Mapping multivalued onto Boolean dynamics

Phenotype Control techniques for Boolean gene regulatory networks

Modeling cell signal transduction pathways via Boolean networks (BNs) has become an established method for analyzing intracellular communications over the last few decades. What's more, BNs provide a course-grained approach, not only to understanding molecular communications, but also for targeting pathway components that alter the long-term outcomes of the system. This has come to be known as phenotype control theory. In this review we study the interplay of various approaches for controlling gene regulatory networks such as: algebraic methods, control kernel, feedback vertex set, and stable motifs. The study will also include comparative discussion between the methods, using an established cancer model of T-Cell Large Granular Lymphocyte Leukemia. Further, we explore possible options for making the control search more efficient using reduction and modularity. Finally, we will include challenges presented such as the complexity and the availability of software for implementing each of these control techniques.

Trap spaces of multi-valued networks: definition, computation, and applications

MOTIVATION

Boolean networks are simple but efficient mathematical formalism for modelling complex biological systems. However, having only two levels of activation is sometimes not enough to fully capture the dynamics of real-world biological systems. Hence, the need for multi-valued networks (MVNs), a generalization of Boolean networks. Despite the importance of MVNs for modelling biological systems, only limited progress has been made on developing theories, analysis methods, and tools that can support them. In particular, the recent use of trap spaces in Boolean networks made a great impact on the field of systems biology, but there has been no similar concept defined and studied for MVNs to date.

RESULTS

In this work, we generalize the concept of trap spaces in Boolean networks to that in MVNs. We then develop the theory and the analysis methods for trap spaces in MVNs. In particular, we implement all proposed methods in a Python package called trapmvn. Not only showing the applicability of our approach via a realistic case study, we also evaluate the time efficiency of the method on a large collection of real-world models. The experimental results confirm the time efficiency, which we believe enables more accurate analysis on larger and more complex multi-valued models.

AVAILABILITY AND IMPLEMENTATION

Source code and data are freely available at https://github.com/giang-trinh/trap-mvn.

Phenotype control techniques for Boolean gene regulatory networks

Modeling cell signal transduction pathways via Boolean networks (BNs) has become an established method for analyzing intracellular communications over the last few decades. What’s more, BNs provide a course-grained approach, not only to understanding molecular communications, but also for targeting pathway components that alter the long-term outcomes of the system. This has come to be known as phenotype control theory . In this review we study the interplay of various approaches for controlling gene regulatory networks such as: algebraic methods, control kernel, feedback vertex set, and stable motifs. The study will also include comparative discussion between the methods, using an established cancer model of T-Cell Large Granular Lymphocyte (T-LGL) Leukemia. Further, we explore possible options for making the control search more efficient using reduction and modularity. Finally, we will include challenges presented such as the complexity and the availability of software for implementing each of these control techniques.

Boolean Models of the Transport, Synthesis, and Metabolism of Tryptophan in Escherichia coli

The tryptophan (trp) operon in Escherichia coli codes for the proteins responsible for the synthesis of the amino acid tryptophan from chorismic acid, and has been one of the most well-studied gene networks since its discovery in the 1960s. The tryptophanase (tna) operon codes for proteins needed to transport and metabolize it. Both of these have been modeled individually with delay differential equations under the assumption of mass-action kinetics. Recent work has provided strong evidence for bistable behavior of the tna operon. The authors of Orozco-Gómez et al. (Sci Rep 9(1):5451, 2019) identified a medium range of tryptophan in which the system has two stable steady-states, and they reproduced these experimentally. In this paper, we will show how a Boolean model can capture this bistability. We will also develop and analyze a Boolean model of the trp operon. Finally, we will combine these two to create a single Boolean model of the transport, synthesis, and metabolism of tryptophan. In this amalgamated model, the bistability disappears, presumably reflecting the ability of the trp operon to produce tryptophan and drive the system toward homeostasis. All of these models have longer attractors that we call "artifacts of synchrony", which disappear in the asynchronous automata. This curiously matches the behavior of a recent Boolean model of the arabinose operon in E. coli, and we discuss some open-ended questions that arise along these lines.

WebMaBoSS: A Web Interface for Simulating Boolean Models Stochastically

WebMaBoSS is an easy-to-use web interface for conversion, storage, simulation and analysis of Boolean models that allows to get insight from these models without any specific knowledge of modeling or coding. It relies on an existing software, MaBoSS, which simulates Boolean models using a stochastic approach: it applies continuous time Markov processes over the Boolean network. It was initially built to fill the gap between Boolean and continuous formalisms, i.e., providing semi-quantitative results using a simple representation with a minimum number of parameters to fit. The goal of WebMaBoSS is to simplify the use and the analysis of Boolean models coping with two main issues: 1) the simulation of Boolean models of intracellular processes with MaBoSS, or any modeling tool, may appear as non-intuitive for non-experts; 2) the simulation of already-published models available in current model databases (e.g., Cell Collective, BioModels) may require some extra steps to ensure compatibility with modeling tools such as MaBoSS. With WebMaBoSS, new models can be created or imported directly from existing databases. They can then be simulated, modified and stored in personal folders. Model simulations are performed easily, results visualized interactively, and figures can be exported in a preferred format. Extensive model analyses such as mutant screening or parameter sensitivity can also be performed. For all these tasks, results are stored and can be subsequently filtered to look for specific outputs. This web interface can be accessed at the address: https://maboss.curie.fr/webmaboss/ and deployed locally using docker. This application is open-source under LGPL license, and available at https://github.com/sysbio-curie/WebMaBoSS.

Computing Integrated Information (Φ) in Discrete Dynamical Systems with Multi-Valued Elements

Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (Φ). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.

Reconciling qualitative, abstract, and scalable modeling of biological networks

Predicting biological systems' behaviors requires taking into account many molecular and genetic elements for which limited information is available past a global knowledge of their pairwise interactions. Logical modeling, notably with Boolean Networks (BNs), is a well-established approach that enables reasoning on the qualitative dynamics of networks. Several dynamical interpretations of BNs have been proposed. The synchronous and (fully) asynchronous ones are the most prominent, where the value of either all or only one component can change at each step. Here we prove that, besides being costly to analyze, these usual interpretations can preclude the prediction of certain behaviors observed in quantitative systems. We introduce an execution paradigm, the Most Permissive Boolean Networks (MPBNs), which offers the formal guarantee not to miss any behavior achievable by a quantitative model following the same logic. Moreover, MPBNs significantly reduce the complexity of dynamical analysis, enabling to model genome-scale networks.

Bisimilar Booleanization of multivalued networks

Discrete modelling frameworks of Biological networks can be divided in two distinct categories: Boolean and multivalued. Although multivalued networks are more expressive for qualifying the regulatory behaviours modelled by more than two values, the ability to automatically convert them to Boolean network with an equivalent behaviour breaks down the fundamental borders between the two approaches. Theoretically investigating the conversion process provides relevant insights into bridging the gap between them. Basically, the conversion aims at finding a Boolean network bisimulating a multivalued one. In this article, we investigate the bisimilar conversion where the Boolean integer coding is a parameter that can be freely modified. Based on this analysis, we define a computational method automatically inferring a bisimilar Boolean network from a given multivalued one.

Basins of Attraction, Commitment Sets, and Phenotypes of Boolean Networks

The attractors of Boolean networks and their basins have been shown to be highly relevant for model validation and predictive modeling, e.g., in systems biology. Yet, there are currently very few tools available that are able to compute and visualize not only attractors but also their basins. In the realm of asynchronous, non-deterministic modeling not only is the repertoire of software even more limited, but also the formal notions for basins of attraction are often lacking. In this setting, the difficulty both for theory and computation arises from the fact that states may be elements of several distinct basins. In this paper, we address this topic by partitioning the state space into sets that are committed to the same attractors. These commitment sets can easily be generalized to sets that are equivalent w.r.t. the long-term behaviors of pre-selected nodes which leads us to the notions of markers and phenotypes which we illustrate in a case study on bladder tumorigenesis. For every concept, we propose equivalent CTL model checking queries and an extension of the state of the art model checking software NuSMV is made available that is capable of computing the respective sets. All notions are fully integrated as three new modules in our Python package PyBoolNet, including functions for visualizing the basins, commitment sets, and phenotypes as quotient graphs and pie charts.

Control of Intracellular Molecular Networks Using Algebraic Methods

Many problems in biology and medicine have a control component. Often, the goal might be to modify intracellular networks, such as gene regulatory networks or signaling networks, in order for cells to achieve a certain phenotype, what happens in cancer. If the network is represented by a mathematical model for which mathematical control approaches are available, such as systems of ordinary differential equations, then this problem might be solved systematically. Such approaches are available for some other model types, such as Boolean networks, where structure-based approaches have been developed, as well as stable motif techniques. However, increasingly many published discrete models are mixed-state or multistate, that is, some or all variables have more than two states, and thus the development of control strategies for multistate networks is needed. This paper presents a control approach broadly applicable to general multistate models based on encoding them as polynomial dynamical systems over a finite algebraic state set, and using computational algebra for finding appropriate intervention strategies. To demonstrate the feasibility and applicability of this method, we apply it to a recently developed multistate intracellular model of E2F-mediated bladder cancerous growth and to a model linking intracellular iron metabolism and oncogenic pathways. The control strategies identified for these published models are novel in some cases and represent new hypotheses, or are supported by the literature in others as potential drug targets. Our Macaulay2 scripts to find control strategies are publicly available through GitHub at https://github.com/luissv7/multistatepdscontrol.

Repairing Boolean logical models from time-series data using Answer Set Programming

Background

Boolean models of biological signalling-regulatory networks are increasingly used to formally describe and understand complex biological processes. These models may become inconsistent as new data become available and need to be repaired. In the past, the focus has been shed on the inference of (classes of) models given an interaction network and time-series data sets. However, repair of existing models against new data is still in its infancy, where the process is still manually performed and therefore slow and prone to errors.

Results

In this work, we propose a method with an associated tool to suggest repairs over inconsistent Boolean models, based on a set of atomic repair operations. Answer Set Programming is used to encode the minimal repair problem as a combinatorial optimization problem. In particular, given an inconsistent model, the tool provides the minimal repairs that render the model capable of generating dynamics coherent with a (set of) time-series data set(s), considering either a synchronous or an asynchronous updating scheme.

Conclusions

The method was validated using known biological models from different species, as well as synthetic models obtained from randomly generated networks. We discuss the method’s limitations regarding each of the updating schemes and the considered minimization algorithm.

Electronic supplementary material

The online version of this article (10.1186/s13015-019-0145-8) contains supplementary material, which is available to authorized users.

Computation of Robust Minimal Intervention Sets in Multi-Valued Biological Regulatory Networks

Enabled by rapid advances in computational sciences, in silico logical modeling of complex and large biological networks is more and more feasible making it an increasingly popular approach among biologists. Automated high-throughput, drug target identification is one of the primary goals of this in silico network biology. Targets identified in this way are then used to mine a library of drug chemical compounds in order to identify appropriate therapies. While identification of drug targets is exhaustively feasible on small networks, it remains computationally difficult on moderate and larger models. Moreover, there are several important constraints such as off-target effects, efficacy and safety that should be integrated into the identification of targets if the intention is translation to the clinical space. Here we introduce numerical constraints whereby efficacy is represented by efficiency in response and robustness of outcome. This paper introduces an algorithm that relies on a Constraint Satisfaction (CS) technique to efficiently compute the Minimal Intervention Sets (MIS) within a set of often complex clinical safety constraints with the aim of identifying the smallest least invasive set of targets pharmacologically accessible for therapy that most efficiently and reliably achieve the desired outcome.

BioLQM: A Java Toolkit for the Manipulation and Conversion of Logical Qualitative Models of Biological Networks

Here we introduce bioLQM, a new Java software toolkit for the conversion, modification, and analysis of Logical Qualitative Models of biological regulatory networks. BioLQM provides core modeling operations as building blocks for the development of integrated modeling software, or for the assembly of heterogeneous analysis workflows involving several complementary tools. Based on the definition of multi-valued logical models, bioLQM implements import and export facilities, notably for the recent SBML qual exchange format, as well as for formats used by several popular tools, facilitating the design of workflows combining these tools. Model modifications enable the definition of various perturbations, as well as model reduction, easing the analysis of large models. Another modification enables the study of multi-valued models with tools limited to the Boolean case. Finally, bioLQM provides a framework for the development of novel analysis tools. The current version implements various updating modes for model simulation (notably synchronous, asynchronous, and random asynchronous), as well as some static analysis features for the identification of attractors. The bioLQM software can be integrated into analysis workflows through command line and scripting interfaces. As a Java library, it further provides core data structures to the GINsim and EpiLog interactive tools, which supply graphical interfaces and additional analysis methods for cellular and multi-cellular qualitative models.

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.

Identification of control targets in Boolean molecular network models via computational algebra

BACKGROUND

Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system.

RESULTS

This paper presents a method for the identification of potential intervention targets in Boolean molecular network models using algebraic techniques. The approach exploits an algebraic representation of Boolean networks to encode the control candidates in the network wiring diagram as the solutions of a system of polynomials equations, and then uses computational algebra techniques to find such controllers. The control methods in this paper are validated through the identification of combinatorial interventions in the signaling pathways of previously reported control targets in two well studied systems, a p53-mdm2 network and a blood T cell lymphocyte granular leukemia survival signaling network. Supplementary data is available online and our code in Macaulay2 and Matlab are available via http://www.ms.uky.edu/~dmu228/ControlAlg .

CONCLUSIONS

This paper presents a novel method for the identification of intervention targets in Boolean network models. The results in this paper show that the proposed methods are useful and efficient for moderately large networks.

Analysis of a dynamic model of guard cell signaling reveals the stability of signal propagation

BACKGROUND

Analyzing the long-term behaviors (attractors) of dynamic models of biological systems can provide valuable insight into biological phenotypes and their stability. In this paper we identify the allowed long-term behaviors of a multi-level, 70-node dynamic model of the stomatal opening process in plants.

RESULTS

We start by reducing the model's huge state space. We first reduce unregulated nodes and simple mediator nodes, then simplify the regulatory functions of selected nodes while keeping the model consistent with experimental observations. We perform attractor analysis on the resulting 32-node reduced model by two methods: 1. converting it into a Boolean model, then applying two attractor-finding algorithms; 2. theoretical analysis of the regulatory functions. We further demonstrate the robustness of signal propagation by showing that a large percentage of single-node knockouts does not affect the stomatal opening level.

CONCLUSIONS

Combining both methods with analysis of perturbation scenarios, we conclude that all nodes except two in the reduced model have a single attractor; and only two nodes can admit oscillations. The multistability or oscillations of these four nodes do not affect the stomatal opening level in any situation. This conclusion applies to the original model as well in all the biologically meaningful cases. In addition, the stomatal opening level is resilient against single-node knockouts. Thus, we conclude that the complex structure of this signal transduction network provides multiple information propagation pathways while not allowing extensive multistability or oscillations, resulting in robust signal propagation. Our innovative combination of methods offers a promising way to analyze multi-level models.

Controllability of Boolean networks via input controls under Harvey's update scheme

In this article, the controllability of Boolean networks via input controls under Harvey's update scheme is investigated. First, the model of Boolean control networks under Harvey's stochastic update is proposed, by means of semi-tensor product approach, which is converted into discrete-time linear representation. And, a general formula of control-depending network transition matrix is provided. Second, based on discrete-time dynamics, controllability of the proposed model is analytically discussed by revealing the necessary and sufficient conditions of the reachable sets, respectively, for three kinds of controls, i.e., free Boolean control sequence, input control networks, and close-loop control. Examples are showed to demonstrate the effectiveness and feasibility of the proposed scheme.

Analysis and characterization of asynchronous state transition graphs using extremal states

Logical modeling of biological regulatory networks gives rise to a representation of the system's dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multivalued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step toward meaningful network reduction as well as model refinement methods.

Analysis of cell adhesion during early stages of colon cancer based on an extended multi-valued logic approach

Cell adhesion in the normal colon is typically associated with differentiated cells, whereas in cancerous colon it is associated with advanced tumors. For advanced tumors growing evidence supports the existence of stem-like cells that have originated from transdifferentiation. Because stem cells can also be transformed in their own niche, at the base of the Lieberkühn's crypts, we conjectured that cell adhesion can also be critical in early tumorigenesis. To assess this hypothesis we built an annotated, multi-valued logic model addressing cell adhesion of normal and tumorigenic stem cells in the human colon. The model accounts for (i) events involving intercellular adhesion structures, (ii) interactions involving cytoskeleton-related structures, (iii) compartmental distribution of α/β/γ/δ-catenins, and (iv) variations in critical cell adhesion regulators (e.g., ILK, FAK, IQGAP, SNAIL, Caveolin). We developed a method that can deal with graded multiple inhibitions, something which is not possible with conventional logical approaches. The model comprises 315 species (including 26 genes), interconnected by 269 reactions. Simulations of the model covered six scenarios, which considered two types of colonic cells (stem vs. differentiated cells), under three conditions (normal, stressed and tumor). Each condition results from the combination of 92 inputs. We compared our multi-valued logic approach with the conventional Boolean approach for one specific example and validated the predictions against published data. Our analysis suggests that stem cells in their niche synthesize high levels of cytoplasmatic E-cadherin and CdhEP(Ser684,686,692), even under normal-mitogenic stimulus or tumorigenic conditions. Under these conditions, E-cadherin would be incorporated into the plasmatic membrane, but only as a non-adhesive CdhE_β-catenin_IQGAP complex. Under stress conditions, however, this complex could be displaced, yielding adhesive CdhE_β-catenin((cis/trans)) complexes. In the three scenarios tested with stem cells, desmosomes or tight junctions were not assembled. Other model predictions include expected levels of the nuclear complex β-catenin_TCF4 and the anti-apoptotic protein Survivin for both normal and tumorigenic colonic stem cells.

Dynamic network-based epistasis analysis: boolean examples

In this article we focus on how the hierarchical and single-path assumptions of epistasis analysis can bias the inference of gene regulatory networks. Here we emphasize the critical importance of dynamic analyses, and specifically illustrate the use of Boolean network models. Epistasis in a broad sense refers to gene interactions, however, as originally proposed by Bateson, epistasis is defined as the blocking of a particular allelic effect due to the effect of another allele at a different locus (herein, classical epistasis). Classical epistasis analysis has proven powerful and useful, allowing researchers to infer and assign directionality to gene interactions. As larger data sets are becoming available, the analysis of classical epistasis is being complemented with computer science tools and system biology approaches. We show that when the hierarchical and single-path assumptions are not met in classical epistasis analysis, the access to relevant information and the correct inference of gene interaction topologies is hindered, and it becomes necessary to consider the temporal dynamics of gene interactions. The use of dynamical networks can overcome these limitations. We particularly focus on the use of Boolean networks that, like classical epistasis analysis, relies on logical formalisms, and hence can complement classical epistasis analysis and relax its assumptions. We develop a couple of theoretical examples and analyze them from a dynamic Boolean network model perspective. Boolean networks could help to guide additional experiments and discern among alternative regulatory schemes that would be impossible or difficult to infer without the elimination of these assumption from the classical epistasis analysis. We also use examples from the literature to show how a Boolean network-based approach has resolved ambiguities and guided epistasis analysis. Our article complements previous accounts, not only by focusing on the implications of the hierarchical and single-path assumption, but also by demonstrating the importance of considering temporal dynamics, and specifically introducing the usefulness of Boolean network models and also reviewing some key properties of network approaches.

Reduction of Boolean network models

Boolean networks have been successfully used in modelling gene regulatory networks. However, for large networks, analysis by simulation becomes unfeasible. In this paper we propose a reduction method for Boolean networks that decreases the size of the network, while preserving important dynamical properties and topological features. As a result, the reduced network can be used to infer properties about the original network and to gain a better understanding of the role of network topology on the dynamics. In particular, we use the reduction method to study steady states of Boolean networks and apply our results to models of Th-lymphocyte differentiation and the lac operon.