Interacting biological systems at all organizational levels display emergent behavior. Modeling these systems is made challenging by the number and variety of biological components and interactions (from molecules in gene regulatory networks to species in ecological networks) and the often-incomplete state of system knowledge (e.g., the unknown values of kinetic parameters for biochemical reactions). Boolean networks have emerged as a powerful tool for modeling these systems. We provide a methodological overview of Boolean network models of biological systems. After a brief introduction, we describe the process of building, analyzing, and validating a Boolean model. We then present the use of the model to make predictions about the system's response to perturbations and about how to control (or at least influence) its behavior. We emphasize the interplay between structural and dynamical properties of Boolean networks and illustrate them in three case studies from disparate levels of biological organization.
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 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 o…
Boolean networks have long been used as models of molecular networks and play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.
A Novel Antifragility Measure Based on Satisfaction and Its Application to Random and Biological Boolean Networks
Universidad Nacional Autónoma de México（墨西哥国立自治大学）；Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México（墨西哥国立自治大学复杂性科学中心）；Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México（墨西哥国立自治大学应用数学与系统研究所）；ITMO University（ITMO大学）
Despite the large quantity of information available, thorough researches in various biological databases are still needed in order to reconstruct and understand the steps that lead to known or new phenomena. By using protein-protein interaction networks and algorithms to extract relevant interconnections among proteins of interest, it is possible to assemble subnetworks from global interactomes. Using these extracted networks it is possible to use algorithms to predict signal directions while activation and inhibition effects can be predicted using RNA interference screenings. The result of th…
One successful model of interacting biological systems is the Boolean network. The dynamics of a Boolean network, controlled with Boolean functions, is usually considered to be a Markovian (memory-less) process. However, both self organizing features of biological phenomena and their intelligent nature should raise some doubt about ignoring the history of their time evolution. Here, we extend the Boolean network Markovian approach: we involve the effect of memory on the dynamics. This can be explored by modifying Boolean functions into non-Markovian functions, for example, by investigating the…
This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the li…
Genetic interaction can be defined as a deviation of the phenotypic quantitative effect of a double gene mutation from the effect predicted from single mutations using a simple (e.g., multiplicative or linear additive) statistical model. Experimentally characterized genetic interaction networks in model organisms provide important insights into relationships between different biological functions. We describe a computational methodology allowing to systematically and quantitatively characterize a Boolean mathematical model of a biological network in terms of genetic interactions between all lo…
Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables. We bring clarity to such results using the formalism of Boolean dynamical networks, analyzing the effectiveness of external control in selecting a desired final state when that state is among the original attractors of the dynamics. Analyzing 49 existing biological network models, we find strong numerical evidence that the average number of nodes that must be forced scales logarithmically with the number of original attractors. This suggests that biological ne…
This paper presents the foundation for a decomposition theory for Boolean networks, a type of discrete dynamical system that has found a wide range of applications in the life sciences, engineering, and physics. Given a Boolean network satisfying certain conditions, there is a unique collection of subnetworks so that the network can be reconstructed from these subnetworks by an extension operation. The main result of the paper is that this structural decomposition induces a corresponding decomposition of the network dynamics. The theory is motivated by the search for a mathematical framework t…
This article presents an algorithm that allows modeling of biological networks in a qualitative framework with continuous time. Mathematical modeling is used as a systems biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and to predict the effect of perturbations. We propose a modeling approach that is intrinsically continuous in time. The algorithm presented here fills the gap between qualitative and quantitative modeling. It is based on continuous time Markov process applied on a Boolean state space. In order to desc…
Computational models of biological processes provide one of the most powerful methods for a detailed analysis of the mechanisms that drive the behavior of complex systems. Logic-based modeling has enhanced our understanding and interpretation of those systems. Defining rules that determine how the output activity of biological entities is regulated by their respective inputs has proven to be challenging, due to increasingly larger models and the presence of noise in data, allowing multiple model parameterizations to fit the experimental observations. We present several Boolean function metrics…
Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analysing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous updating 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.
LRI, CNRS, U. Paris-Sud U. Paris-Saclay（巴黎-萨克雷大学南巴黎校区法国国家科学研究中心计算机研究所）；LaBRI, CNRS, U. Bordeaux Bordeaux INP（波尔多大学波尔多国立理工学院法国国家科学研究中心计算机与自动化研究所）；Institut Curie, INSERM U. PSL, Mines ParisTech（巴黎文理研究大学法国国家健康与医学研究院巴黎高等矿业学院居里研究所）
Boolean networks model finite discrete dynamical systems with complex behaviours. The state of each component is determined by a Boolean function of the state of (a subset of) the components of the network. This paper addresses the synthesis of these Boolean functions from constraints on their domain and emerging dynamical properties of the resulting network. The dynamical properties relate to the existence and absence of trajectories between partially observed configurations, and to the stable behaviours (fixpoints and cyclic attractors). The synthesis is expressed as a Boolean satisfiability…
Many biological systems, such as metabolic pathways, exhibit bistability behavior: these biological systems exhibit two distinct stable states with switching between the two stable states controlled by certain conditions. Since understanding bistability is key for understanding these biological systems, mathematical modeling of the bistability phenomenon has been at the focus of researches in quantitative and system biology. Recent study shows that Boolean networks offer relative simple mathematical models that are capable of capturing these essential information. Thus a better understanding o…
Recent developments in Omics-technologies revolutionized the investigation of biology by producing molecular data in multiple dimensions and scale. This breakthrough in biology raises the crucial issue of their interpretation based on modelling. In this undertaking, network provides a suitable framework for modelling the interactions between molecules. Basically a Biological network is composed of nodes referring to the components such as genes or proteins, and the edges/arcs formalizing interactions between them. The evolution of the interactions is then modelled by the definition of a dynami…
The control of biological systems presents interesting applications such as cell reprogramming or drug target identification. A common type of control strategy consists in a set of interventions that, by fixing the values of some variables, force the system to evolve to a desired state. This work presents a new approach for finding control strategies in biological systems modeled by Boolean networks. In this context, we explore the properties of trap spaces, subspaces of the state space which the dynamics cannot leave. Trap spaces for biological networks can often be efficiently computed, and …
In a previous article, an algorithm for identifying therapeutic targets in Boolean networks modeling 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 modeled biological system. Given a logic-based model of patho…
Boolean threshold networks have recently been proposed as useful tools to model the dynamics of genetic regulatory networks, and have been successfully applied to describe the cell cycles of \textit{S. cerevisiae} and \textit{S. pombe}. Threshold networks assume that gene regulation processes are additive. This, however, contrasts with the mechanism proposed by S. Kauffman in which each of the logic functions must be carefully constructed to accurately take into account the combinatorial nature of gene regulation. While Kauffman Boolean networks have been extensively studied and proved to have…
Understanding control mechanisms in biological systems plays a crucial role in important applications, for instance in cell reprogramming. Boolean modeling allows the identification of possible efficient strategies, helping to reduce the usually high and time-consuming experimental efforts. Available approaches to control strategy identification usually focus either on attractor or phenotype control, and are unable to deal with more complex control problems, for instance phenotype avoidance. They also fail to capture, in many situations, all possible minimal strategies, finding instead only su…
Due to the scarcity of quantitative details about biological phenomena, quantitative modeling in systems biology can be compromised, especially at the subcellular scale. One way to get around this is qualitative modeling because it requires few to no quantitative information. One of the most popular qualitative modeling approaches is the Boolean network formalism. However, Boolean models allow variables to take only two values, which can be too simplistic in some cases. The present work proposes a modeling approach derived from Boolean networks where continuous logical operators are used and w…
Finite dynamical systems (e.g. Boolean networks and logical models) have been used in modeling biological systems to focus attention on the qualitative features of the system, such as the wiring diagram. Since the analysis of such systems is hard, it is necessary to focus on subclasses that have the properties of being general enough for modeling and simple enough for theoretical analysis. In this paper we propose the class of AND-NOT networks for modeling biological systems and show that it provides several advantages. Some of the advantages include: Any finite dynamical system can be written…
To model biological systems using networks, it is desirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have been proposed in the literature. However, analytical study of these more general Boolean networks models is lagging. This paper aims to develop a concise theory for these different Boolean logic based modeling methods. Boolean models for networks where each node can have more than two levels of expression and B…
The study of control mechanisms of biological systems allows for interesting applications in bioengineering and medicine, for instance in cell reprogramming or drug target identification. A control strategy often consists of a set of interventions that, by fixing the values of some components, ensure that the long term dynamics of the controlled system is in a desired state. A common approach to control in the Boolean framework consists in checking how the fixed values propagate through the network, to establish whether the effect of percolating the interventions is sufficient to induce the ta…
Boolean networks (BNs) are widely used to model the qualitative dynamics of biological systems. Besides the logical rules determining the evolution of each component with respect to the state of its regulators, the scheduling of component updates can have a dramatic impact on the predicted behaviours. In this paper, we explore the use of Read (contextual) Petri Nets (RPNs) to study dynamics of BNs from a concurrency theory perspective. After showing bi-directional translations between RPNs and BNs and analogies between results on synchronism sensitivity, we illustrate that usual updating modes…
Progress in cell type reprogramming has revived the interest in Waddington's concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington's landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system. However, quasi-potential landscapes, originally developed for continuous systems, are not suitable for discrete-valued networks which are im…
Complex diseases can be modeled as damage to intracellular networks that results in abnormal cell behaviors. Network-based dynamic models such as Boolean models have been employed to model a variety of biological systems including those corresponding to disease. Previous work designed compensatory interactions to stabilize an attractor of a Boolean network after single node damage. We generalize this method to a multinode damage scenario and to the simultaneous stabilization of multiple steady state attractors. We classify the emergent situations, with a special focus on combinatorial effects,…
We study the target control problem of asynchronous Boolean networks, to identify a set of nodes, the perturbation of which can drive the dynamics of the network from any initial state to the desired steady state (or attractor). We are particularly interested in temporary perturbations, which are applied for sufficient time and then released to retrieve the original dynamics. Temporary perturbations have the apparent advantage of averting unforeseen consequences, which might be induced by permanent perturbations. Despite the infamous state-space explosion problem, in this work, we develop an e…
Background. Dynamical models of gene regulatory networks (GRNs) are highly effective in describing complex biological phenomena and processes, such as cell differentiation and cancer development. Yet, the topological and functional characterization of real GRNs is often still partial and an exhaustive picture of their functioning is missing. Motivation. We here introduce CABeRNET, a Cytoscape app for the generation, simulation and analysis of Boolean models of GRNs, specifically focused on their augmentation when a only partial topological and functional characterization of the network is avai…
Characterization of the differences between biological and random networks can reveal the design principles that enable the robust realization of crucial biological functions including the establishment of different cell types. Previous studies, focusing on identifying topological features that are present in biological networks but not in random networks, have, however, provided few functional insights. We use a Boolean modeling framework and ideas from spin glass literature to identify functional differences between five real biological networks and random networks with similar topological f…
Boolean networks serve as discrete models of regulation and signaling in biological cells. Identifying the key controllers of such processes is important for understanding the dynamical systems and planning further analysis. Here we quantify the dynamical impact of a node as the probability of damage spreading after switching the node's state. We find that the leading eigenvector of the adjacency matrix is a good predictor of dynamical impact in the case of long-term spreading. This so-called eigenvector centrality is also a good proxy measure of the influence a node's initial state has on the…
Emergence of robustness in biological networks is a paramount feature of evolving organisms, but a study of this property in vivo, for any level of representation such as Genetic, Metabolic, or Neuronal Networks, is a very hard challenge. In the case of Genetic Networks, mathematical models have been used in this context to provide insights on their robustness, but even in relatively simple formulations, such as Boolean Networks (BN), it might not be feasible to compute some measures for large system sizes. We describe in this work a Monte Carlo approach to calculate the size of the largest ba…
A Multilayer Structure Facilitates the Production of Antifragile Systems in Boolean Network Models
Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México（墨西哥国立自治大学复杂性科学中心）；Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México（墨西哥国立自治大学计算机科学与工程研究生院）；Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México（墨西哥国立自治大学应用数学与系统研究所）；ITMO University（ITMO大学）
In the applications of Boolean networks to modeling biological systems, an important computational problem is the detection of the fixed points of these networks. This is an NP-complete problem in general. There have been various attempts to develop algorithms to address the computation need for large size Boolean networks. The existing methods are usually based on known algorithms and thus limited to the situations where these known algorithms can apply. In this paper, we propose a novel approach to this problem. We show that any system of Boolean equations is equivalent to one Boolean equati…
In this study, we explored the utility of a descriptive and predictive bionetwork model for phospholipase C-coupled calcium signaling pathways, built with non-kinetic experimental information. Boolean models generated from these data yield oscillatory activity patterns for both the endoplasmic reticulum resident inositol-1,4,5-trisphosphate receptor (IP3R) and the plasma-membrane resident canonical transient receptor potential channel 3 (TRPC3). These results are specific as randomization of the Boolean operators ablates oscillatory pattern formation. Furthermore, knock-out simulations of the …
Gene regulatory networks (GRNs) are increasingly used for explaining biological processes with complex transcriptional regulation. A GRN links the expression levels of a set of genes via regulatory controls that gene products exert on one another. Boolean networks are a common modeling choice since they balance between detail and ease of analysis. However, even for Boolean networks the problem of fitting a given network model to an expression dataset is NP-Complete. Previous methods have addressed this issue heuristically or by focusing on acyclic networks and specific classes of regulation fu…
Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, wh…
The complex dynamics of gene expression in living cells can be well-approximated using Boolean networks. The average sensitivity is a natural measure of stability in these systems: values below one indicate typically stable dynamics associated with an ordered phase, whereas values above one indicate chaotic dynamics. This yields a theoretically motivated adaptive advantage to being near the critical value of one, at the boundary between order and chaos. Here, we measure average sensitivity for 66 publicly available Boolean network models describing the function of gene regulatory circuits acro…
Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are small…
Nested canalizing Boolean (NCF) functions play an important role in biological motivated regulative networks and in signal processing, in particular describing stack filters. It has been conjectured that NCFs have a stabilizing effect on the network dynamics. It is well known that the average sensitivity plays a central role for the stability of (random) Boolean networks. Here we provide a tight upper bound on the average sensitivity for NCFs as a function of the number of relevant input variables. As conjectured in literature this bound is smaller than 4/3 This shows that a large number of fu…
Graph Neural Network-Based Reinforcement Learning for Controlling Biological Networks - the GATTACA Framework
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（数学、信息学与力学学院，华沙大学）；IDEAS Research Institute（IDEAS研究机构）；IDEAS NCBR Sp. z o.o.（IDEAS NCBR公司）
