Algebraic Model Checking for Boolean Gene Regulatory Networks

Gene regulatory networks can be successfully modeled as Boolean networks. A much discussed hypothesis says that such model networks reproduce empirical findings the best if they are tuned to operate at criticality, i.e. at the borderline between their ordered and disordered phases. Critical networks have been argued to lead to a number of functional advantages such as maximal dynamical range, maximal sensitivity to environmental changes, as well as to an excellent tradeoff between stability and flexibility. Here, we study the effect of noise within the context of Boolean networks trained to learn complex tasks under supervision. We verify that quasi-critical networks are the ones learning in the fastest possible way –even for asynchronous updating rules– and that the larger the task complexity the smaller the distance to criticality. On the other hand, when additional sources of intrinsic noise in the network states and/or in its wiring pattern are introduced, the optimally performing networks become clearly subcritical. These results suggest that in order to compensate for inherent stochasticity, regulatory and other type of biological networks might become subcritical rather than being critical, all the most if the task to be performed has limited complexity.

A natural language for implementing algebraically Expert Systems

Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steady-state analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steady-state distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.

Boolean networks are currently receiving considerable attention as a computational scheme for system level analysis and modeling of biological systems. Studying control-related problems in Boolean networks may reveal new insights into the intrinsic control in complex biological systems and enable us to develop strategies for manipulating biological systems using exogenous inputs. This paper considers controllability and observability of Boolean biological networks. We propose a new approach, which draws from the rich theory of symbolic computation, to solve the problems. Consequently, simple necessary and sufficient conditions for reachability, controllability, and observability are obtained, and algorithmic tests for controllability and observability which are based on the Gröbner basis method are presented. As practical applications, we apply the proposed approach to several different biological systems, namely, the mammalian cell-cycle network, the T-cell activation network, the large granular lymphocyte survival signaling network, and the Drosophila segment polarity network, gaining novel insights into the control and/or monitoring of the specific biological systems.

IQCELL is a platform that infers Boolean gene regulatory networks from single-cell RNA sequencing data. Boolean networks can be simulated under normal and perturbed conditions. In this chapter, we provide a detailed guideline for implementing IQCELL from a raw dataset. The steps include processing data, inferring informative genes, inferring gene regulatory network, and simulating the resulted network under normal and perturbed conditions.

The management of uncertainty and imprecision is becoming essential for modeling many real problems. Multivalued model checking, a generalization of classical model checking, is useful for analyzing models that contain uncertainty or inconsistency. This paper shows that Groebner bases can provide canonical symbolic representations for multi-valued logics, and therefore, can be applied to symbolic multi-valued model checking. We propose a general polynomial form of multi-valued logics, and then present a modified model checking algorithm based on the algebraic representations of mv-Kripke structures as well as mv-CTL formulas. This algebraic approach to multi-valued model checking could serve as a useful supplement to the existent method based on Multi-valued Decision Diagrams, and may give new guidance for verifying properties of systems containing uncertain or inconsistent information.

The general singular Boolean networks are proposed in this study, motivated by the algebraic form of dynamic‐algebraic Boolean networks via the semi‐tensor product of matrices. First, one of the most important problems for this kind of networks, solvability problem, is discussed. Then, in order to calculate the fixed points and cycles, the transition matrix of a singular Boolean network is defined, which contains all the state transferring information. At last, the general singular Boolean control networks are considered with their solvability and the disturbance decoupling problem is presented and solved by a constant control. Illustrative examples are given to show the feasibility of the results.

Disturbance decoupling of Boolean networks via robust indistinguishability method

A novel algebraic approach is proposed to study dynamics of asynchronous random Boolean networks where a random number of nodes can be updated at each time step (ARBNs). In this article, the logical equations of ARBNs are converted into the discrete-time linear representation and dynamical behaviors of systems are investigated. We provide a general formula of network transition matrices of ARBNs as well as a necessary and sufficient algebraic criterion to determine whether a group of given states compose an attractor of length in ARBNs. Consequently, algorithms are achieved to find all of the attractors and basins in ARBNs. Examples are showed to demonstrate the feasibility of the proposed scheme.

This paper considers a problem of finding control strategies for Boolean networks, where Boolean networks have been used as a model of genetic networks. This paper shows that finding a control strategy leading to the desired global state is NP-hard even if there is only one control node in the network. This result justifies existing exponential time algorithms for finding control strategies for probabilistic Boolean networks. On the other hand, this paper shows that the problem can be solved in polynomial time if the network has a tree structure.

A new approach for global controllability of higher order Boolean control network

This paper is dedicated to solving the model reconstruction problem of Boolean networks (BNs) based on incomplete information. By resorting to semi‐tensor product, the issue of finding logical functions is converted into designing corresponding structure matrices. In line with desired attractors, two circumstances, standard BNs and delayed BNs, are taken into consideration, and several new results are presented. For the model reconstruction of standard BNs, an algebraic condition is put forward and an algorithm is developed for reconstructing unknown logical functions. For the model reconstruction of delayed BNs, an augmented system is deduced, based on which, several criteria and the corresponding algorithm are proposed. Finally, as an application to gene regulatory networks, the results obtained in this paper are used to analyze the WNT5A network and the apoptosis network.

Blockchain systems based on smart contracts are critical systems that have to be verified in order to ensure their reliability and efficiency. Verifying these systems is a major challenge that is still an active topic of research in different domains. In this paper, we focus on verifying these systems that we model using trust protocols under uncertainty. Specifically, we address the problem using an effective verification approach called three-valued model checking. We introduce a new logic by extending the recently proposed Computation Tree Logic of Trust (TCTL) to the three-valued case (\(3v-TCTL\)) to reason about trust with uncertainty over smart contract-based systems. We also propose a new transformation approach to reduce the \(3v-TCTL\) model checking problem to the classical case. We apply our approach to a smart contract-based drug traceability system in the healthcare supply chain. The approach is implemented using a Java toolkit that automatically interacts with the NuSMV model checker. We verify this system against a set of specifications and report the results of our experiments.KeywordsSmart contractBlockchainThree-valued model checkingTCTLTrustUncertainty

—Bounded model checking has proven to be an efficient method for finding bugs in system designs. In this paper, we present an incremental semantic translation for Bounded model checking and give an incremental algorithm. We implement this method in NuSMV model checker and report encouraging results. Keywords-bounded model checking; semantic encoding; incremental; NuSMV I. I NTRODUCTION Bounded Model Checking (BMC) [1] is an efficient method for finding bugs in system designs. The basic idea of BMC is to convert the model checking problem to a Boolean satisfiability problem, i.e., given a Kripke structure M representing the system, an LTL formula ' and a bound k, a propositional formula M;:' k is created and it is satisfiable if and only if M involves a counterexample of ', and the length of such counterexample is bounded by k. BMC has established itself as a complementary method to BDD [2] based approach. There are some advantages for BMC:  First, it leverages the impressive achievements in SAT solver technology and several industry applications have proven that BMC seems to be able to solve certain problems that are not feasible for BDDs [4].

The problem of formal software specification has been addressed and discussed since the infancy of software engineering. However, among all the proposed solutions, none is universally accepted yet. Many different formal descriptions can in fact be given for the same software component; thus, the problem of determining the consistency relation among those descriptions becomes relevant and potentially critical. In this work, we propose a method for comparing two specific kinds of formal specifications of containers. In particular, we check the consistency of intensional behavior models with algebraic specifications. The consistency check is performed by generating a behavioral equivalence model from the intensional model, converting the algebraic axioms into temporal logic formulae, and then checking them against the model by using the NuSMV model checker. An automated software tool which encodes the problem as model checking has been implemented to check the consistency of recovered specifications of relevant Java classes.