Structural stability analysis of gene regulatory networks modeled by Boolean networks

This paper studies the structural monostability and structural cycle‐stability of Boolean networks (BNs). Firstly, the structural‐equivalent Boolean networks are converted to the algebraic forms by using the semitensor product of matrices. Secondly, the concepts of structural monostability and structural cycle‐stability for Boolean networks are proposed. On the basis of the algebraic forms of structural‐equivalent Boolean networks, some necessary and sufficient conditions are presented for the structural monostability and structural cycle‐stability of Boolean networks. Finally, an illustrative example is worked out to show the effectiveness of the obtained results.     

Modeling pathways of differentiation in genetic regulatory networks with Boolean networks

We have carried out the first examination of pathways of cell differentiation in model genetic networks in which cell types are assumed to be attractors of the nonlinear dynamics, and differentiation corresponds to a transition of the cell to a new basin of attraction, which may be induced by a signal or noise perturbation. The associated flow along a transient to a new attractor corresponds to a pathway of differentiation. We have measured a variety of features of such model pathways of differentiation, most of which should be observable using gene array techniques. © 2005 Wiley Periodicals, Inc. Complexity 11: 52–60, 2005     

Identification of predictors of Boolean networks from observed attractor states

Predictors of Boolean networks are of significance for biologists to target their research on gene regulation and control. This paper aims to investigate how to determine predictors of Boolean networks from observed attractor states by solving logical equations. The proposed method consists of four steps. First, all possible cycles formed by known attractor states are constructed. Then, for each possible cycle, all data‐permitted predictors of each node are identified according to the known attractor states. Subsequently, the data‐permitted predictors are incorporated with some common biological constraints to generate logical equations that describe whether such possible predictors can ultimately be chosen as valid ones by the biological constraints. Finally, solve the logical equations; the solutions determine a family of predictors satisfying the known attractor states. The approach is quite different from others such as computer algorithm‐based and provides a new angle and means to understand and analyze the structures of Boolean networks.     

Persistence of activity in threshold contact processes,an “Annealed approximation” of random Boolean networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437–467 random Boolean networks. There are three parameters: $n = {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs to each node, and $p = {\rm the}$ expected fraction of 1'sin the Boolean functions at each node. Following a standard practice in thephysics literature, we use a threshold contact process on a random graph on n nodes, in which each node has in degree r, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$ , then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$ , and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Stability of stationary solutions of piecewise affine differential equations describing gene regulatory networks

We study stability properties of a class of piecewise affine systems of ordinary differential equations arising in the modeling of gene regulatory networks. Our method goes back to the concept of a Filippov stationary solution (in the narrow sense) to a differential inclusion corresponding to the system in question. The main result of the paper justifies a reduction principle in the stability analysis enabling to omit the variables that are not singular, i.e. that stay away from the discontinuity set of the system. We suggest also “the first approximation method” to study asymptotic stability of stationary solutions based on calculating the principal part of the system, which is 0-homogeneous rather than linear. This leads to an efficient algorithm of how to check asymptotic stability without calculating the eigenvalues of the system?s Jacobian. In Appendix A we discuss and compare two other concepts of stationary solutions to the system in question.     

On computation of the steady-state probability distribution of probabilistic Boolean networks with gene perturbation

Given a Probabilistic Boolean Network (PBN), an important problem is to study its steady-state probability distribution for network analysis. In this paper, we present a new perturbation bound of the steady-state probability distribution of PBNs with gene perturbation. The main contribution of our results is that this new bound is established without additional condition required by the existing method. The other contribution of this paper is to propose a fast algorithm based on the special structure of a transition probability matrix of PBNs with gene perturbation to compute its steady-state probability distribution. Experimental results are given to demonstrate the effectiveness of the new bound, and the efficiency of the proposed method.     

Robustness in complex information systems: The role of information “barriers” in Boolean networks

In this supposed “information age,” a high premium is put on the widespread availability of information. Access to as much information as possible is often cited as key to the making of effective decisions. While it would be foolish to deny the central role that information and its flow has in effective decision‐making processes, this chapter explores the equally important role of “barriers” to information flows in the robustness of complex systems. The analysis demonstrates that (for simple Boolean networks at least) a complex system's ability to filter out, i.e., block, certain information flows is essential if it is not to be beholden to every external signal. The reduction of information is as important as the availability of information. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

A preferential attachment model with Poisson growth for scale-free networks

We propose a scale-free network model with a tunable power-law exponent. The Poisson growth model, as we call it, is an offshoot of the celebrated model of Barabási and Albert where a network is generated iteratively from a small seed network; at each step a node is added together with a number of incident edges preferentially attached to nodes already in the network. A key feature of our model is that the number of edges added at each step is a random variable with Poisson distribution, and, unlike the Barabási–Albert model where this quantity is fixed, it can generate any network. Our model is motivated by an application in Bayesian inference implemented as Markov chain Monte Carlo to estimate a network; for this purpose, we also give a formula for the probability of a network under our model.     

Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework

To each Boolean function and each x{0,1}ⁿ, we associate a signed directed graph G(x), and we show that the existence of a positive circuit in G(x) for some x is a necessary condition for the existence of several fixed points in the dynamics (the sign of a circuit being defined as the product of the signs of its edges), and that the existence of a negative circuit is a necessary condition for the existence of an attractive cycle. These two results are inspired by rules for discrete models of genetic regulatory networks proposed by the biologist R. Thomas. The proof of the first result is modelled after a recent proof of the discrete Jacobian conjecture.     

Modeling gene regulatory networks with piecewise linear differential equations

Microarray chips generate large amounts of data about a cell’s state. In our work we want to analyze these data in order to describe the regulation processes within a cell. Therefore, we build a model which is capable of capturing the most relevant regulating interactions and present an approach how to calculate the parameters for the model from time-series data. This approach uses the discrete approximation method of least squares to solve a data fitting modeling problem. Furthermore, we analyze the features of our proposed system, i.e., which kinds of dynamical behaviors the system is able to show.     

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.     

Robust stability in distribution of Boolean networks under multi-bits stochastic function perturbations

In this work, robust stability in distribution of Boolean networks (BNs) is studied under multi-bits probabilistic and markovian function perturbations. Firstly, definition of multi-bits stochastic function perturbations is given and an identification matrix is introduced to present each case. Then, by viewing each case as a switching subsystem, BNs under multi-bits stochastic function perturbations can be equivalently converted into stochastic switching systems. After constructing respective transition probability matrices which can unify multi-bits probabilistic and markovian function perturbations in a consolidated framework, robust stability in distribution can be handled. On such basis, necessary and sufficient conditions for robust stability in distribution of BNs under stochastic function perturbations are given respectively. Finally, two numerical examples are presented to verify the validity of our theoretical results.     

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

The use of “control parameters” as applied to describe the dynamics of complex mathematical systems within models of real social systems is discussed. Whereas single control parameters cannot sufficiently characterize the dynamics of such systems it is suggested that domains of values of certain sets of parameters are appropriately denoting necessary conditions for highly disordered dynamics of social systems. Various of those control parameters permit a straightforward interpretation in terms of properties of social rules and structures. © 1999 John Wiley & Sons, Inc.     

Stability analysis of the differential genetic regulatory networks model with time-varying delays and Markovian jumping parameters

In this paper, we investigate the stability and robust stability criteria for genetic regulatory networks with interval time-varying delays and Markovian jumping parameters. The genetic regulatory networks have a finite number of modes, which may jump from one mode to another according to the Markov process. By using Lyapunov–Krasovskii functional, some sufficient conditions are derived in terms of linear matrix inequalities to achieve the global asymptotic stability in the mean square of the considered genetic regulatory networks. Two numerical examples are provided to illustrate the usefulness of the obtained theoretical results.     

Communicability graph and community structures in complex networks

We use the concept of the network communicability [E. Estrada, N. Hatano, Communicability in complex networks, Phys. Rev. E 77 (2008) 036111] to define communities in a complex network. The communities are defined as the cliques of a “communicability graph”, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. We discuss the efficiency of this algorithm of community detection. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. Finally, we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. We then analyze the USA airport network, for which we successfully detect two big communities of the eastern airports and of the western/central airports as well as two bridging central communities. In striking contrast, a well-known algorithm groups all but two of the continental airports into one community.