Modeling stochasticity and variability in gene regulatory networks

Cancer mutationscape: revealing the link between modular restructuring and intervention efficacy among mutations

There is increasing evidence that biological systems are modular in both structure and function. Complex biological signaling networks such as gene regulatory networks (GRNs) are proving to be composed of subcategories that are interconnected and hierarchically ranked. These networks contain highly dynamic processes that ultimately dictate cellular function over time, as well as influence phenotypic fate transitions. In this work, we use a stochastic multicellular signaling network of pancreatic cancer (PC) to show that the variance in topological rankings of the most phenotypically influential modules implies a strong relationship between structure and function. We further show that induction of mutations alters the modular structure, which analogously influences the aggression and controllability of the disease in silico. We finally present evidence that the impact and location of mutations with respect to PC modular structure directly corresponds to the efficacy of single agent treatments in silico, because topologically deep mutations require deep targets for control.

Pancreatic cancer mutationscape: revealing the link between modular restructuring and intervention efficacy amidst common mutations

There is increasing evidence that biological systems are modular in both structure and function. Complex biological signaling networks such as gene regulatory networks (GRNs) are proving to be composed of subcategories that are interconnected and hierarchically ranked. These networks contain highly dynamic processes that ultimately dictate cellular function over time, as well as influence phenotypic fate transitions. In this work, we use a stochastic multicellular signaling network of pancreatic cancer (PC) to show that the variance in topological rankings of the most phenotypically influential modules implies a strong relationship between structure and function. We further show that induction of mutations alters the modular structure, which analogously influences the aggression and controllability of the disease in silico. We finally present evidence that the impact and location of mutations with respect to PC modular structure directly corresponds to the efficacy of single agent treatments in silico, because topologically deep mutations require deep targets for control.

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.

Simulating the Evolution of Signaling Signatures During CART-Cell and Tumor Cell Interactions

Immunotherapies have been proven to have significant therapeutic efficacy in the treatment of cancer. The last decade has seen adoptive cell therapies, such as chimeric antigen receptor T-cell (CART-cell) therapy, gain FDA approval against specific cancers. Additionally, there are numerous clinical trials ongoing investigating additional designs and targets. Nevertheless, despite the excitement and promising potential of CART-cell therapy, response rates to therapy vary greatly between studies, patients, and cancers. There remains an unmet need to develop computational frameworks that more accurately predict CART-cell function and clinical efficacy. Here we present a coarse-grained model simulated with logical rules that demonstrates the evolution of signaling signatures following the interaction between CART-cells and tumor cells and allows for in silico based prediction of CART-cell functionality prior to experimentation.Clinical Relevance- Analysis of CART-cell signaling signatures can inform future CAR receptor design and combination therapy approaches aimed at improving therapy response.

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.

Simulating the Evolution of Signaling Signatures during CART-Cell - Tumor Cell Interactions

Immunotherapies have been proven to have significant therapeutic efficacy in the treatment of cancer. The last decade has seen adoptive cell therapies, such as chimeric antigen receptor T-cell (CART-cell) therapy, gain FDA approval against specific cancers. Additionally, there are numerous clinical trials ongoing investigating additional designs and targets. Nevertheless, despite the excitement and promising potential of CART-cell therapy, response rates to therapy vary greatly between studies, patients, and cancers. There remains an unmet need to develop computational frameworks that more accurately predict CART-cell function and clinical efficacy. Here we present a coarse-grained model simulated with logical rules that demonstrates the evolution of signaling signatures following the interaction between CART-cells and tumor cells and allows for in silico based prediction of CART-cell functionality prior to experimentation.

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.

Uncovering potential interventions for pancreatic cancer patients via mathematical modeling

Pancreatic Ductal Adenocarcinoma (PDAC) is widely known for its poor prognosis because it is often diagnosed when the cancer is in a later stage. We built a Boolean model to analyze the microenvironment of pancreatic cancer in order to better understand the interplay between pancreatic cancer, stellate cells, and their signaling cytokines. Specifically, we have used our model to study the impact of inducing four common mutations: KRAS, TP53, SMAD4, and CDKN2A. After implementing the various mutation combinations, we used our stochastic simulator to derive aggressiveness scores based on simulated attractor probabilities and long-term trajectory approximations. These aggression scores were then corroborated with clinical data. Moreover, we found sets of control targets that are effective among common mutations. These control sets contain nodes within both the pancreatic cancer cell and the pancreatic stellate cell, including PIP3, RAF, PIK3 and BAX in pancreatic cancer cell as well as ERK and PIK3 in the pancreatic stellate cell. Many of these nodes were found to be differentially expressed among pancreatic cancer patients in the TCGA database. Furthermore, literature suggests that many of these nodes can be targeted by drugs currently in circulation. The results herein help provide a proof of concept in the path towards personalized medicine through a means of mathematical systems biology. All data and code used for running simulations, statistical analysis, and plotting is available on a GitHub repository athttps://github.com/drplaugher/PCC_Mutations.

Boolean modelling as a logic-based dynamic approach in systems medicine

Molecular mechanisms of health and disease are often represented as systems biology diagrams, and the coverage of such representation constantly increases. These static diagrams can be transformed into dynamic models, allowing for in silico simulations and predictions. Boolean modelling is an approach based on an abstract representation of the system. It emphasises the qualitative modelling of biological systems in which each biomolecule can take two possible values: zero for absent or inactive, one for present or active. Because of this approximation, Boolean modelling is applicable to large diagrams, allowing to capture their dynamic properties. We review Boolean models of disease mechanisms and compare a range of methods and tools used for analysis processes. We explain the methodology of Boolean analysis focusing on its application in disease modelling. Finally, we discuss its practical application in analysing signal transduction and gene regulatory pathways in health and disease.

Modeling the Pancreatic Cancer Microenvironment in Search of Control Targets

Pancreatic ductal adenocarcinoma is among the leading causes of cancer-related deaths globally due to its extreme difficulty to detect and treat. Recently, research focus has shifted to analyzing the microenvironment of pancreatic cancer to better understand its key molecular mechanisms. This microenvironment can be represented with a multi-scale model consisting of pancreatic cancer cells (PCCs) and pancreatic stellate cells (PSCs), as well as cytokines and growth factors which are responsible for intercellular communication between the PCCs and PSCs. We have built a stochastic Boolean network (BN) model, validated by literature and clinical data, in which we probed for intervention strategies that force this gene regulatory network (GRN) from a diseased state to a healthy state. To do so, we implemented methods from phenotype control theory to determine a procedure for regulating specific genes within the microenvironment. We identified target genes and molecules, such that the application of their control drives the GRN to the desired state by suppression (or expression) and disruption of specific signaling pathways that may eventually lead to the eradication of the cancer cells. After applying well-studied control methods such as stable motifs, feedback vertex sets, and computational algebra, we discovered that each produces a different set of control targets that are not necessarily minimal nor unique. Yet, we were able to gain more insight about the performance of each process and the overlap of targets discovered. Nearly every control set contains cytokines, KRas, and HER2/neu, which suggests they are key players in the system's dynamics. To that end, this model can be used to produce further insight into the complex biological system of pancreatic cancer with hopes of finding new potential targets.

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.

Mayer-Type Optimal Control of Probabilistic Boolean Control Network With Uncertain Selection Probabilities

This article considers a Mayer-type optimal control problem of probabilistic Boolean control networks (PBCNs) with uncertainty on selection probabilities which obey Beta probabilistic distributions. The expectation with respect to both the selection probabilities and the transitions of state variables is set as a cost function, and it deduces an equivalent formulation as a multistage decision problem. Furthermore, the dynamic programming technique is applied to solve the problem and performs a novel optimization algorithm in the fashion of semitensor product. A numerical example of a biological model of apoptosis protein demonstrates the effectiveness and feasibility of the proposed framework and algorithms.

Mathematical modeling of the Candida albicans yeast to hyphal transition reveals novel control strategies

Candida albicans, an opportunistic fungal pathogen, is a significant cause of human infections, particularly in immunocompromised individuals. Phenotypic plasticity between two morphological phenotypes, yeast and hyphae, is a key mechanism by which C. albicans can thrive in many microenvironments and cause disease in the host. Understanding the decision points and key driver genes controlling this important transition and how these genes respond to different environmental signals is critical to understanding how C. albicans causes infections in the host. Here we build and analyze a Boolean dynamical model of the C. albicans yeast to hyphal transition, integrating multiple environmental factors and regulatory mechanisms. We validate the model by a systematic comparison to prior experiments, which led to agreement in 17 out of 22 cases. The discrepancies motivate alternative hypotheses that are testable by follow-up experiments. Analysis of this model revealed two time-constrained windows of opportunity that must be met for the complete transition from the yeast to hyphal phenotype, as well as control strategies that can robustly prevent this transition. We experimentally validate two of these control predictions in C. albicans strains lacking the transcription factor UME6 and the histone deacetylase HDA1, respectively. This model will serve as a strong base from which to develop a systems biology understanding of C. albicans morphogenesis.

Evolution of Cellular Differentiation: From Hypotheses to Models

Cellular differentiation is one of the hallmarks of complex multicellularity, allowing individual organisms to capitalize on among-cell functional diversity. The evolution of multicellularity is a major evolutionary transition that allowed for the increase of organismal complexity in multiple lineages, a process that relies on the functional integration of cell-types within an individual. Multiple hypotheses have been proposed to explain the origins of cellular differentiation, but we lack a general understanding of what makes one cell-type distinct from others, and how such differentiation arises. Here, we describe how the use of Boolean networks (BNs) can aid in placing empirical findings into a coherent conceptual framework, and we emphasize some of the standing problems when interpreting data and model behaviors.

Asymptotical Feedback Set Stabilization of Probabilistic Boolean Control Networks

In this article, we investigate the asymptotical feedback set stabilization in distribution of probabilistic Boolean control networks (PBCNs). We prove that a PBCN is asymptotically feedback stabilizable to a given subset if and only if (iff) it constitutes asymptotically feedback stabilizable to the largest control-invariant subset (LCIS) contained in this subset. We proposed an algorithm to calculate the LCIS contained in any given subset with the necessary and sufficient condition for asymptotical set stabilizability in terms of obtaining the reachability matrix. In addition, we propose a method to design stabilizing feedback based on a state-space partition. Finally, the results were applied to solve asymptotical feedback output tracking and asymptotical feedback synchronization of PBCNs. Examples were detailed to demonstrate the feasibility of the proposed method and results.

On Optimal Time-Varying Feedback Controllability for Probabilistic Boolean Control Networks

This brief studies controllability for probabilistic Boolean control network (PBCN) with time-varying feedback control laws. The concept of feedback controllability with an arbitrary probability for PBCNs is formulated first, and a control problem to maximize the probability of time-varying feedback controllability is investigated afterward. By introducing semitensor product (STP) technique, an equivalent multistage decision problem is deduced, and then a novel optimization algorithm is proposed to obtain the maximum probability of controllability and the corresponding optimal feedback law simultaneously. The advantages of the time-varying optimal controller obtained by the proposed algorithm, compared to the time-invariant one, are illustrated by numerical simulations.

Systems biology of ferroptosis: A modeling approach

Ferroptosis is a recently discovered form of iron-dependent regulated cell death (RCD) that occurs via peroxidation of phospholipids containing polyunsaturated fatty acid (PUFA) moieties. Activating this form of cell death is an emerging strategy in cancer treatment. Because multiple pathways and molecular species contribute to the ferroptotic process, predicting which tumors will be sensitive to ferroptosis is a challenge. We thus develop a mathematical model of several critical pathways to ferroptosis in order to perform a systems-level analysis of the process. We show that sensitivity to ferroptosis depends on the activity of multiple upstream cascades, including PUFA incorporation into the phospholipid membrane, and the balance between levels of pro-oxidant factors (reactive oxygen species, lipoxogynases) and antioxidant factors (GPX4). We perform a systems-level analysis of ferroptosis sensitivity as an outcome of five input variables (ACSL4, SCD1, ferroportin, transferrin receptor, and p53) and organize the resulting simulations into 'high' and 'low' ferroptosis sensitivity groups. We make a novel prediction corresponding to the combinatorial requirements of ferroptosis sensitivity to SCD1 and ACSL4 activity. To validate our prediction, we model the ferroptotic response of an ovarian cancer stem cell line following single- and double-knockdown of SCD1 and ACSL4. We find that the experimental outcomes are consistent with our simulated predictions. This work suggests that a systems-level approach is beneficial for understanding the complex combined effects of ferroptotic input, and in predicting cancer susceptibility to ferroptosis.

A Near-Optimal Control Method for Stochastic Boolean Networks

One of the ultimate goals in systems biology is to develop control strategies to find efficient medical treatments. One step towards this goal is to develop methods for changing the state of a cell into a desirable state. We propose an efficient method that determines combinations of network perturbations to direct the system towards a predefined state. The method requires a set of control actions such as the silencing of a gene or the disruption of the interaction between two genes. An optimal control policy defined as the best intervention at each state of the system can be obtained using existing methods. However, these algorithms are computationally prohibitive for models with tens of nodes. Our method generates control actions that approximates the optimal control policy with high probability with a computational efficiency that does not depend on the size of the state space. Our C++ code is available at https://github.com/boaguilar/SDDScontrol.

A Boolean network control algorithm guided by forward dynamic programming

Control problem in a biological system is the problem of finding an interventional policy for changing the state of the biological system from an undesirable state, e.g. disease, into a desirable healthy state. Boolean networks are utilized as a mathematical model for gene regulatory networks. This paper provides an algorithm to solve the control problem in Boolean networks. The proposed algorithm is implemented and applied on two biological systems: T-cell receptor network and Drosophila melanogaster network. Results show that the proposed algorithm works faster in solving the control problem over these networks, while having similar accuracy, in comparison to previous exact methods. Source code and a simple web service of the proposed algorithm is available at http://goliaei.ir/net-control/www/.

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.

Policy Iteration Algorithm for Optimal Control of Stochastic Logical Dynamical Systems

This brief investigates the infinite horizon optimal control problem for stochastic multivalued logical dynamical systems with discounted cost. Applying the equivalent descriptions of stochastic logical dynamics in term of Markov decision process, the discounted infinite horizon optimal control problem is presented in an algebraic form. Then, employing the method of semitensor product of matrices and the increasing-dimension technique, a succinct algebraic form of the policy iteration algorithm is derived to solve the optimal control problem. To show the effectiveness of the proposed policy iteration algorithm, an optimization problem of p53-Mdm2 gene network is investigated.

Dynamics of gene regulatory networks with stochastic propensities

Gene regulatory networks (GRNs) control the production of proteins in cells. It is well-known that this process is not deterministic. Numerous studies employed a non-deterministic transition structure to model these networks. However, it is not realistic to expect state-to-state transition probabilities to remain constant throughout an organism’s lifetime. In this work, we focus on modeling GRN state transition (edge) variability using an ever-changing set of propensities. We suspect that the source of this variation is due to internal noise at the molecular level and can be modeled by introducing additional stochasticity into GRN models. We employ a beta distribution, whose parameters are estimated to capture the pattern inherent in edge behavior with minimum error. Additionally, we develop a method for obtaining propensities from a pre-determined network.

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.

Extreme calorie restriction in yeast retentostats induces uniform non-quiescent growth arrest

Non-dividing Saccharomyces cerevisiae cultures are highly relevant for fundamental and applied studies. However, cultivation conditions in which non-dividing cells retain substantial metabolic activity are lacking. Unlike stationary-phase (SP) batch cultures, the current experimental paradigm for non-dividing yeast cultures, cultivation under extreme calorie restriction (ECR) in retentostat enables non-dividing yeast cells to retain substantial metabolic activity and to prevent rapid cellular deterioration. Distribution of F-actin structures and single-cell copy numbers of specific transcripts revealed that cultivation under ECR yields highly homogeneous cultures, in contrast to SP cultures that differentiate into quiescent and non-quiescent subpopulations. Combined with previous physiological studies, these results indicate that yeast cells subjected to ECR survive in an extended G₁ phase. This study demonstrates that yeast cells exposed to ECR differ from carbon-starved cells and offer a promising experimental model for studying non-dividing, metabolically active, and robust eukaryotic cells.

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.

Data Requirements for Model-Based Cancer Prognosis Prediction

Cancer prognosis prediction is typically carried out without integrating scientific knowledge available on genomic pathways, the effect of drugs on cell dynamics, or modeling mutations in the population. Recent work addresses some of these problems by formulating an uncertainty class of Boolean regulatory models for abnormal gene regulation, assigning prognosis scores to each network based on intervention outcomes, and partitioning networks in the uncertainty class into prognosis classes based on these scores. For a new patient, the probability distribution of the prognosis class was evaluated using optimal Bayesian classification, given patient data. It was assumed that (1) disease is the result of several mutations of a known healthy network and that these mutations and their probability distribution in the population are known and (2) only a single snapshot of the patient's gene activity profile is observed. It was shown that, even in ideal settings where cancer in the population and the effect of a drug are fully modeled, a single static measurement is typically not sufficient. Here, we study what measurements are sufficient to predict prognosis. In particular, we relax assumption (1) by addressing how population data may be used to estimate network probabilities, and extend assumption (2) to include static and time-series measurements of both population and patient data. Furthermore, we extend the prediction of prognosis classes to optimal Bayesian regression of prognosis metrics. Even when time-series data is preferable to infer a stochastic dynamical network, we show that static data can be superior for prognosis prediction when constrained to small samples. Furthermore, although population data is helpful, performance is not sensitive to inaccuracies in the estimated network probabilities.

Molecular network control through boolean canalization

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions. The method is based on identifying appropriate input-output combinations on undesirable transitions that can be modified using the edges in the wiring diagram of the network. Moreover, a method for estimating the number of changed transitions in the state space of the system as a result of an edge deletion in the wiring diagram is presented. The control methods of this paper were applied to a mutated cell-cycle model and to a p53-mdm2 model to identify potential control targets.

Networkcontrology

An increasing number of complex systems are now modeled as networks of coupled dynamical entities. Nonlinearity and high-dimensionality are hallmarks of the dynamics of such networks but have generally been regarded as obstacles to control. Here, I discuss recent advances on mathematical and computational approaches to control high-dimensional nonlinear network dynamics under general constraints on the admissible interventions. I also discuss the potential of network control to address pressing scientific problems in various disciplines.

Iron acquisition and oxidative stress response in aspergillus fumigatus

BACKGROUND

Aspergillus fumigatus is a ubiquitous airborne fungal pathogen that presents a life-threatening health risk to individuals with weakened immune systems. A. fumigatus pathogenicity depends on its ability to acquire iron from the host and to resist host-generated oxidative stress. Gaining a deeper understanding of the molecular mechanisms governing A. fumigatus iron acquisition and oxidative stress response may ultimately help to improve the diagnosis and treatment of invasive aspergillus infections.

RESULTS

This study follows a systems biology approach to investigate how adaptive behaviors emerge from molecular interactions underlying A. fumigatus iron regulation and oxidative stress response. We construct a Boolean network model from known interactions and simulate how changes in environmental iron and superoxide levels affect network dynamics. We propose rules for linking long term model behavior to qualitative estimates of cell growth and cell death. These rules are used to predict phenotypes of gene deletion strains. The model is validated on the basis of its ability to reproduce literature data not used in model generation.

CONCLUSIONS

The model reproduces gene expression patterns in experimental time course data when A. fumigatus is switched from a low iron to a high iron environment. In addition, the model is able to accurately represent the phenotypes of many knockout strains under varying iron and superoxide conditions. Model simulations support the hypothesis that intracellular iron regulates A. fumigatus transcription factors, SreA and HapX, by a post-translational, rather than transcriptional, mechanism. Finally, the model predicts that blocking siderophore-mediated iron uptake reduces resistance to oxidative stress. This indicates that combined targeting of siderophore-mediated iron uptake and the oxidative stress response network may act synergistically to increase fungal cell killing.

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

Cell fate reprogramming by control of intracellular network dynamics

Identifying control strategies for biological networks is paramount for practical applications that involve reprogramming a cell’s fate, such as disease therapeutics and stem cell reprogramming. Here we develop a novel network control framework that integrates the structural and functional information available for intracellular networks to predict control targets. Formulated in a logical dynamic scheme, our approach drives any initial state to the target state with 100% effectiveness and needs to be applied only transiently for the network to reach and stay in the desired state. We illustrate our method’s potential to find intervention targets for cancer treatment and cell differentiation by applying it to a leukemia signaling network and to the network controlling the differentiation of helper T cells. We find that the predicted control targets are effective in a broad dynamic framework. Moreover, several of the predicted interventions are supported by experiments.

Practical applications in modern molecular and systems biology such as the search for new therapeutic targets for diseases and stem cell reprogramming have generated a great interest in controlling the internal dynamics of a cell. Here we present a network control approach that integrates the structural and functional information of the network. We show that stabilizing the expression or activity of a few select components can drive the cell towards a desired fate or away from an undesired fate. We demonstrate our method’s effectiveness by applying it to a type of blood cell cancer and to the differentiation of a type of immune cell. Overall, our approach provides new insights into how to control the dynamics of intracellular networks.

A network biology approach to denitrification in Pseudomonas aeruginosa

Pseudomonas aeruginosa is a metabolically flexible member of the Gammaproteobacteria. Under anaerobic conditions and the presence of nitrate, P. aeruginosa can perform (complete) denitrification, a respiratory process of dissimilatory nitrate reduction to nitrogen gas via nitrite (NO₂), nitric oxide (NO) and nitrous oxide (N₂O). This study focuses on understanding the influence of environmental conditions on bacterial denitrification performance, using a mathematical model of a metabolic network in P. aeruginosa. To our knowledge, this is the first mathematical model of denitrification for this bacterium. Analysis of the long-term behavior of the network under changing concentration levels of oxygen (O₂), nitrate (NO₃), and phosphate (PO₄) suggests that PO₄ concentration strongly affects denitrification performance. The model provides three predictions on denitrification activity of P. aeruginosa under various environmental conditions, and these predictions are either experimentally validated or supported by pertinent biological literature. One motivation for this study is to capture the effect of PO₄ on a denitrification metabolic network of P. aeruginosa in order to shed light on mechanisms for greenhouse gas N₂O accumulation during seasonal oxygen depletion in aquatic environments such as Lake Erie (Laurentian Great Lakes, USA). Simulating the microbial production of greenhouse gases in anaerobic aquatic systems such as Lake Erie allows a deeper understanding of the contributing environmental effects that will inform studies on, and remediation strategies for, other hypoxic sites worldwide.

Steady state analysis of Boolean molecular network models via model reduction and computational algebra

Background

A key problem in the analysis of mathematical models of molecular networks is the determination of their steady states. The present paper addresses this problem for Boolean network models, an increasingly popular modeling paradigm for networks lacking detailed kinetic information. For small models, the problem can be solved by exhaustive enumeration of all state transitions. But for larger models this is not feasible, since the size of the phase space grows exponentially with the dimension of the network. The dimension of published models is growing to over 100, so that efficient methods for steady state determination are essential. Several methods have been proposed for large networks, some of them heuristic. While these methods represent a substantial improvement in scalability over exhaustive enumeration, the problem for large networks is still unsolved in general.

Results

This paper presents an algorithm that consists of two main parts. The first is a graph theoretic reduction of the wiring diagram of the network, while preserving all information about steady states. The second part formulates the determination of all steady states of a Boolean network as a problem of finding all solutions to a system of polynomial equations over the finite number system with two elements. This problem can be solved with existing computer algebra software. This algorithm compares favorably with several existing algorithms for steady state determination. One advantage is that it is not heuristic or reliant on sampling, but rather determines algorithmically and exactly all steady states of a Boolean network. The code for the algorithm, as well as the test suite of benchmark networks, is available upon request from the corresponding author.

Conclusions

The algorithm presented in this paper reliably determines all steady states of sparse Boolean networks with up to 1000 nodes. The algorithm is effective at analyzing virtually all published models even those of moderate connectivity. The problem for large Boolean networks with high average connectivity remains an open problem.

Asynchronous stochastic Boolean networks as gene network models

Logical models have widely been used to gain insights into the biological behavior of gene regulatory networks (GRNs). Most logical models assume a synchronous update of the genes' states in a GRN. However, this may not be appropriate, because each gene may require a different period of time for changing its state. In this article, asynchronous stochastic Boolean networks (ASBNs) are proposed for investigating various asynchronous state-updating strategies in a GRN. As in stochastic computation, ASBNs use randomly permutated stochastic sequences to encode probability. Investigated by several stochasticity models, a GRN is considered to be subject to noise and external perturbation. Hence, both stochasticity and asynchronicity are considered in the state evolution of a GRN. As a case study, ASBNs are utilized to investigate the dynamic behavior of a T helper network. It is shown that ASBNs are efficient in evaluating the steady-state distributions (SSDs) of the network with random gene perturbation. The SSDs found by using ASBNs show the robustness of the attractors of the T helper network, when various stochasticity and asynchronicity models are considered to investigate its dynamic behavior.

Gene perturbation and intervention in context-sensitive stochastic Boolean networks

Background

In a gene regulatory network (GRN), gene expressions are affected by noise, and stochastic fluctuations exist in the interactions among genes. These stochastic interactions are context dependent, thus it becomes important to consider noise in a context-sensitive manner in a network model. As a logical model, context-sensitive probabilistic Boolean networks (CSPBNs) account for molecular and genetic noise in the temporal context of gene functions. In a CSPBN with n genes and k contexts, however, a computational complexity of O(nk²2²ⁿ) (or O(nk2ⁿ)) is required for an accurate (or approximate) computation of the state transition matrix (STM) of the size (2ⁿ ∙ k) × (2ⁿ ∙ k) (or 2ⁿ × 2ⁿ). The evaluation of a steady state distribution (SSD) is more challenging. Recently, stochastic Boolean networks (SBNs) have been proposed as an efficient implementation of an instantaneous PBN.

Results

The notion of stochastic Boolean networks (SBNs) is extended for the general model of PBNs, i.e., CSPBNs. This yields a novel structure of context-sensitive SBNs (CSSBNs) for modeling the stochasticity in a GRN. A CSSBN enables an efficient simulation of a CSPBN with a complexity of O(nLk2ⁿ) for computing the state transition matrix, where L is a factor related to the required sequence length in CSSBN for achieving a desired accuracy. A time-frame expanded CSSBN can further efficiently simulate the stationary behavior of a CSPBN and allow for a tunable tradeoff between accuracy and efficiency. The CSSBN approach is more efficient than an analytical method and more accurate than an approximate analysis.

Conclusions

Context-sensitive stochastic Boolean networks (CSSBNs) are proposed as an efficient approach to modeling the effects of gene perturbation and intervention in gene regulatory networks. A CSSBN analysis provides biologically meaningful insights into the oscillatory dynamics of the p53-Mdm2 network in a context-switching environment. It is shown that random gene perturbation has a greater effect on the final distribution of the steady state of a network compared to context switching activities. The CSSBN approach can further predict the steady state distribution of a glioma network under gene intervention. Ultimately, this will help drug discovery and develop effective drug intervention strategies.

Stabilization of perturbed Boolean network attractors through compensatory interactions

Background

Understanding and ameliorating the effects of network damage are of significant interest, due in part to the variety of applications in which network damage is relevant. For example, the effects of genetic mutations can cascade through within-cell signaling and regulatory networks and alter the behavior of cells, possibly leading to a wide variety of diseases. The typical approach to mitigating network perturbations is to consider the compensatory activation or deactivation of system components. Here, we propose a complementary approach wherein interactions are instead modified to alter key regulatory functions and prevent the network damage from triggering a deregulatory cascade.

Results

We implement this approach in a Boolean dynamic framework, which has been shown to effectively model the behavior of biological regulatory and signaling networks. We show that the method can stabilize any single state (e.g., fixed point attractors or time-averaged representations of multi-state attractors) to be an attractor of the repaired network. We show that the approach is minimalistic in that few modifications are required to provide stability to a chosen attractor and specific in that interventions do not have undesired effects on the attractor. We apply the approach to random Boolean networks, and further show that the method can in some cases successfully repair synchronous limit cycles. We also apply the methodology to case studies from drought-induced signaling in plants and T-LGL leukemia and find that it is successful in both stabilizing desired behavior and in eliminating undesired outcomes. Code is made freely available through the software package BooleanNet.

Conclusions

The methodology introduced in this report offers a complementary way to manipulating node expression levels. A comprehensive approach to evaluating network manipulation should take an "all of the above" perspective; we anticipate that theoretical studies of interaction modification, coupled with empirical advances, will ultimately provide researchers with greater flexibility in influencing system behavior.

Stochastic multiple-valued gene networks

Among various approaches to modeling gene regulatory networks (GRNs), Boolean networks (BNs) and its probabilistic extension, probabilistic Boolean networks (PBNs), have been studied to gain insights into the dynamics of GRNs. To further exploit the simplicity of logical models, a multiple-valued network employs gene states that are not limited to binary values, thus providing a finer granularity in the modeling of GRNs. In this paper, stochastic multiple-valued networks (SMNs) are proposed for modeling the effects of noise and gene perturbation in a GRN. An SMN enables an accurate and efficient simulation of a probabilistic multiple-valued network (as an extension of a PBN). In a k-level SMN of n genes, it requires a complexity of O(nLk(n)) to compute the state transition matrix, where L is a factor related to the minimum sequence length in the SMN for achieving a desired accuracy. The use of randomly permuted stochastic sequences further increases computational efficiency and allows for a tunable tradeoff between accuracy and efficiency. The analysis of a p53-Mdm2 network and a WNT5A network shows that the proposed SMN approach is efficient in evaluating the network dynamics and steady state distribution of gene networks under random gene perturbation.

Majority rules with random tie-breaking in Boolean gene regulatory networks

We consider threshold Boolean gene regulatory networks, where the update function of each gene is described as a majority rule evaluated among the regulators of that gene: it is turned ON when the sum of its regulator contributions is positive (activators contribute positively whereas repressors contribute negatively) and turned OFF when this sum is negative. In case of a tie (when contributions cancel each other out), it is often assumed that the gene keeps it current state. This framework has been successfully used to model cell cycle control in yeast. Moreover, several studies consider stochastic extensions to assess the robustness of such a model.

Here, we introduce a novel, natural stochastic extension of the majority rule. It consists in randomly choosing the next value of a gene only in case of a tie. Hence, the resulting model includes deterministic and probabilistic updates. We present variants of the majority rule, including alternate treatments of the tie situation. Impact of these variants on the corresponding dynamical behaviours is discussed. After a thorough study of a class of two-node networks, we illustrate the interest of our stochastic extension using a published cell cycle model. In particular, we demonstrate that steady state analysis can be rigorously performed and can lead to effective predictions; these relate for example to the identification of interactions whose addition would ensure that a specific state is absorbing.

Stabilizing gene regulatory networks through feedforward loops

The global dynamics of gene regulatory networks are known to show robustness to perturbations in the form of intrinsic and extrinsic noise, as well as mutations of individual genes. One molecular mechanism underlying this robustness has been identified as the action of so-called microRNAs that operate via feedforward loops. We present results of a computational study, using the modeling framework of stochastic Boolean networks, which explores the role that such network motifs play in stabilizing global dynamics. The paper introduces a new measure for the stability of stochastic networks. The results show that certain types of feedforward loops do indeed buffer the network against stochastic effects.

On the relationship of steady states of continuous and discrete models arising from biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.