Attractor analysis of asynchronous Boolean models of signal transduction networks

Models of Cell Processes are Far from the Edge of Chaos

Complex living systems are thought to exist at the "edge of chaos" separating the ordered dynamics of robust function from the disordered dynamics of rapid environmental adaptation. Here, a deeper inspection of 72 experimentally supported discrete dynamical models of cell processes reveals previously unobserved order on long time scales, suggesting greater rigidity in these systems than was previously conjectured. We find that propagation of internal perturbations is transient in most cases, and that even when large perturbation cascades persist, their phenotypic effects are often minimal. Moreover, we find evidence that stochasticity and desynchronization can lead to increased recovery from regulatory perturbation cascades. Our analysis relies on new measures that quantify the tendency of perturbations to spread through a discrete dynamical system. Computing these measures was not feasible using current methodology; thus, we developed a multipurpose CUDA-based simulation tool, which we have made available as the open-source Python library cubewalkers. Based on novel measures and simulations, our results suggest that-contrary to current theory-cell processes are ordered and far from the edge of chaos.

Modulating the dynamics of NFκB and PI3K enhances the ensemble-level TNFR1 signaling mediated apoptotic response

Cell-to-cell variability during TNFα stimulated Tumor Necrosis Factor Receptor 1 (TNFR1) signaling can lead to single-cell level pro-survival and apoptotic responses. This variability stems from the heterogeneity in signal flow through intracellular signaling entities that regulate the balance between these two phenotypes. Using systematic Boolean dynamic modeling of a TNFR1 signaling network, we demonstrate that the signal flow path variability can be modulated to enable cells favour apoptosis. We developed a computationally efficient approach "Boolean Modeling based Prediction of Steady-state probability of Phenotype Reachability (BM-ProSPR)" to accurately predict the network's ability to settle into different phenotypes. Model analysis juxtaposed with the experimental observations revealed that NFκB and PI3K transient responses guide the XIAP behaviour to coordinate the crucial dynamic cross-talk between the pro-survival and apoptotic arms at the single-cell level. Model predicted the experimental observations that ~31% apoptosis increase can be achieved by arresting Comp1 - IKK* activity which regulates the NFκB and PI3K dynamics. Arresting Comp1 - IKK* activity causes signal flow path re-wiring towards apoptosis without significantly compromising NFκB levels, which govern adequate cell survival. Priming an ensemble of cancerous cells with inhibitors targeting the specific interaction involving Comp1 and IKK* prior to TNFα exposure could enable driving them towards apoptosis.

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.

Weak Stabilization of Boolean Networks Under State-Flipped Control

In this brief, stabilization of Boolean networks (BNs) by flipping a subset of nodes is considered, here we call such action state-flipped control. The state-flipped control implies that the logical variables of certain nodes are flipped from 1 to 0 or 0 to 1 as time flows. Under state-flipped control on certain nodes, a state-flipped-transition matrix is defined to describe the impact on the state transition space. Weak stabilization is first defined and then some criteria are presented to judge the same. An algorithm is proposed to find a stabilizing kernel such that BNs can achieve weak stabilization to the desired state with in-degree more than 0. By defining a reachable set, another approach is proposed to verify weak stabilization, and an algorithm is given to obtain a flip sequence steering an initial state to a given target state. Subsequently, the issue of finding flip sequences to steer BNs from weak stabilization to global stabilization is addressed. In addition, a model-free reinforcement algorithm, namely the Q -learning ( [Formula: see text]) algorithm, is developed to find flip sequences to achieve global stabilization. Finally, several numerical examples are given to illustrate the obtained theoretical results.

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.

Statistical control of structural networks with limited interventions to minimize cellular phenotypic diversity represented by point attractors

The underlying genetic networks of cells give rise to diverse behaviors known as phenotypes. Control of this cellular phenotypic diversity (CPD) may reveal key targets that govern differentiation during development or drug resistance in cancer. This work establishes an approach to control CPD that encompasses practical constraints, including model limitations, the number of simultaneous control targets, which targets are viable for control, and the granularity of control. Cellular networks are often limited to the structure of interactions, due to the practical difficulty of modeling interaction dynamics. However, these dynamics are essential to CPD. In response, our statistical control approach infers the CPD directly from the structure of a network, by considering an ensemble average function over all possible Boolean dynamics for each node in the network. These ensemble average functions are combined with an acyclic form of the network to infer the number of point attractors. Our approach is applied to several known biological models and shown to outperform existing approaches. Statistical control of CPD offers a new avenue to contend with systemic processes such as differentiation and cancer, despite practical limitations in the field.

Probabilistic edge weights fine-tune Boolean network dynamics

Biological systems are noisy by nature. This aspect is reflected in our experimental measurements and should be reflected in the models we build to better understand these systems. Noise can be especially consequential when trying to interpret specific regulatory interactions, i.e. regulatory network edges. In this paper, we propose a method to explicitly encode edge-noise in Boolean dynamical systems by probabilistic edge-weight (PEW) operators. PEW operators have two important features: first, they introduce a form of edge-weight into Boolean models through the noise, second, the noise is dependent on the dynamical state of the system, which enables more biologically meaningful modeling choices. Moreover, we offer a simple-to-use implementation in the already well-established BooleanNet framework. In two application cases, we show how the introduction of just a few PEW operators in Boolean models can fine-tune the emergent dynamics and increase the accuracy of qualitative predictions. This includes fine-tuning interactions which cause non-biological behaviors when switching between asynchronous and synchronous update schemes in dynamical simulations. Moreover, PEW operators also open the way to encode more exotic cellular dynamics, such as cellular learning, and to implementing edge-weights for regulatory networks inferred from omics data.

Cyclic attractors of nonexpanding q-ary networks

Discrete dynamical systems in which model components take on categorical values have been successfully applied to biological networks to study their global dynamic behavior. Boolean models in particular have been used extensively. However, multi-state models have also emerged as effective computational tools for the analysis of complex mechanisms underlying biological networks. Models in which variables assume more than two discrete states provide greater resolution, but this scheme introduces discontinuities. In particular, variables can increase or decrease by more than one unit in one time step. This can be corrected, without changing fixed points of the system, by applying an additional rule to each local activation function. On the other hand, if one is interested in cyclic attractors of their system, then this rule can potentially introduce new cyclic attractors that were not observed previously. This article makes some advancements in understanding the state space dynamics of multi-state network models with synchronous, sequential, or block-sequential update schedules and establishes conditions under which no new cyclic attractors are added to networks when the additional rule is applied. Our analytical results have the potential to be incorporated into modeling software and aid researchers in their analyses of biological multi-state networks.

SNAIL driven by a feed forward loop motif promotes TGFβinduced epithelial to mesenchymal transition

Epithelial to Mesenchymal Transition (EMT) plays an important role in tissue regeneration, embryonic development, and cancer metastasis. Several signaling pathways are known to regulate EMT, among which the modulation of TGFβ(Transforming Growth Factor-β) induced EMT is crucial in several cancer types. Several mathematical models were built to explore the role of core regulatory circuit of ZEB/miR-200, SNAIL/miR-34 double negative feedback loops in modulating TGFβinduced EMT. Different emergent behavior including tristability, irreversible switching, existence of hybrid EMT states were inferred though these models. Some studies have explored the role of TGFβreceptor activation, SMADs nucleocytoplasmic shuttling and complex formation. Recent experiments have revealed that MDM2 along with SMAD complex regulates SNAIL expression driven EMT. Encouraged by this, in the present study we developed a mathematical model for p53/MDM2 dependent TGFβinduced EMT regulation. Inclusion of p53 brings in an additional mechanistic perspective in exploring the EM transition. The network formulated comprises a C1FFL moderating SNAIL expression involving MDM2 and SMAD complex, which functions as a noise filter and persistent detector. The C1FFL was also observed to operate as a coincidence detector driving the SNAIL dependent downstream signaling into phenotypic switching decision. Systems modelling and analysis of the devised network, displayed interesting dynamic behavior, systems response to various inputs stimulus, providing a better understanding of p53/MDM2 dependent TGF-βinduced Epithelial to Mesenchymal Transition.

Exploring attractor bifurcations in Boolean networks

Background

Boolean networks (BNs) provide an effective modelling formalism for various complex biochemical phenomena. Their long term behaviour is represented by attractors–subsets of the state space towards which the BN eventually converges. These are then typically linked to different biological phenotypes. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a methodology for analysing bifurcations in asynchronous parametrised Boolean networks.

Results

In this paper, we propose a computational framework employing advanced symbolic graph algorithms that enable the analysis of large networks with hundreds of Boolean variables. To visualise the results of this analysis, we developed a novel interactive presentation technique based on decision trees, allowing us to quickly uncover parameters crucial to the changes in the attractor landscape. As a whole, the methodology is implemented in our tool AEON. We evaluate the method’s applicability on a complex human cell signalling network describing the activity of type-1 interferons and related molecules interacting with SARS-COV-2 virion. In particular, the analysis focuses on explaining the potential suppressive role of the recently proposed drug molecule GRL0617 on replication of the virus.

Conclusions

The proposed method creates a working analogy to the concept of bifurcation analysis widely used in kinetic modelling to reveal the impact of parameters on the system’s stability. The important feature of our tool is its unique capability to work fast with large-scale networks with a relatively large extent of unknown information. The results obtained in the case study are in agreement with the recent biological findings.

On Attractor Detection and Optimal Control of Deterministic Generalized Asynchronous Random Boolean Networks

Deterministic asynchronous Boolean networks play a crucial role in modeling and analysis of gene regulatory networks. In this paper, we focus on a typical type of deterministic asynchronous Boolean networks called deterministic generalized asynchronous random Boolean networks (DGARBNs). We first formulate the extended state transition graph, which captures the whole dynamics of a DGARBN and paves potential ways to analyze this DGARBN. We then propose two SMT-based methods for attractor detection and optimal control of DGARBNs. These methods are implemented in a JAVA tool called DABoolNet. Two experiments are designed to highlight the scalability of the proposed methods. We also formally state and prove several relations between DGARBNs and other models including deterministic asynchronous models, block-sequential Boolean networks, generalized asynchronous random Boolean networks, and mixed-context random Boolean networks. Several case studies are presented to show the applications of our methods.

Modelling of glucose repression signalling in yeast Saccharomyces cerevisiae

Saccharomyces cerevisiae has a sophisticated signalling system that plays a crucial role in cellular adaptation to changing environments. The SNF1 pathway regulates energy homeostasis upon glucose derepression; hence, it plays an important role in various processes, such as metabolism, cell cycle and autophagy. To unravel its behaviour, SNF1 signalling has been extensively studied. However, the pathway components are strongly interconnected and inconstant; therefore, elucidating its dynamic behaviour based on experimental data only is challenging. To tackle this complexity, systems biology approaches have been successfully employed. This review summarizes the progress, advantages and disadvantages of the available mathematical modelling frameworks covering Boolean, dynamic kinetic, single-cell models, which have been used to study processes and phenomena ranging from crosstalks to sources of cell-to-cell variability in the context of SNF1 signalling. Based on the lessons from existing models, we further discuss how to develop a consensus dynamic mechanistic model of the entire SNF1 pathway that can provide novel insights into the dynamics of nutrient signalling.

An FVS-Based Approach to Attractor Detection in Asynchronous Random Boolean Networks

Boolean networks (BNs)play a crucial role in modeling and analyzing biological systems. One of the central issues in the analysis of BNs is attractor detection, i.e., identification of all possible attractors. This problem becomes more challenging for large asynchronous random Boolean networks (ARBNs)because of the asynchronous and non-deterministic updating scheme. In this paper, we present and formally prove several relations between feedback vertex sets (FVSs)and dynamics of BNs. From these relations, we propose an FVS-based method for detecting attractors in ARBNs. Our approach relies on the principle of removing arcs in the state transition graph to get a candidate set and the reachability property to filter the candidate set. We formally prove the correctness of our method and show its efficiency by conducting experiments on real biological networks and randomly generated N- K networks. The obtained results are very promising since our method can handle large networks whose sizes are up to 101 without using any network reduction technique.

Boolean modelling in plant biology

Signalling and genetic networks underlie most biological processes and are often complex, containing many highly connected components. Modelling these networks can provide insight into mechanisms but is challenging given that rate parameters are often not well defined. Boolean modelling, in which components can only take on a binary value with connections encoded by logic equations, is able to circumvent some of these challenges, and has emerged as a viable tool to probe these complex networks. In this review, we will give an overview of Boolean modelling, with a specific emphasis on its use in plant biology. We review how Boolean modelling can be used to describe biological networks and then discuss examples of its applications in plant genetics and plant signalling.

Boolean factor graph model for biological systems: the yeast cell-cycle network

Background

The desire to understand genomic functions and the behavior of complex gene regulatory networks has recently been a major research focus in systems biology. As a result, a plethora of computational and modeling tools have been proposed to identify and infer interactions among biological entities. Here, we consider the general question of the effect of perturbation on the global dynamical network behavior as well as error propagation in biological networks to incite research pertaining to intervention strategies.

Results

This paper introduces a computational framework that combines the formulation of Boolean networks and factor graphs to explore the global dynamical features of biological systems. A message-passing algorithm is proposed for this formalism to evolve network states as messages in the graph. In addition, the mathematical formulation allows us to describe the dynamics and behavior of error propagation in gene regulatory networks by conducting a density evolution (DE) analysis. The model is applied to assess the network state progression and the impact of gene deletion in the budding yeast cell cycle. Simulation results show that our model predictions match published experimental data. Also, our findings reveal that the sample yeast cell-cycle network is not only robust but also consistent with real high-throughput expression data. Finally, our DE analysis serves as a tool to find the optimal values of network parameters for resilience against perturbations, especially in the inference of genetic graphs.

Conclusion

Our computational framework provides a useful graphical model and analytical tools to study biological networks. It can be a powerful tool to predict the consequences of gene deletions before conducting wet bench experiments because it proves to be a quick route to predicting biologically relevant dynamic properties without tunable kinetic parameters.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04361-8.

Unsupervised logic-based mechanism inference for network-driven biological processes

Modern analytical techniques enable researchers to collect data about cellular states, before and after perturbations. These states can be characterized using analytical techniques, but the inference of regulatory interactions that explain and predict changes in these states remains a challenge. Here we present a generalizable, unsupervised approach to generate parameter-free, logic-based models of cellular processes, described by multiple discrete states. Our algorithm employs a Hamming-distance based approach to formulate, test, and identify optimized logic rules that link two states. Our approach comprises two steps. First, a model with no prior knowledge except for the mapping between initial and attractor states is built. We then employ biological constraints to improve model fidelity. Our algorithm automatically recovers the relevant dynamics for the explored models and recapitulates key aspects of the biochemical species concentration dynamics in the original model. We present the advantages and limitations of our work and discuss how our approach could be used to infer logic-based mechanisms of signaling, gene-regulatory, or other input-output processes describable by the Boolean formalism.

Control for multifunctionality: bioinspired control based on feeding in Aplysia californica

Animals exhibit remarkable feats of behavioral flexibility and multifunctional control that remain challenging for robotic systems. The neural and morphological basis of multifunctionality in animals can provide a source of bioinspiration for robotic controllers. However, many existing approaches to modeling biological neural networks rely on computationally expensive models and tend to focus solely on the nervous system, often neglecting the biomechanics of the periphery. As a consequence, while these models are excellent tools for neuroscience, they fail to predict functional behavior in real time, which is a critical capability for robotic control. To meet the need for real-time multifunctional control, we have developed a hybrid Boolean model framework capable of modeling neural bursting activity and simple biomechanics at speeds faster than real time. Using this approach, we present a multifunctional model of Aplysia californica feeding that qualitatively reproduces three key feeding behaviors (biting, swallowing, and rejection), demonstrates behavioral switching in response to external sensory cues, and incorporates both known neural connectivity and a simple bioinspired mechanical model of the feeding apparatus. We demonstrate that the model can be used for formulating testable hypotheses and discuss the implications of this approach for robotic control and neuroscience.

Concepts in Boolean network modeling: What do they all mean?

Boolean network models are one of the simplest models to study complex dynamic behavior in biological systems. They can be applied to unravel the mechanisms regulating the properties of the system or to identify promising intervention targets. Since its introduction by Stuart Kauffman in 1969 for describing gene regulatory networks, various biologically based networks and tools for their analysis were developed. Here, we summarize and explain the concepts for Boolean network modeling. We also present application examples and guidelines to work with and analyze Boolean network models.

Pinning Controllers for Activation Output Tracking of Boolean Network Under One-Bit Perturbation

This paper studies pinning controllers for activation output tracking (AOT) of Boolean network under one-bit perturbation, based on the semitensor product of matrices. First, the definition of AOT with respect to an activation number is presented, where the activation number means the number of active outputs whose logical variables are 1 s. Then, several criteria are established for AOT issue. Further, the impact of one-bit perturbation on AOT is studied, where one-bit perturbation means that only one logical function has one-bit change of its truth table by flipping the value from 1 to 0 or 0 to 1. In addition, if a one-bit perturbation is a valid perturbation on AOT, an output feedback pinning control is designed to recover AOT. The obtained results are effectively illustrated by a D. melanogaster segmentation polarity gene network and a reduced signal transduction network.

Boolean model of growth signaling, cell cycle and apoptosis predicts the molecular mechanism of aberrant cell cycle progression driven by hyperactive PI3K

The PI3K/AKT signaling pathway plays a role in most cellular functions linked to cancer progression, including cell growth, proliferation, cell survival, tissue invasion and angiogenesis. It is generally recognized that hyperactive PI3K/AKT1 are oncogenic due to their boost to cell survival, cell cycle entry and growth-promoting metabolism. That said, the dynamics of PI3K and AKT1 during cell cycle progression are highly nonlinear. In addition to negative feedback that curtails their activity, protein expression of PI3K subunits has been shown to oscillate in dividing cells. The low-PI3K/low-AKT1 phase of these oscillations is required for cytokinesis, indicating that oncogenic PI3K may directly contribute to genome duplication. To explore this, we construct a Boolean model of growth factor signaling that can reproduce PI3K oscillations and link them to cell cycle progression and apoptosis. The resulting modular model reproduces hyperactive PI3K-driven cytokinesis failure and genome duplication and predicts the molecular drivers responsible for these failures by linking hyperactive PI3K to mis-regulation of Polo-like kinase 1 (Plk1) expression late in G2. To do this, our model captures the role of Plk1 in cell cycle progression and accurately reproduces multiple effects of its loss: G2 arrest, mitotic catastrophe, chromosome mis-segregation / aneuploidy due to premature anaphase, and cytokinesis failure leading to genome duplication, depending on the timing of Plk1 inhibition along the cell cycle. Finally, we offer testable predictions on the molecular drivers of PI3K oscillations, the timing of these oscillations with respect to division, and the role of altered Plk1 and FoxO activity in genome-level defects caused by hyperactive PI3K. Our model is an important starting point for the predictive modeling of cell fate decisions that include AKT1-driven senescence, as well as the non-intuitive effects of drugs that interfere with mitosis.

Edgetic perturbations to eliminate fixed-point attractors in Boolean regulatory networks

The dynamics of complex biological networks may be modeled in a Boolean framework, where the state of each system component is either abundant (ON) or scarce/absent (OFF), and each component's dynamic trajectory is determined by a logical update rule involving the state(s) of its regulator(s). It is possible to encode the update rules in the topology of the so-called expanded graph, analysis of which reveals the long-term behavior, or attractors, of the network. Here, we develop an algorithm to perturb the expanded graph (or, equivalently, the logical update rules) to eliminate stable motifs: subgraphs that cause a subset of components to stabilize to one state. Depending on the topology of the expanded graph, these perturbations lead to the modification or loss of the corresponding attractor. While most perturbations of biological regulatory networks in the literature involve the knockout (fixing to OFF) or constitutive activation (fixing to ON) of one or more nodes, we here consider edgetic perturbations, where a node's update rule is modified such that one or more of its regulators is viewed as ON or OFF regardless of its actual state. We apply the methodology to two biological networks. In a network representing T-LGL leukemia, we identify edgetic perturbations that eliminate the cancerous attractor, leaving only the healthy attractor representing cell death. In a network representing drought-induced closure of plant stomata, we identify edgetic perturbations that modify the single attractor such that stomata, instead of being fixed in the closed state, oscillates between the open and closed states.

Taming Asynchrony for Attractor Detection in Large Boolean Networks

Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analyzing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous scheme. Existing methods are prohibited due to the well-known state-space explosion problem in large Boolean networks. In this paper, we tackle this challenge by proposing a SCC-based decomposition method. We prove the correctness of our proposed method and demonstrate its efficiency with two real-life biological networks.

Boolean Calcium Signalling Model Predicts Calcium Role in Acceleration and Stability of Abscisic Acid-Mediated Stomatal Closure

Inconsistent hypotheses have proposed Ca²⁺ as either being essential or irrelevant and redundant in ABA induced stomatal closure. This study integrates all available information from literature to define ABA signalling pathway and presents it in a systems view for clearer understanding of the role of Ca²⁺ in stomatal closure. Importantly, it incorporates into an Asynchronous Boolean model time delays sourced from an extensive literature search. The model predicted the timing of ABA events and mutant behaviour close to biology. It revealed biologically reported timing for Ca²⁺ activation and Ca²⁺ dynamics consistent with biology. It also predicts that Ca²⁺ elevation is not essential in stomatal closure but it can accelerate closure, consistent with previous findings, but our model further explains that acting as a mediator, Ca²⁺ accelerates stomatal closure by enhancing plasma membrane slowly activating anion channel SLAC1 and actin rearrangement. It shows statistical significance of Ca²⁺ induced acceleration of closure and that of Ca²⁺ induced acceleration of SLAC1 activation. Further, the model demonstrates that Ca²⁺ enhances resilience of closure to perturbation of important elements; especially, ROS pathway, as did previous ABA model, and even to the ABA signal disruption. It goes further to elucidate the mechanisms by which Ca²⁺ engenders stomatal closure in these perturbations.

General method to find the attractors of discrete dynamic models of biological systems

Analyzing the long-term behaviors (attractors) of dynamic models of biological networks can provide valuable insight. We propose a general method that can find the attractors of multilevel discrete dynamical systems by extending a method that finds the attractors of a Boolean network model. The previous method is based on finding stable motifs, subgraphs whose nodes' states can stabilize on their own. We extend the framework from binary states to any finite discrete levels by creating a virtual node for each level of a multilevel node, and describing each virtual node with a quasi-Boolean function. We then create an expanded representation of the multilevel network, find multilevel stable motifs and oscillating motifs, and identify attractors by successive network reduction. In this way, we find both fixed point attractors and complex attractors. We implemented an algorithm, which we test and validate on representative synthetic networks and on published multilevel models of biological networks. Despite its primary motivation to analyze biological networks, our motif-based method is general and can be applied to any finite discrete dynamical system.

Determining Relative Dynamic Stability of Cell States Using Boolean Network Model

Cell state transition is at the core of biological processes in metazoan, which includes cell differentiation, epithelial-to-mesenchymal transition (EMT) and cell reprogramming. In these cases, it is important to understand the molecular mechanism of cellular stability and how the transitions happen between different cell states, which is controlled by a gene regulatory network (GRN) hard-wired in the genome. Here we use Boolean modeling of GRN to study the cell state transition of EMT and systematically compare four available methods to calculate the cellular stability of three cell states in EMT in both normal and genetically mutated cases. The results produced from four methods generally agree but do not totally agree with each other. We show that distribution of one-degree neighborhood of cell states, which are the nearest states by Hamming distance, causes the difference among the methods. From that, we propose a new method based on one-degree neighborhood, which is the simplest one and agrees with other methods to estimate the cellular stability in all scenarios of our EMT model. This new method will help the researchers in the field of cell differentiation and cell reprogramming to calculate cellular stability using Boolean model, and then rationally design their experimental protocols to manipulate the cell state transition.

Target Control in Logical Models Using the Domain of Influence of Nodes

Dynamical models of biomolecular networks are successfully used to understand the mechanisms underlying complex diseases and to design therapeutic strategies. Network control and its special case of target control, is a promising avenue toward developing disease therapies. In target control it is assumed that a small subset of nodes is most relevant to the system's state and the goal is to drive the target nodes into their desired states. An example of target control would be driving a cell to commit to apoptosis (programmed cell death). From the experimental perspective, gene knockout, pharmacological inhibition of proteins, and providing sustained external signals are among practical intervention techniques. We identify methodologies to use the stabilizing effect of sustained interventions for target control in Boolean network models of biomolecular networks. Specifically, we define the domain of influence (DOI) of a node (in a certain state) to be the nodes (and their corresponding states) that will be ultimately stabilized by the sustained state of this node regardless of the initial state of the system. We also define the related concept of the logical domain of influence (LDOI) of a node, and develop an algorithm for its identification using an auxiliary network that incorporates the regulatory logic. This way a solution to the target control problem is a set of nodes whose DOI can cover the desired target node states. We perform greedy randomized adaptive search in node state space to find such solutions. We apply our strategy to in silico biological network models of real systems to demonstrate its effectiveness.

Dynamics in Epistasis Analysis

Finding regulatory relationships between genes, including the direction and nature of influence between them, is a fundamental challenge in the field of molecular genetics. One classical approach to this problem is epistasis analysis. Broadly speaking, epistasis analysis infers the regulatory relationships between a pair of genes in a genetic pathway by considering the patterns of change in an observable trait resulting from single and double deletion of genes. While classical epistasis analysis has yielded deep insights on numerous genetic pathways, it is not without limitations. Here, we explore the possibility of dynamic epistasis analysis, in which, in addition to performing genetic perturbations of a pathway, we drive the pathway by a time-varying upstream signal. We explore the theoretical power of dynamical epistasis analysis by conducting an identifiability analysis of Boolean models of genetic pathways, comparing static and dynamic approaches. We find that even relatively simple input dynamics greatly increases the power of epistasis analysis to discriminate alternative network structures. Further, we explore the question of experiment design, and show that a subset of short time-varying signals, which we call dynamic primitives, allow maximum discriminative power with a reduced number of experiments.

Identifying a Probabilistic Boolean Threshold Network From Samples

This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and -tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

Griffin: A Tool for Symbolic Inference of Synchronous Boolean Molecular Networks

Boolean networks are important models of biochemical systems, located at the high end of the abstraction spectrum. A number of Boolean gene networks have been inferred following essentially the same method. Such a method first considers experimental data for a typically underdetermined “regulation” graph. Next, Boolean networks are inferred by using biological constraints to narrow the search space, such as a desired set of (fixed-point or cyclic) attractors. We describe Griffin, a computer tool enhancing this method. Griffin incorporates a number of well-established algorithms, such as Dubrova and Teslenko's algorithm for finding attractors in synchronous Boolean networks. In addition, a formal definition of regulation allows Griffin to employ “symbolic” techniques, able to represent both large sets of network states and Boolean constraints. We observe that when the set of attractors is required to be an exact set, prohibiting additional attractors, a naive Boolean coding of this constraint may be unfeasible. Such cases may be intractable even with symbolic methods, as the number of Boolean constraints may be astronomically large. To overcome this problem, we employ an Artificial Intelligence technique known as “clause learning” considerably increasing Griffin's scalability. Without clause learning only toy examples prohibiting additional attractors are solvable: only one out of seven queries reported here is answered. With clause learning, by contrast, all seven queries are answered. We illustrate Griffin with three case studies drawn from the Arabidopsis thaliana literature. Griffin is available at: http://turing.iimas.unam.mx/griffin.

A systems pharmacology model for inflammatory bowel disease

Motivation

The literature on complex diseases is abundant but not always quantitative. This is particularly so for Inflammatory Bowel Disease (IBD), where many molecular pathways are qualitatively well described but this information cannot be used in traditional quantitative mathematical models employed in drug development. We propose the elaboration and validation of a logic network for IBD able to capture the information available in the literature that will facilitate the identification/validation of therapeutic targets.

Results

In this article, we propose a logic model for Inflammatory Bowel Disease (IBD) which consists of 43 nodes and 298 qualitative interactions. The model presented is able to describe the pathogenic mechanisms of the disorder and qualitatively describes the characteristic chronic inflammation. A perturbation analysis performed on the IBD network indicates that the model is robust. Also, as described in clinical trials, a simulation of anti-TNFα, anti-IL2 and Granulocyte and Monocyte Apheresis showed a decrease in the Metalloproteinases node (MMPs), which means a decrease in tissue damage. In contrast, as clinical trials have demonstrated, a simulation of anti-IL17 and anti-IFNγ or IL10 overexpression therapy did not show any major change in MMPs expression, as corresponds to a failed therapy. The model proved to be a promising in silico tool for the evaluation of potential therapeutic targets, the identification of new IBD biomarkers, the integration of IBD polymorphisms to anticipate responders and non-responders and can be reduced and transformed in quantitative model/s.

Method for identification of sensitive nodes in Boolean models of biological networks

Biological systems are often represented as Boolean networks and analysed to identify sensitive nodes which on perturbation disproportionately change a predefined output. There exist different kinds of perturbation methods: perturbation of function, perturbation of state and perturbation in update scheme. Nodes may have defects in interpretation of the inputs from other nodes and calculation of the node output. To simulate these defects and systematically assess their effect on the system output, two new function perturbations, referred to as ‘not of function’ and ‘function of not’, are introduced. In the former, the inputs are assumed to be correctly interpreted but the output of the update rule is perturbed; and in the latter, each input is perturbed but the correct update rule is applied. These and previously used perturbation methods were applied to two existing Boolean models, namely the human melanogenesis signalling network and the fly segment polarity network. Through mathematical simulations, it was found that these methods successfully identified nodes earlier found to be sensitive using other methods, and were also able to identify sensitive nodes which were previously unreported.

Therapeutic target discovery using Boolean network attractors: improvements of kali

In a previous article, an algorithm for identifying therapeutic targets in Boolean networks modelling pathological mechanisms was introduced. In the present article, the improvements made on this algorithm, named kali, are described. These improvements are (i) the possibility to work on asynchronous Boolean networks, (ii) a finer assessment of therapeutic targets and (iii) the possibility to use multivalued logic. kali assumes that the attractors of a dynamical system, such as a Boolean network, are associated with the phenotypes of the modelled biological system. Given a logic-based model of pathological mechanisms, kali searches for therapeutic targets able to reduce the reachability of the attractors associated with pathological phenotypes, thus reducing their likeliness. kali is illustrated on an example network and used on a biological case study. The case study is a published logic-based model of bladder tumorigenesis from which kali returns consistent results. However, like any computational tool, kali can predict but cannot replace human expertise: it is a supporting tool for coping with the complexity of biological systems in the field of drug discovery.

ViSiBooL-visualization and simulation of Boolean networks with temporal constraints

Summary

Mathematical models and their simulation are increasingly used to gain insights into cellular pathways and regulatory networks. Dynamics of regulatory factors can be modeled using Boolean networks (BNs), among others. Text-based representations of models are precise descriptions, but hard to understand and interpret. ViSiBooL aims at providing a graphical way of modeling and simulating networks. By providing visualizations of static and dynamic network properties simultaneously, it is possible to directly observe the effects of changes in the network structure on the behavior. In order to address the challenges of clear design and a user-friendly graphical user interface (GUI), ViSiBooL implements visual representations of BNs. Additionally temporal extensions of the BNs for the modeling of regulatory time delays are incorporated. The GUI of ViSiBooL allows to model, organize, simulate and visualize BNs as well as corresponding simulation results such as attractors. Attractor searches are performed in parallel to the modeling process. Hence, changes in the network behavior are visualized at the same time.

Availability and Implementation

ViSiBooL (Java 8) is freely available at http://sysbio.uni-ulm.de/?Software:ViSiBooL .

Contact

hans.kestler@uni-ulm.de.

Developing Network Models of Multiscale Host Responses Involved in Infections and Diseases

Complex interactions involved in host response to infections and diseases require advanced analytical tools to infer drivers of the response in order to develop strategies for intervention. This chapter discusses approaches to assemble interactions ranging from molecular to cellular levels and their analysis to investigate the cross talk between immune pathways. Particularly, construction of immune networks by either data-driven or literature-driven methods is explained. Next, graph theoretic approaches for probing static network properties as well as visualization of networks are discussed. Finally, development of Boolean models for simulation of network dynamics to investigate cross talk and emergent properties are considered along with Boolean-like models that may compensate for some of the limitations encountered in Boolean simulations. In conclusion, the chapter will allow readers to construct and analyze multiscale networks involved in immune responses.

A network modeling approach to elucidate drug resistance mechanisms and predict combinatorial drug treatments in breast cancer

Background

Mechanistic models of within-cell signal transduction networks can explain how these networks integrate internal and external inputs to give rise to the appropriate cellular response. These models can be fruitfully used in cancer cells, whose aberrant decision-making regarding their survival or death, proliferation or quiescence can be connected to errors in the state of nodes or edges of the signal transduction network.

Results

Here we present a comprehensive network, and discrete dynamic model, of signal transduction in ER+ breast cancer based on the literature of ER+, HER2+, and PIK3CA-mutant breast cancers. The network model recapitulates known resistance mechanisms to PI3K inhibitors and suggests other possibilities for resistance. The model also reveals known and novel combinatorial interventions that are more effective than PI3K inhibition alone.

Conclusions

The use of a logic-based, discrete dynamic model enables the identification of results that are mainly due to the organization of the signaling network, and those that also depend on the kinetics of individual events. Network-based models such as this will play an increasing role in the rational design of high-order therapeutic combinations.

Electronic supplementary material

The online version of this article (10.1186/s41236-017-0007-6) contains supplementary material, which is available to authorized users.

A logic-based dynamic modeling approach to explicate the evolution of the central dogma of molecular biology

It is nearly half a century past the age of the introduction of the Central Dogma (CD) of molecular biology. This biological axiom has been developed and currently appears to be all the more complex. In this study, we modified CD by adding further species to the CD information flow and mathematically expressed CD within a dynamic framework by using Boolean network based on its present-day and 1965 editions. We show that the enhancement of the Dogma not only now entails a higher level of complexity, but it also shows a higher level of robustness, thus far more consistent with the nature of biological systems. Using this mathematical modeling approach, we put forward a logic-based expression of our conceptual view of molecular biology. Finally, we show that such biological concepts can be converted into dynamic mathematical models using a logic-based approach and thus may be useful as a framework for improving static conceptual models in biology.

A Network Model to Explore the Effect of the Micro-environment on Endothelial Cell Behavior during Angiogenesis

Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-function mutations. These properties make our model a suitable platform for the understanding of the molecular mechanisms underlying some pathologies. For example, it is possible to follow the changes in the activation patterns caused by mutations that promote Tip EC behavior and inhibit Phalanx EC behavior, that lead to the conditions associated with retinal vascular disorders and tumor vascularization. Moreover, the model describes how mutations that promote Phalanx EC behavior are associated with the development of arteriovenous and venous malformations. These results suggest that the network model that we propose has the potential to be used in the study of how the modulation of the EC extracellular micro-environment may improve the outcome of vascular disease treatments.

A new discrete dynamic model of ABA-induced stomatal closure predicts key feedback loops

Stomata, microscopic pores in leaf surfaces through which water loss and carbon dioxide uptake occur, are closed in response to drought by the phytohormone abscisic acid (ABA). This process is vital for drought tolerance and has been the topic of extensive experimental investigation in the last decades. Although a core signaling chain has been elucidated consisting of ABA binding to receptors, which alleviates negative regulation by protein phosphatases 2C (PP2Cs) of the protein kinase OPEN STOMATA 1 (OST1) and ultimately results in activation of anion channels, osmotic water loss, and stomatal closure, over 70 additional components have been identified, yet their relationships with each other and the core components are poorly elucidated. We integrated and processed hundreds of disparate observations regarding ABA signal transduction responses underlying stomatal closure into a network of 84 nodes and 156 edges and, as a result, established those relationships, including identification of a 36-node, strongly connected (feedback-rich) component as well as its in- and out-components. The network's domination by a feedback-rich component may reflect a general feature of rapid signaling events. We developed a discrete dynamic model of this network and elucidated the effects of ABA plus knockout or constitutive activity of 79 nodes on both the outcome of the system (closure) and the status of all internal nodes. The model, with more than 1024 system states, is far from fully determined by the available data, yet model results agree with existing experiments in 82 cases and disagree in only 17 cases, a validation rate of 75%. Our results reveal nodes that could be engineered to impact stomatal closure in a controlled fashion and also provide over 140 novel predictions for which experimental data are currently lacking. Noting the paucity of wet-bench data regarding combinatorial effects of ABA and internal node activation, we experimentally confirmed several predictions of the model with regard to reactive oxygen species, cytosolic Ca2+ (Ca2+c), and heterotrimeric G-protein signaling. We analyzed dynamics-determining positive and negative feedback loops, thereby elucidating the attractor (dynamic behavior) repertoire of the system and the groups of nodes that determine each attractor. Based on this analysis, we predict the likely presence of a previously unrecognized feedback mechanism dependent on Ca2+c. This mechanism would provide model agreement with 10 additional experimental observations, for a validation rate of 85%. Our research underscores the importance of feedback regulation in generating robust and adaptable biological responses. The high validation rate of our model illustrates the advantages of discrete dynamic modeling for complex, nonlinear systems common in biology.

Bistability and Asynchrony in a Boolean Model of the L-arabinose Operon in Escherichia coli

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism for gene repression, and the lack of inducer exclusion by glucose. For 11 out of 12 choices of initial conditions, we use computational algebra and Sage to verify that the state space contains a single fixed point that correctly matches the biology. The final initial condition, medium levels of arabinose and no glucose, successfully predicts the system's bistability. Finally, we compare the state space under synchronous and asynchronous update and see that the former has several artificial cycles that go away under a general asynchronous update.

Advanced Boolean modeling of biological networks applied to systems pharmacology

Motivation

Literature on complex diseases is abundant but not always quantitative. Many molecular pathways are qualitatively well described but this information cannot be used in traditional quantitative mathematical models employed in drug development. Tools for analysis of discrete networks are useful to capture the available information in the literature but have not been efficiently integrated by the pharmaceutical industry. We propose an expansion of the usual analysis of discrete networks that facilitates the identification/validation of therapeutic targets.

Results

In this article, we propose a methodology to perform Boolean modeling of Systems Biology/Pharmacology networks by using SPIDDOR (Systems Pharmacology for effIcient Drug Development On R) R package. The resulting models can be used to analyze the dynamics of signaling networks associated to diseases to predict the pathogenesis mechanisms and identify potential therapeutic targets.

Availability and Implementation

The source code is available at https://github.com/SPIDDOR/SPIDDOR .

Contact

itzirurzun@alumni.unav.es , itroconiz@unav.es.

Supplementary information

Supplementary data are available at Bioinformatics online.

COMPUTATIONAL METHODS FOR ASYNCHRONOUS BASINS

For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.

Estimating Propensity Parameters Using Google PageRank and Genetic Algorithms

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) are a modeling framework that considers two propensity parameters for updating each node and uses one when the update has a positive impact on the variable, that is, when the update causes the variable to increase its value, and uses the other when the update has a negative impact, that is, when the update causes it to decrease its value. This framework offers additional features for simulations but also adds a complexity in parameter estimation of the propensities. This paper presents a method for estimating the propensity parameters for SDDS. The method is based on adding noise to the system using the Google PageRank approach to make the system ergodic and thus guaranteeing the existence of a stationary distribution. Then with the use of a genetic algorithm, the propensity parameters are estimated. Approximation techniques that make the search algorithms efficient are also presented and Matlab/Octave code to test the algorithms are available at http://www.ms.uky.edu/~dmu228/GeneticAlg/Code.html.

An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network

Background

Boolean network modeling has been widely used to model large-scale biomolecular regulatory networks as it can describe the essential dynamical characteristics of complicated networks in a relatively simple way. When we analyze such Boolean network models, we often need to find out attractor states to investigate the converging state features that represent particular cell phenotypes. This is, however, very difficult (often impossible) for a large network due to computational complexity.

Results

There have been some attempts to resolve this problem by partitioning the original network into smaller subnetworks and reconstructing the attractor states by integrating the local attractors obtained from each subnetwork. But, in many cases, the partitioned subnetworks are still too large and such an approach is no longer useful. So, we have investigated the fundamental reason underlying this problem and proposed a novel efficient way of hierarchically partitioning a given large network into smaller subnetworks by focusing on some attractors corresponding to a particular phenotype of interest instead of considering all attractors at the same time. Using the definition of attractors, we can have a simplified update rule with fixed state values for some nodes. The resulting subnetworks were small enough to find out the corresponding local attractors which can be integrated for reconstruction of the global attractor states of the original large network.

Conclusions

The proposed approach can substantially extend the current limit of Boolean network modeling for converging state analysis of biological networks.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0338-4) contains supplementary material, which is available to authorized users.

Boolean regulatory network reconstruction using literature based knowledge with a genetic algorithm optimization method

Background

Prior knowledge networks (PKNs) provide a framework for the development of computational biological models, including Boolean models of regulatory networks which are the focus of this work. PKNs are created by a painstaking process of literature curation, and generally describe all relevant regulatory interactions identified using a variety of experimental conditions and systems, such as specific cell types or tissues. Certain of these regulatory interactions may not occur in all biological contexts of interest, and their presence may dramatically change the dynamical behaviour of the resulting computational model, hindering the elucidation of the underlying mechanisms and reducing the usefulness of model predictions. Methods are therefore required to generate optimized contextual network models from generic PKNs.

Results

We developed a new approach to generate and optimize Boolean networks, based on a given PKN. Using a genetic algorithm, a model network is built as a sub-network of the PKN and trained against experimental data to reproduce the experimentally observed behaviour in terms of attractors and the transitions that occur between them under specific perturbations. The resulting model network is therefore contextualized to the experimental conditions and constitutes a dynamical Boolean model closer to the observed biological process used to train the model than the original PKN. Such a model can then be interrogated to simulate response under perturbation, to detect stable states and their properties, to get insights into the underlying mechanisms and to generate new testable hypotheses.

Conclusions

Generic PKNs attempt to synthesize knowledge of all interactions occurring in a biological process of interest, irrespective of the specific biological context. This limits their usefulness as a basis for the development of context-specific, predictive dynamical Boolean models. The optimization method presented in this article produces specific, contextualized models from generic PKNs. These contextualized models have improved utility for hypothesis generation and experimental design. The general applicability of this methodological approach makes it suitable for a variety of biological systems and of general interest for biological and medical research. Our method was implemented in the software optimusqual, available online at http://www.vital-it.ch/software/optimusqual/.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1287-z) contains supplementary material, which is available to authorized users.

A Qualitative Model of the Differentiation Network in Chondrocyte Maturation: A Holistic View of Chondrocyte Hypertrophy

Differentiation of chondrocytes towards hypertrophy is a natural process whose control is essential in endochondral bone formation. It is additionally thought to play a role in several pathophysiological processes, with osteoarthritis being a prominent example. We perform a dynamic analysis of a qualitative mathematical model of the regulatory network that directs this phenotypic switch to investigate the influence of the individual factors holistically. To estimate the stability of a SOX9 positive state (associated with resting/proliferation chondrocytes) versus a RUNX2 positive one (associated with hypertrophy) we employ two measures. The robustness of the state in canalisation (size of the attractor basin) is assessed by a Monte Carlo analysis and the sensitivity to perturbations is assessed by a perturbational analysis of the attractor. Through qualitative predictions, these measures allow for an in silico screening of the effect of the modelled factors on chondrocyte maintenance and hypertrophy. We show how discrepancies between experimental data and the model’s results can be resolved by evaluating the dynamic plausibility of alternative network topologies. The findings are further supported by a literature study of proposed therapeutic targets in the case of osteoarthritis.