifespan expansion in wild‐type and HER2–neu mice. Using the model, we have finally performed an in silico prospective screen of the risk–benefit ratio of rapamycin dosage for healthy lifespan expansion strategies. We present here a comprehensive prognostic and predictive systems biology tool for human aging.

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.

Mathematical Modelling of Molecular Pathways Enabling Tumour Cell Invasion and Migration

Understanding the etiology of metastasis is very important in clinical perspective, since it is estimated that metastasis accounts for 90% of cancer patient mortality. Metastasis results from a sequence of multiple steps including invasion and migration. The early stages of metastasis are tightly controlled in normal cells and can be drastically affected by malignant mutations; therefore, they might constitute the principal determinants of the overall metastatic rate even if the later stages take long to occur. To elucidate the role of individual mutations or their combinations affecting the metastatic development, a logical model has been constructed that recapitulates published experimental results of known gene perturbations on local invasion and migration processes, and predict the effect of not yet experimentally assessed mutations. The model has been validated using experimental data on transcriptome dynamics following TGF-β-dependent induction of Epithelial to Mesenchymal Transition in lung cancer cell lines. A method to associate gene expression profiles with different stable state solutions of the logical model has been developed for that purpose. In addition, we have systematically predicted alleviating (masking) and synergistic pairwise genetic interactions between the genes composing the model with respect to the probability of acquiring the metastatic phenotype. We focused on several unexpected synergistic genetic interactions leading to theoretically very high metastasis probability. Among them, the synergistic combination of Notch overexpression and p53 deletion shows one of the strongest effects, which is in agreement with a recent published experiment in a mouse model of gut cancer. The mathematical model can recapitulate experimental mutations in both cell line and mouse models. Furthermore, the model predicts new gene perturbations that affect the early steps of metastasis underlying potential intervention points for innovative therapeutic strategies in oncology.

GDSCalc: A Web-Based Application for Evaluating Discrete Graph Dynamical Systems

Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interactions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neighborhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a discrete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web application; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.

A Modeling Approach to Explain Mutually Exclusive and Co-Occurring Genetic Alterations in Bladder Tumorigenesis

Relationships between genetic alterations, such as co-occurrence or mutual exclusivity, are often observed in cancer, where their understanding may provide new insights into etiology and clinical management. In this study, we combined statistical analyses and computational modeling to explain patterns of genetic alterations seen in 178 patients with bladder tumors (either muscle-invasive or non-muscle-invasive). A statistical analysis on frequently altered genes identified pair associations, including co-occurrence or mutual exclusivity. Focusing on genetic alterations of protein-coding genes involved in growth factor receptor signaling, cell cycle, and apoptosis entry, we complemented this analysis with a literature search to focus on nine pairs of genetic alterations of our dataset, with subsequent verification in three other datasets available publicly. To understand the reasons and contexts of these patterns of associations while accounting for the dynamics of associated signaling pathways, we built a logical model. This model was validated first on published mutant mice data, then used to study patterns and to draw conclusions on counter-intuitive observations, allowing one to formulate predictions about conditions where combining genetic alterations benefits tumorigenesis. For example, while CDKN2A homozygous deletions occur in a context of FGFR3-activating mutations, our model suggests that additional PIK3CA mutation or p21CIP deletion would greatly favor invasiveness. Furthermore, the model sheds light on the temporal orders of gene alterations, for example, showing how mutual exclusivity of FGFR3 and TP53 mutations is interpretable if FGFR3 is mutated first. Overall, our work shows how to predict combinations of the major gene alterations leading to invasiveness through two main progression pathways in bladder cancer.

A Semiquantitative Framework for Gene Regulatory Networks: Increasing the Time and Quantitative Resolution of Boolean Networks

Boolean models have been instrumental in predicting general features of gene networks and more recently also as explorative tools in specific biological applications. In this study we introduce a basic quantitative and a limited time resolution to a discrete (Boolean) framework. Quantitative resolution is improved through the employ of normalized variables in unison with an additive approach. Increased time resolution stems from the introduction of two distinct priority classes. Through the implementation of a previously published chondrocyte network and T helper cell network, we show that this addition of quantitative and time resolution broadens the scope of biological behaviour that can be captured by the models. Specifically, the quantitative resolution readily allows models to discern qualitative differences in dosage response to growth factors. The limited time resolution, in turn, can influence the reachability of attractors, delineating the likely long term system behaviour. Importantly, the information required for implementation of these features, such as the nature of an interaction, is typically obtainable from the literature. Nonetheless, a trade-off is always present between additional computational cost of this approach and the likelihood of extending the model’s scope. Indeed, in some cases the inclusion of these features does not yield additional insight. This framework, incorporating increased and readily available time and semi-quantitative resolution, can help in substantiating the litmus test of dynamics for gene networks, firstly by excluding unlikely dynamics and secondly by refining falsifiable predictions on qualitative behaviour.

Predicting genetic interactions from Boolean models of biological networks

Genetic interaction can be defined as a deviation of the phenotypic quantitative effect of a double gene mutation from the effect predicted from single mutations using a simple (e.g., multiplicative or linear additive) statistical model. Experimentally characterized genetic interaction networks in model organisms provide important insights into relationships between different biological functions. We describe a computational methodology allowing us to systematically and quantitatively characterize a Boolean mathematical model of a biological network in terms of genetic interactions between all loss of function and gain of function mutations with respect to all model phenotypes or outputs. We use the probabilistic framework defined in MaBoSS software, based on continuous time Markov chains and stochastic simulations. In addition, we suggest several computational tools for studying the distribution of double mutants in the space of model phenotype probabilities. We demonstrate this methodology on three published models for each of which we derive the genetic interaction networks and analyze their properties. We classify the obtained interactions according to their class of epistasis, dependence on the chosen initial conditions and the phenotype. The use of this methodology for validating mathematical models from experimental data and designing new experiments is discussed.

Model checking to assess T-helper cell plasticity

Computational modeling constitutes a crucial step toward the functional understanding of complex cellular networks. In particular, logical modeling has proven suitable for the dynamical analysis of large signaling and transcriptional regulatory networks. In this context, signaling input components are generally meant to convey external stimuli, or environmental cues. In response to such external signals, cells acquire specific gene expression patterns modeled in terms of attractors (e.g., stable states). The capacity for cells to alter or reprogram their differentiated states upon changes in environmental conditions is referred to as cell plasticity. In this article, we present a multivalued logical framework along with computational methods recently developed to efficiently analyze large models. We mainly focus on a symbolic model checking approach to investigate switches between attractors subsequent to changes of input conditions. As a case study, we consider the cellular network regulating the differentiation of T-helper (Th) cells, which orchestrate many physiological and pathological immune responses. To account for novel cellular subtypes, we present an extended version of a published model of Th cell differentiation. We then use symbolic model checking to analyze reachability properties between Th subtypes upon changes of environmental cues. This allows for the construction of a synthetic view of Th cell plasticity in terms of a graph connecting subtypes with arcs labeled by input conditions. Finally, we explore novel strategies enabling specific Th cell polarizing or reprograming events.

Cooperative development of logical modelling standards and tools with CoLoMoTo

The identification of large regulatory and signalling networks involved in the control of crucial cellular processes calls for proper modelling approaches. Indeed, models can help elucidate properties of these networks, understand their behaviour and provide (testable) predictions by performing in silico experiments. In this context, qualitative, logical frameworks have emerged as relevant approaches, as demonstrated by a growing number of published models, along with new methodologies and software tools. This productive activity now requires a concerted effort to ensure model reusability and interoperability between tools. Following an outline of the logical modelling framework, we present the most important achievements of the Consortium for Logical Models and Tools, along with future objectives. Our aim is to advertise this open community, which welcomes contributions from all researchers interested in logical modelling or in related mathematical and computational developments.

Integrative modelling of the influence of MAPK network on cancer cell fate decision

The Mitogen-Activated Protein Kinase (MAPK) network consists of tightly interconnected signalling pathways involved in diverse cellular processes, such as cell cycle, survival, apoptosis and differentiation. Although several studies reported the involvement of these signalling cascades in cancer deregulations, the precise mechanisms underlying their influence on the balance between cell proliferation and cell death (cell fate decision) in pathological circumstances remain elusive. Based on an extensive analysis of published data, we have built a comprehensive and generic reaction map for the MAPK signalling network, using CellDesigner software. In order to explore the MAPK responses to different stimuli and better understand their contributions to cell fate decision, we have considered the most crucial components and interactions and encoded them into a logical model, using the software GINsim. Our logical model analysis particularly focuses on urinary bladder cancer, where MAPK network deregulations have often been associated with specific phenotypes. To cope with the combinatorial explosion of the number of states, we have applied novel algorithms for model reduction and for the compression of state transition graphs, both implemented into the software GINsim. The results of systematic simulations for different signal combinations and network perturbations were found globally coherent with published data. In silico experiments further enabled us to delineate the roles of specific components, cross-talks and regulatory feedbacks in cell fate decision. Finally, tentative proliferative or anti-proliferative mechanisms can be connected with established bladder cancer deregulations, namely Epidermal Growth Factor Receptor (EGFR) over-expression and Fibroblast Growth Factor Receptor 3 (FGFR3) activating mutations.

Depending on environmental conditions, strongly intertwined cellular signalling pathways are activated, involving activation/inactivation of proteins and genes in response to external and/or internal stimuli. Alterations of some components of these pathways can lead to wrong cell behaviours. For instance, cancer-related deregulations lead to high proliferation of malignant cells enabling sustained tumour growth. Understanding the precise mechanisms underlying these pathways is necessary to delineate efficient therapeutical approaches for each specific tumour type. We particularly focused on the Mitogen-Activated Protein Kinase (MAPK) signalling network, whose involvement in cancer is well established, although the precise conditions leading to its positive or negative influence on cell proliferation are still poorly understood. We tackled this problem by first collecting sparse published biological information into a comprehensive map describing the MAPK network in terms of stylised chemical reactions. This information source was then used to build a dynamical Boolean model recapitulating network responses to characteristic stimuli observed in selected bladder cancers. Systematic model simulations further allowed us to link specific network components and interactions with proliferative/anti-proliferative cell responses.

From a biological hypothesis to the construction of a mathematical model

Mathematical models serve to explain complex biological phenomena and provide predictions that can be tested experimentally. They can provide plausible scenarios of a complex biological behavior when intuition is not sufficient anymore. The process from a biological hypothesis to a mathematical model might be challenging for biologists that are not familiar with mathematical modeling. In this chapter we discuss a possible workflow that describes the steps to be taken starting from a biological hypothesis on a biochemical cellular mechanism to the construction of a mathematical model using the appropriate formalism. An important part of this workflow is formalization of biological knowledge, which can be facilitated by existing tools and standards developed by the systems biology community. This chapter aims at introducing modeling to experts in molecular biology that would like to convert their hypotheses into mathematical models.