Pinning controllability of autonomous Boolean control networks

Autonomous Boolean networks (ABNs), which are developed to model the Boolean networks (BNs) with regulatory delays, are well known for their advantages of characterizing the intrinsic evolution rules of biological systems such as the gene regulatory networks. As a special type of ABNs with binary inputs, the autonomous Boolean control networks (ABCNs) are introduced for designing and analyzing the therapeutic intervention strategies where the binary inputs represent whether a certain medicine is dominated or not. An important problem in the therapeutic intervention is to design a control sequence steering an ABCN from an undesirable location (implying a diseased state) to a desirable one (corresponding to a healthy state). Motivated by such background, this paper aims to investigate the reachability and controllability of ABCNs with pinning controllers. Several necessary and sufficient criteria are provided by resorting to the semi-tensor product techniques of matrices. Moreover, an effective pinning control algorithm is presented for steering an ABCN from any given states to the desired state in the shortest time period. Numerical examples are also presented to demonstrate the results obtained.This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61329301, 61273156), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20130017), Six Talent Peaks Project for the High Level Personnel from the Jiangsu Province of China (Grant No. 2015- DZXX-003), Scientific Research Foundations of Graduate School of Southeast University (Grant No. YBJJ1560), and Innovation Program of Jiangsu Province (Grant No. KYZZ15 0050)

On the effects of firing memory in the dynamics of conjunctive networks

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete

A weighted pair graph representation for reconstructibility of Boolean control networks

A new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility definition given in [Fornasini & Valcher, TAC2013, Def. 4]. Based on the WPG representation, an effective algorithm for determining the new reconstructibility notion for BCNs is designed with the help of the theories of finite automata and formal languages. We prove that a BCN is not reconstructible iff its WPG has a complete subgraph. Besides, we prove that a BCN is reconstructible in the sense of [Fornasini & Valcher, TAC2013, Def. 4] iff its WPG has no cycles, which is simpler to be checked than the condition in [Fornasini & Valcher, TAC2013, Thm. 4].Comment: 20 pages, 10 figures, accepted by SIAM Journal on Control and Optimizatio

Dynamical pathway analysis

<p>Abstract</p> <p>Background</p> <p>Although a great deal is known about one gene or protein and its functions under different environmental conditions, little information is available about the complex behaviour of biological networks subject to different environmental perturbations. Observing differential expressions of one or more genes between normal and abnormal cells has been a mainstream method of discovering pertinent genes in diseases and therefore valuable drug targets. However, to date, no such method exists for elucidating and quantifying the differential dynamical behaviour of genetic regulatory networks, which can have greater impact on phenotypes than individual genes.</p> <p>Results</p> <p>We propose to redress the deficiency by formulating the functional study of biological networks as a control problem of dynamical systems. We developed mathematical methods to study the stability, the controllability, and the steady-state behaviour, as well as the transient responses of biological networks under different environmental perturbations. We applied our framework to three real-world datasets: the SOS DNA repair network in <it>E. coli </it>under different dosages of radiation, the GSH redox cycle in mice lung exposed to either poisonous air or normal air, and the MAPK pathway in mammalian cell lines exposed to three types of HIV type I Vpr, a wild type and two mutant types; and we found that the three genetic networks exhibited fundamentally different dynamical properties in normal and abnormal cells.</p> <p>Conclusion</p> <p>Difference in stability, relative stability, degrees of controllability, and transient responses between normal and abnormal cells means considerable difference in dynamical behaviours and different functioning of cells. Therefore differential dynamical properties can be a valuable tool in biomedical research.</p

Synchronization in output-coupled temporal Boolean networks

This paper presents an analytical study of synchronization in an array of output-coupled temporal Boolean networks. A temporal Boolean network (TBN) is a logical dynamic system developed to model Boolean networks with regulatory delays. Both state delay and output delay are considered, and these two delays are assumed to be different. By referring to the algebraic representations of logical dynamics and using the semi-tensor product of matrices, the output-coupled TBNs are firstly converted into a discrete-time algebraic evolution system, and then the relationship between the states of coupled TBNs and the initial state sequence is obtained. Then, some necessary and sufficient conditions are derived for the synchronization of an array of TBNs with an arbitrary given initial state sequence. Two numerical examples including one epigenetic model are finally given to illustrate the obtained results