Controllability of asynchronous Boolean multiplex control networks , Chao Luo , Xingyuan Wang, and Hong Liu

Citation: Chaos 24, 033108 (2014); doi: 10.1063/1.4887278 View online: http://dx.doi.org/10.1063/1.4887278 View Table of Contents: http://aip.scitation.org/toc/cha/24/3 Published by the American Institute of Physics

CHAOS 24, 033108 (2014)

Controllability of asynchronous Boolean multiplex control networks Chao Luo,1,2,a) Xingyuan Wang,3 and Hong Liu1,2 1

School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan 250014, China 3 Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China 2

(Received 20 May 2014; accepted 25 June 2014; published online 9 July 2014) In this article, the controllability of asynchronous Boolean multiplex control networks (ABMCNs) is studied. First, the model of Boolean multiplex control networks under Harvey’ asynchronous update is presented. By means of semi-tensor product approach, the logical dynamics is converted into linear representation, and a generalized formula of control-depending network transition matrices is achieved. Second, a necessary and sufficient condition is proposed to verify that only control-depending fixed points of ABMCNs can be controlled with probability one. Third, using two types of controls, the controllability of system is studied and formulae are given to show: (a) when an initial state is given, the reachable set at time s under a group of specified controls; (b) the reachable set at time s under arbitrary controls; (c) the specific probability values from a given initial state to destination states. Based on the above formulae, an algorithm to calculate overall reachable states from a specified initial state is presented. Moreover, we also discuss an approach to find the particular control sequence which steers the system between two states with maximum probability. Examples are shown to illustrate the C 2014 AIP Publishing LLC. feasibility of the proposed scheme. V [http://dx.doi.org/10.1063/1.4887278]

The control of genetic regulatory networks (GRNs) has received considerable attention in the past few years, which provides some potential routes to work out the problems in life science, such as therapeutic intervention for disease and the prolonging of lifespan. Meanwhile, with the development of system biology, integrating research on multiple biological systems becomes the trend in this research field. Hence, based on the concept of multiplex networks, the studies on Boolean control networks are novel and meaningful. And for a more realistic analysis, asynchronous update is considered in the proposed model.

I. INTRODUCTION

As a kind of time- and state-discrete models, Boolean networks (BNs) are used to study the dynamic properties of GRNs.1 Based on an assumption that small differences in concentration levels are irrelevant, the expressions of genes are roughly divided into two levels: active and repressing states represented by logical 0 and 1. Each gene of GRNs is represented by a node of a directed graph, and Boolean function assigned to each node is employed to indicate the mutual regulation among genes. Random Boolean networks (RBNs) proposed by Kauffman are the simplest and most widely investigated models, where the connections and update rules of nodes (genes) on networks are randomly a)

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assigned. The studies on dynamics of BNs,2,3 algorithms to find the attractors in state space4–7 and some applications in biological research8–10 have been presented during the past few decades. Moreover, some extended models of BNs, such as asynchronous BNs (ABNs),11 probabilistic BNs,12 and switched BNs13 have also attracted considerable attention of researchers. In 2010, a novel kind of network architecture named multiplex networks was first proposed by Mucha,14 which can be seen as integrated networks of several parallel systems combined by some identical nodes as shown in Fig. 1. Different from the usual concept, sub-networks are not connected by the traditional way of coupling. Each layer of multiplex networks represents an independent system, and nodes of system acting as different roles participate in one or more layers of interactions. When a node is involved in more than one layer of multiplex networks, its final state at each time step will be constrained by all of involved sub-networks. In the last three years, a variety of studies based on multiplex networks have been achieved, including network topology and dynamic properties,15 diffusion dynamics16,17 and game theory,18,19 etc. It is noteworthy that, with the in-depth study of biology, the architecture of multiplex networks is consistent with the concept of integration of system biology. As mentioned in Ref. 20, “the same gene or biochemical species can be involved in a regulatory interaction, in a metabolic reaction, or in another signaling pathway,” biologists put forward the idea on the research of integrated biological networks containing gene regulatory networks, protein-protein interactions and other types of interactions such as RNA binding proteins (RBPs).21 Hence, multiplex networks

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FIG. 1. Illustration of multiplex networks with two layers. Nodes a–e are identical in both layer 1 and layer 2.

provide a novel way to construct the abstract models of biochemical networks, which can better depict the interactions and relationships of genes on a holistic level. As to abstract models, a certain degree of idealization is necessary. For classical BNs, researchers assume that all of nodes on networks are updated in parallel, which is based on an assumption that update scheme is irrelevant to dynamical behaviors of networks. However, in reality, we cannot find a synchronized clock in biological systems. As the studies in Ref. 22, “factors such as mRNA and protein synthesis, degradation and transport times mean that the system is replete with delays of varying amounts, and genes are activated or inhibited in a fundamentally asynchronous manner.” More and more evidences have shown that considerable divergences exist in the dynamics between synchronous and asynchronous systems. BNs under asynchronous update were first proposed by Harvey et al.11 and some generic properties of asynchronous system were discussed. In Ref. 23, Greil et al. showed that the mean number of attractors in an asynchronous critical Boolean network grows like a power law and the mean size of the attractors increases as a stretched exponential with the system size, which is in strong contrast to the synchronous case; in Ref. 24, Saadatpour et al., implemented a comparative study on the dynamic behaviors of a Boolean model of a signal transduction network using synchronous and asynchronous updating schemes; in Ref. 25, for the connectivity K ¼ 1 of asynchronous Boolean networks, Shreim et al. analytically studied the number of attractors and basin sizes in state space; in Ref. 26, Jack et al. developed an asynchronous threshold Boolean network simulation algorithm to model signal transduction in a single cell; in Refs. 6 and 27, algorithms were presented to find the attractors in state space of ABNs. Based on the above results, in many biological phenomena, ABNs are more plausible compared with the synchronous cases. Hence, it is meaningful to study the properties of BNs under asynchronous update. Accompany with the development of biology, control of biological system become a hot topic in recent years.28–33 As to the research of GRNs, one of the major goals is to carry

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out the therapeutic intervention strategies for diseased targets.34,35 Correspondingly, Boolean control networks (BCNs) as a theoretical branch of the above studies provide an efficient way to investigate the control of GRNs based on abstract models.36 When control theory is applied into biological systems, a fundamental problem is to determine whether a specific system can be controlled. In recent five years, controllability of BCNs has become hot topic and attracted interest of researchers in this research field. In Ref. 37, Cheng et al. studied the controllability of BCNs and provided the method to calculate the reachable sets; based on Cheng’s studies, the results of controllability of BCNs under time-invariant and -variant delay were proposed.38,39 Moreover, the controllability of l-th order BCNs, i.e., the updated states of controlled systems depend on the last l values, also have been discussed;40 based on the results of Ref. 40, the conditions to determine the global controllability of BCNs with avoiding set were presented,41 furthermore, a way to transform l-th order BCNs to equivalent time-variant BCNs was achieved;42 to avoid a set of forbidden states, the controllability of BCNs were discussed via the PerronFrobenius theory.43 The above studies have achieved reasonable results and provided an insight into the intrinsic control in biological systems. However, to the best of our knowledge, few literature was related to the controllability of BCNs involving the consideration of multiplex architecture. Compared with the classic BNs, Boolean multiplex networks have more complex topology structure and higher holistic level which can provide a generalized model to be better in conformity with the development of biology. Furthermore, most of the previous studies were based on synchronous and deterministic models. As the above discussion, asynchronous systems are closer to the reality, based on which the studies on BNs have access to the generic properties of biological systems. So, we think it is valuable to extend the related research into the field of Boolean multiplex networks under asynchronous update. In this article, we focus on controllability of asynchronous Boolean multiplex control networks (ABMCNs). In Ref. 20, Cozzo et al. presented the model of Boolean multiplex (multilevel) networks, based on which we construct Boolean multiplex control networks by introducing the input controls. And Harvey’s update scheme is implemented into system, which means only one node would be randomly chosen for update at each time step. By means of algebraic representation, the controllability of ABMCNs is to be discussed under two types of controls, i.e., free Boolean control sequences and the controls satisfying certain logical rule. Since asynchronous systems are nondeterministic, we first proof that only control-depending fixed points of ABMCNs can be controllable with probability one, which implies, in most cases, the discussion of controllability of ABMCNs is based on probability. Respectively, for each situation, the generalized formulae are proposed to calculate the reachable sets at time step s from specified initial states under given control sequences or arbitrary controls, and we will show how to calculate the specific probability value from an initial state to a destination state under a certain control sequence. Moreover, we are to discuss an approach which can find a

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precise control sequence steering the system between two states with maximum probability. This article is organized as follows. In preliminaries, some introductions on mathematic tools used in this article are provided. In main results, the model of ABMCNs is first proposed and the linear representation of system is achieved. Consequently, by means of two types of controls, the controllability of BMCNs is discussed. Some examples are shown to illustrate the main results. Finally, a concluding remark is given. II. PRELIMINARIES

In this article, studies on controllability of Boolean multiplex control networks are based on the algebraic representation of logical dynamics by using the semi-tensor product (STP). Some concepts and properties of STP are briefly introduced. Definition 1 (Ref. 36): 1) Let X be a row vector of dimension np, and Y¼ ½y1 ; y2 ; …; yp T be a column vector of dimension p. Then, we split X into p equal-size blocks as X 1 ; X2 ; …; X p , where X i 2 Rn , i ¼ 1; 2; …; p. The STP of X and Y, denoted by X3 Y, can be defined as a row vector X3 Y ¼

p X

X i yi 2 R n :

i¼1

Similarly, the STP of YT and X T can be defined as a column vector YT 3 X T ¼

p X

yi ðX i ÞT 2 Rn :

i¼1

2) Let A 2 Mmn and B 2 Mpq , where Mxy represents the real matrix set of dimension xy. If either n is a factor of p, say nt ¼ p and denote it as A t B, or p is a factor of n, say n ¼ pt and denote it as A t B, then we define the STP of A and B, denoted by C ¼ A3 B, as the following: C consists of m  q blocks as C ¼ ðCij Þ and each block is Cij ¼ Ai 3 Bj ;

i ¼ 1; …; m; j ¼ 1; …; q;

where Ai is the i-th row of A and Bj is the j-th column of B. It is easy to verify that for two column vectors X 2 Rm and Y 2 Rn , X3 Y 2 Rmn . Note that STP of matrix can be seen as a generalization of the conventional matrix product, so the properties of matrix product, such as distributive rule, associative rule, still hold. For statement ease, some notations in this article are defined. the r-th column 1) drn denotes   of the n  n identity matrix In and Dn :¼ drn j1  r  n , which is the set of all n colKðdrn Þ ¼ r, furtherumns of In . We define  i i an operation  ip 0 1 2 more, when C ¼ dn ; dn ; …; dn , KðC0 Þ ¼ fi1 ; i2 ; …; ip g. r   Correspondingly,  i i  KðrÞn ¼ dn and K ðfi1 ; i2 ; …; ip gÞn ip 1 2 ¼ dn ; dn ; …; dn .

2) A matrix A 2 Mnm   can be called a logical matrix if A ¼ din1 ; din2 ; …; dinm , which is briefly denoted by A ¼ dn ½i1 ; i2 ; …; im . And the set of n  m logical matrices is denoted by Lnm . 3) Let matrix A 2 Mnm , RowðAÞi and ColðAÞj represent the i-th row and j-th column of matrix A. And, Ai;j is the element at position ði; jÞ of matrix A. 4) Let column vector X 2 Rm , all of row indies of X in which row elements are not equal to zero compose a set denoted by XðXÞ. For example, X ¼ ½1; 0; 2; 1T and XðXÞ ¼ f1; 3; 4g. Assume X n ¼ x1 3 x2 3 …3 xn ¢3 ni¼1 xi , where xi 2 D2 ; i ¼ 1; 2; …; n. We can get X 2n ¼ Un X n , where Q   Un ¼ ni¼1 I 2i1  ðI 2  W½2;2Ni  ÞMr and  refers to the Kronecker product. Here, Mr ¼ d4 ½1; 4, which is powerreducing matrix and it can be verified that P2 ¼ Mr P, 8P 2 D2 . Next, we define the swap matrix W½m;n , let X 2 Rm and Y 2 Rn be two column vectors W½m;n XY ¼ YX; where W½m;n is a mn  mn matrix labeled columns by ð11; 12; …; 1n; …; m1; m2; …; mnÞ and rows by ð11; 21; …; m1; …; 1n; 2n; …; mnÞ, the elements in position ððI; JÞ; ði; jÞÞ is  1; I ¼ i and J ¼ j wðI;JÞ;ði;jÞ ¼ 0; otherwise: W ½m;m is briefly denoted by W½m . In order to get the matrix expression of logical dynamics, the Boolean values should be denoted as vectors Ture ¼ 1 d12 and False ¼ 0 d22 . Lemma 1 (Ref. 36): Any logical function f ðx1 ; x2 ; …; xr Þ with logical arguments x1 ; x2 ; …; xr 2 D2 , can be expressed in a multi-linear form as f ðx1 ; x2 ; …; xr Þ ¼ Mf 3 ri¼1 xi ; where Mf 2 2  2r is unique, which is called the structure matrix of logical function f. More details on STP can be found in Ref. 36. In the following, the matrix products are assumed to be STP and the symbol 3 is omitted. III. MAIN RESULTS A. Algebraic expression of asynchronous Boolean multiplex control networks

In this section, we propose the model of asynchronous Boolean multiplex control networks and construct the algebraic form of ABMCNs using STP technique. Different from the signal-layer model, some nodes on multiplex networks exist in multiple layers, the states of which on different layers evolve independently of each other. However, at the end of each time step, there should be unique deterministic states for those joint nodes, which are determined by all of the values on different layers. For a

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Boolean multiplex network with n~ nodes and k~ layers, ~ represents the state of node i assume xli ðtÞ; l 2 f1; 2;…;kg on layer l at time t. One can obtain ~ x 2 ðtÞ;…;~x n~ ðtÞÞ; i¼1;2;…;~ n ; l¼ 1;2;…; k; xli ðtþ1Þ¼fil ð~ x 1 ðtÞ;~ (1) where fil is the update rule of node i on layer l. Furthermore, n g represents the global state of assume ~ x i ðtÞ;i 2 f1; 2;…;~ node i at time t. Refer to Ref. 20, we can get   ~ ~ (2) x i ðtþ1Þ ¼~f i x1i ; x2i ; …; xki ; i ¼ 1; 2; …; n~; where ~f i is the canalizing function. Note that, strictly speaking, the emergence of xli ðtÞ should be ahead of ~x i ðtÞ. Based on an assumption that the interval between the above two states is trivial, the same time step t is used for both of them. ~ controls are introduced into system, Boolean When m multiplex control networks can be described as l ¼ ^f i ðu1 ðtÞ; …; um~ ðtÞ; ~x 1 ðtÞ; ~x 2 ðtÞ; …; ~x n~ ðtÞÞ; ~ i ¼ 1; 2; …; n~; l ¼ 1; 2; …; k;

xli ðtþ1Þ

(3)

^ i ¼ ðI 2i1  Li ÞW½2m~ ;2i1  ðI2m~  Ui1 ÞðI 2iþm~  Un~i Þ where L is the generalized form of control-depending network transition matrix of system (6) when node i is randomly chosen for update at time t, which contains all of the transfer information of a dynamic system. When a specified asynchronous Boolean multiplex control network is given, we can get the corresponding control-depending network transition matrix of network. In the following, respectively, for two kinds of controls, we discuss the controllability of ABMCNs: 1) Controls come from a free Boolean sequence. Precisely, ~ controls are freely designed and described as at time t, m ~ uðtÞ ¼ 3 m i¼1 ui ðtÞ. 2) The controls are determined by certain logical rules, which can be called input control networks 8 > u1 ðt þ 1Þ ¼ g1 ðu1 ðtÞ; u2 ðtÞ; …; um~ ðtÞÞ; > > > > < u2 ðt þ 1Þ ¼ g2 ðu1 ðtÞ; u2 ðtÞ; …; um~ ðtÞÞ; .. > > . > > > : u ðt þ 1Þ ¼ g ðu ðtÞ; u ðtÞ; …; u ðtÞÞ; m ~ m ~ 1 2 m ~

(7)

l ~ are controls and ^f i is the update where ui ðtÞ, i ¼ 1; 2; …; m rule of node i on layer l with controls. l Using Lemma 1, for each logical rule ^f i , a corresponding structure matrix M li can be found, and Eq. (3) can be converted into the algebraic form as

~ are logical rules. where gi : f0; 1gm~ ! f0; 1g; i ¼ 1; 2; …; m

~ (4) i ¼ 1; 2; …; n~; l ¼ 1; 2; …; k;

Under Harvey’s asynchronous update scheme, there are totally n~ different update choices at time t, which are corresponding to n~ different control-depending network transition ^ i , i 2 f1; 2; …; n~g. Assume the same probability matrices L for each  node to be chosen for update at time t, say, ^ i ¼ 1=~ n . Then, one can obtain Pr L

x ðtÞ; xli ðtþ1Þ ¼ Mli uðtÞ~

~ ~ ~ x i ðtÞ, uðtÞ ¼ 3 m where x~ðtÞ ¼ 3 ni¼1 i¼1 ui ðtÞ. Subsequently, the algebraic representation of Eq. (2) can be obtained as ~

~ i M1 uðtÞ~ x~i ðtþ1Þ ¼ M x ðtÞM2i uðtÞ~ x ðtÞ…M ki uðtÞ~ x ðtÞ i x ðtÞ; i ¼ 1; 2; …; n~ ¼ Li uðtÞ~

(5)

~ i is the structure matrix of logical function ~f i and where M ~ ~ k1 ~ nÞ ~ i 3 kj¼1 Li ¼ M ð2ðj1Þðmþ~  M ji ÞUmþ~ ~ n. When Boolean networks are under Harvey’s asynchronous update, at time t, there is only one node chosen randomly for update, i.e.,  x ðtÞ; i 2 f1; 2; …; n~g x~i ðtþ1Þ ¼ Li uðtÞ~ (6) x~j ðtþ1Þ ¼ x~j ðtÞ; j 6¼ i; j ¼ 1; 2; …; n~: Multiplying all the n~ equations of system (6), one can get x~ðt þ 1Þ ¼ x~1 ðtÞ; …; x~i1 ðtÞLi uðtÞ~ x ðtÞ~ x iþ1 ðtÞ; …; x~n~ ðtÞ x 1 ðtÞ; …; x~i1 ðtÞ ¼ ðI2i1  Li ÞW ½2m~ ;2i1  uðtÞ~  x~ðtÞ~ x iþ1 ðtÞ; …; x~n~ ðtÞ ¼ ðI2i1  Li ÞW ½2m~ ;2i1  uðtÞUi1 x~1 ðtÞ; …;  x~i1 ðtÞ~ x i ðtÞUn~i x~iþ1 ðtÞ; …; x~n~ ðtÞ ¼ ðI2i1  Li ÞW ½2m~ ;2i1  ðI 2m~  Ui1 Þ ðI 2iþm~  Un~i ÞuðtÞ~ x ðtÞ ^ x ðtÞ; ¢L i uðtÞ~

B. Controllability of asynchronous Boolean multiplex control networks with probability one

n~ X ^ i: ¼1 L L n~ i¼1

(8)

Definition 2: Consider the asynchronous Boolean multiplex control network (6), the destination state xd 2 D2n~ is said to be deterministically controllable from an initial state x~0 2 D2n~ at time s > 0, if a group of controls  1Þ  ~ uð0Þ; uð1Þ; …; uðs ~ ~ x~i ðsÞ ¼ x~d j3 ni¼1 x i ð0Þ can be found such that Pr 3 ni¼1 ¼ x~0 g ¼ 1. Definition 3: As to the asynchronous Boolean multiplex control network (6), when a control u 2 D2m~ exists such that state ^ i u~ x f ; 8i 2 f1; 2; …; n~g, x~f is said to be x~f 2 D2n~ holds x~f ¼ L a control-depending fixed point. Theorem 1: For system (6), a given state x~c can be controllable with probability one if and only if state x~c is a controldepending fixed point. Proof: (Sufficiency) Assume the state x~c 2 D2n~ is a controldepending fixed point of system (6). According to Definition 3, a control u 2 D2m~ can be found that  ~ ~ ~ Pr 3 ni¼1 x~i ð1Þ ¼ xc j3 ni¼1 x i ð0Þ ¼ xc g ¼ 1. Consequently, we can find a group of controls uðiÞ ¼ u, i ¼ 0; 1; …s  1 such

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 ~  ~ ~ that Pr 3 ni¼1 x~i ðsÞ ¼ x~d j3 ni¼1 x i ð0Þ ¼ x~0 ¼ 1. So, x~c can be controllable from itself with probability one at any time steps. ~ ~ x i ðsÞ is con(Necessity) When state x~c ¼ x~ðsÞ ¼ 3 ni¼1 trollable with probability one from initial state x~0 ¼ x~ð0Þ at time s, x~c should be proved to be a control-depending fixed point. Here, we assume x~ðs  1Þ 6¼ x~ðsÞ. According  ~to Definition 2, we can find a control uðs  1Þ that Pr 3 ni¼1 ~ ~ x~i ðsÞ ¼ x~c j3 ni¼1 x i ðs  1Þ ¼ x~ðs  1Þg ¼ 1, which means ^ ^ i2 uðs  1Þ~ x~ðsÞ ¼ L i1 uðs  1Þ~ x ðs  1Þ ¼ L x ðs  1Þ, where ^ i1 uðs  1Þ i1 ; i2 2 f1; 2; …; n~g and i1 6¼ i2 . When x~ðsÞ ¼ L x~ðs  1Þ, considering the rule of asynchronous update scheme, there should be only one element x~i1 ðs1Þ 6¼ x~i1 ðsÞ and the rest elements x~j ðs1Þ ¼ x~j ðsÞ;j 6¼ i1 ; j ¼ 1;2;…; n~. ^ i2 uðs1Þ~ x ðs1Þ, there should be only And when x~ðsÞ ¼ L one element x~i2 ðs1Þ 6¼ x~i2 ðsÞ and the rest elements x~j0 ðs1Þ ¼ x~j0 ðsÞ;j0 6¼ i2 ;j0 ¼ 1;2;…; n~. The two results are contradictory. So the above assumption cannot be held. Say, x~ðs1Þ should be equal to x~ðsÞ. Deduce the rest from this, we can obtain x~ð0Þ ¼ x~ð1Þ ¼ … ¼ x~ðsÞ. And according to Definition 3, it can be proved that x~c should be a controldepending fixed point of system (6). This completes the proof. The above result shows, given two states x~0 ; x~d 2 D2n in asynchronous system (6), when x~0 6¼ x~d , x~d cannot be controllable from x~0 with probability one. Hence, in the following, the discussion of controllability of ABMCNs is based on probability.

C. Controllability of asynchronous Boolean multiplex control networks via free Boolean sequence

In this section, we discuss the case that controls are free ~ Boolean sequences, which means uðtÞ¼3 m i¼1 ui ðtÞ, ui ðtÞ 2 D2 . Definition 4: Given the initial state x~0 2 D2n~ and the destination state x~d 2 D2n~ , the asynchronous Boolean multiplex control network (6) is said to be controllable with probability from x~0 to x~d at time s > 0, if a control sequence uð0Þ, 1Þ can be  ~ uð1Þ,…, uðs   found such that ~ ~ x~i ðsÞ ¼ x~d j3 ni¼1 x i ð0Þ ¼ x~0 > 0. Pr 3 ni¼1 First, when the initial state and a specified control sequence are given, the reachable set with probability at time s can be obtained. Theorem 2: For system (6), the destination state x~d is reachable with probability from initial state x~0 after s time steps under a series of given controls uð0Þ, uð1Þ,…, uðs  1Þ, if s

 ~ x0 Þ ÞÞ n~ ; L xd 2 KðXðColð u 2

(9)

P u s1 ~ ¼ LW  ½2n~ ;2m~  , L¼  1 n~ L ^ where L i¼1 i , 3 t¼0 uðtÞ ¼ d2m~ s . n~ Proof: By the STP of matrices, system (6) can be rewritten as ^ i W½2n~ ;2m~  x~ðtÞuðtÞ: x~ðt þ 1Þ ¼ L Since each node on network has the same probability to be chosen for update at time t. One can obtain the overall expected value of x~ðtÞ as

Exðt þ 1Þ ¼

n~ 1X ^ i W ½2n~ ;2m~  ExðtÞuðtÞ L n~ i¼1

~ ¢LExðtÞuðtÞ:

(10)

To expand the above formula and yields ~ ~ x ð0Þuð0Þ; Exð1Þ ¼ LExð0Þuð0Þ ¼ L~ 2

~ ~ x~ð0Þuð0Þuð1Þ; Exð2Þ ¼ LExð1Þuð1Þ ¼L .. . ~ s x~ð0Þuð0Þuð1Þ uðs  1Þ: ExðsÞ ¼ L u is given, When the control sequence 3 s1 ~ t¼0 uðtÞ ¼ d2ms s ~ x0 Þ Þ implies the index set of all of the reachable XðColðL u states with probability from the initial state x0 at time s. This completes the proof. We denote by Rð~ x 0 Þs the set of states which are reachable from an initial state x~0 with probability at time s, similarly, by x 0 Þs the overall reachable set with probability. Rð~ x 0 Þ ¼ [1 s¼0 Rð~ Based on the above result, it is easy to get Rð~ x 0 Þs . Lemma 2: For system (6), the set of states which are reachable from an initial state x~0 with probability at time s under a free control sequence is n o ~ s x~0 Þ 6¼ 0 ; (11) Rð~ x 0 Þs ¼ di2n~ jRowðL i

P  1 n~ L ~ LW  ½2n~ ;2m~  , L¼ ^ where L¼ i¼1 i . n~ Furthermore, when a control sequence is given, one can calculate the specific probability value from an initial state x~0 to a destination state x~d at time s. u Lemma 3: For system (6), when control 3 s1 t¼0 uðtÞ ¼ d2m~ s , a the probability from the initial state x~0 ¼ d2n~ to the destination state x~d ¼ db2n~ at time s is ~ s x~0 Þ ; Prf~ x ðsÞ ¼ db2n~ j~ x ð0Þ ¼ da2n~ g ¼ ðL b;u

(12)

P ^  1 n~ L ~ LW  ½2n~ ;2m~  , L¼ where L¼ i¼1 i . n~ In the following, when control is a free Boolean sequence, the overall reachable set Rð~ x 0 Þ is discussed. A condensed version of control-depending network transition matrix L0 2 M2n~ 2n~ is first defined as follows: ~ m

~ ^ ColðÚnp¼1 ColðL0 Þi ¼ Úi2 L p Þj ; i ¼ 1; 2; …; 2n~ ; (13) j¼ði1Þ2m~ þ1

^ p ; p ¼ 1; 2; where Ú is the Boolean operation of disjunction, L …; n~ are the control-depending network transition matrices. Based on the condensed control-depending network transition matrix, one can get the overall reachable set Rð~ x0Þ by using the following procedure, i.e., one can calculate all of the reachable states from a given initial state x~0 under a free Boolean sequence as control regardless of the constraint of time step (Table I). After slightly modification, we also can obtain Rð~ x 0 Þs using the above algorithm. Compared with Lemma 2, Algorithm 1 is totally based on set calculation to avoid the overload matrix computation. However, it just can find the reachable states from specified initial states but cannot provide more information, such as the specific probability

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TABLE I. Algorithm 1 (An algorithm for finding the overall reachable set Rð~ x 0 Þ). Begin: Algorithm 1 0

/* The initial state~ x 0 ¼ da2n~ */

C ¼ XðColðL Þa Þ 0

while [i2C XðColðL Þi Þ6C do 0

C ¼ ð[i2C XðColðL Þi ÞÞ [ C end while  Rx~ ¼ KðCÞ n~ 0

2

/* The overall reachable set Rx~ 0 */

End: Algorithm 1

values to different destination states and the corresponding control sequences. D. Controllability of asynchronous Boolean multiplex control networks via input control networks

By STP of matrix, one can get the linear representation of system (7) as uðt þ 1Þ ¼ GuðtÞ;

(14)

where G 2 L2m~ 2m~ is the network transient matrix of input control network. Definition 5: Consider the asynchronous Boolean multiplex control network (6) with input control network (7), given an initial state x~0 2 D2n~ and a destination state x~d 2 D2n~ , x~d is said to be controllable from x~0 at time s with probability, if an initial control u0 2 D2m~ can be found such that ~ ~ ~ ~ ~ x i ðsÞ ¼ x~d j3 ni¼1 x i ð0Þ ¼ x~0 ; 3 m Prð3 ni¼1 i¼1 ui ð0Þ ¼ u0 Þ > 0. Theorem 3: For system (6) with input control network (7), the destination state x~d is controllable with probability from initial state x~0 under initial control u0 at time s if G  x~d 2 KðXðH ðsÞW½2n~ ;2m~  x~0 u0 ÞÞ2n~ ;

(15)

 si ÞÞ3 s ðI ðsiÞm~  Um~ Þ, where HG ðsÞ ¼ 3 si¼1 ðI2ði1Þm~  ðLG i¼2 2 P  1 n~ Li . L¼ i¼1 n~ Proof: Based on the above discussion, one can obtain

  x ð0Þ ¼ HG ð1ÞW ½2n~ ;2m~  x~ð0Þuð0Þ; Exð1Þ ¼ Luð0ÞExð0Þ ¼ Luð0Þ~ x ð0Þ ¼ HG ð1Þuð0Þ~  ~ uð0Þ~     x ð0Þ ¼ HG ð2ÞW½2n~ ;2m~  x~ð0Þuð0Þ; Exð2Þ ¼ Luð1ÞExð1Þ ¼ LGuð0Þ Luð0Þ~ x ð0Þ ¼ LGðI 2m~  LÞUm .. .   s1 uð0ÞLuðs   s1 uð0ÞLG  s2 uð0Þ…LGuð0Þ   ExðsÞ ¼ Luðs  1ÞExðs  1Þ ¼ LG  2ÞExðs  2Þ ¼ … ¼ LG Luð0Þ~ x ð0Þ s1 s2 s3     x ð0Þ ¼ LG ðI2m~  ðLG ÞÞðI 22m~  ðLG ÞÞ…ðI 2ðs1Þm~  LÞðI 2ðs2Þm~  Um Þ…ðI 2m~  Um~ ÞUm~ uð0Þ~ s si s  ¼ 3 ðI ði1Þm~  ðLG ÞÞ3 ðI ðsiÞm~  Um~ Þuð0Þ~ x ð0Þ i¼1

i¼2

2

2

¢HG ðsÞW ½2n~ ;2m~  x~ð0Þuð0Þ

This completes the proof. And, we can obtain the reachable set Rð~ x 0 Þs as follows. Lemma 4: For system (6) with input control network (7), the set of states which are reachable with probability from an initial state x~0 at time s is    Rð~ x 0 Þs ¼ di2n~ jRow HG ðsÞW ½2n~ ;2m~  x~0 6¼ 0 :

IV. EXAMPLES

Example A Consider an asynchronous Boolean multiplex control network with k~ ¼ 2 layers, n~ ¼ 4 nodes and ~ ¼ 1 control shown in Fig. 2. m

i

Lemma 5: For system (6) with input control network (7), the probability from the initial state x~0 to the destination state x~d under initial control u0 at time s is

~  ~ ~ x~i ðsÞ ¼ x~d j3 ni¼1 x~i ð0Þ ¼ x~0 ; 3 m Pr 3 ni¼1 i¼1 ui ð0Þ ¼ u0   : ¼ HG ðsÞW½2n~ ;2m~  x~0 u0 Kð~ x d Þ;1

Since controls generated by input control network only take some special Boolean values at each time step, Algorithm 1 cannot be applied to find the overall reachable set Rð~ x 0 Þ at this case. Here, we just provide the result to find Rð~ x 0 Þs , and the algorithm for finding the overall reachable set Rð~ x 0 Þ remains for further study.

FIG. 2. An asynchronous Boolean multiplex control network (16).

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8 1 x ðt þ 1Þ ¼ :x13 ðtÞ Ù u1 ðtÞ > > < 1 Layer 1 : x12 ðt þ 1Þ ¼ x13 ðtÞÚ:u1 ðtÞ > > : 1 x3 ðt þ 1Þ ¼ x11 ðtÞ $ x12 ðtÞ 8 2 x ðt þ 1Þ ¼ x23 ðtÞ ! x24 ðtÞ > > < 2 Layer 2 : x23 ðt þ 1Þ ¼ :x24 ðtÞ > > : 2 x4 ðt þ 1Þ ¼ x24 ðtÞ Ú u1 ðtÞ;

(16)

where :, Ú, Ù, !, and $ represent the logical functions of negation, disjunction, conjunction, implication, and equivalence, respectively. ~ ~ ~ ~ ðtÞ ¼ 3 m x i ðtÞ and u Based on the discussion in Sec. III A, we define x~ðtÞ ¼ 3 ni¼1 i¼1 ui ðtÞ, and one can obtain Ú Md ¼ d2 ½1; 1; 1; 2, : Mn ¼ d2 ½1; 2, Ù Mc ¼ d2 ½1; 2; 2; 2, ! Mim ¼ d2 ½1; 2; 1; 1, and $ Me ¼ d2 ½1; 2; 2; 1. As to the canalizing function ~f i ; i 2 f1; 2; …; n~g, without loss of the generality, we choose disjunction function, i.e., ~ i ¼ Md ; i 2 f1; 2; …; n~g. The controls u1 ðtÞ in Eqs. (16) are free Boolean variables. In the following, under Harvey’s asynM chronous update, the controllability of Boolean multiplex control networks (16) is to be discussed. First, we calculate the ~ control-depending network transition matrix of system. It should be noted that, at time t, x~i ðtÞ ¼ xli ðtÞ; l 2 f1; 2; …;kg. Case 1: at time t, when node 1 is selected for update, x 2 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ ¼ Mc Mn ðI2  W ½8;2 ÞW½2 ðI2  Mr ÞW½2;8 Ed W½2 u1 ðtÞ~ x 1 ðtÞ~ x 2 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ x~ðt þ 1Þ ¼ Mc Mn x~13 ðtÞu1 ðtÞ~ ^ x ðtÞ: ¢ L 1 uðtÞ~ Case 2: at time t, when node 2 is selected for update, x 24 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ ¼ ðI 2  Md ÞðI 2  Md ÞðI 4  Mn ÞðI 8  Mi ÞW½2;4 x~ðt þ 1Þ ¼ x~1 ðtÞMd Md x~13 ðtÞMn u1 ðtÞMi x~23 ðtÞ~ ^ 2 uðtÞ~  ðI4  Mr ÞðI 8  W½2 ÞðI4  Mr ÞðI8  Mr ÞEd W½4;2 u1 ðtÞ~ x 1 ðtÞ~ x 2 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ¢ L x ðtÞ: Case 3: at time t, when node 3 is selected for update, x~ðt þ 1Þ ¼ x~1 ðtÞ~ x 2 ðtÞMd Me x~11 ðtÞ~ x 12 ðtÞMn x~24 ðtÞ~ x 4 ðtÞ ¼ ðI4  Md ÞðI4  Me ÞðI 2  W½2 ÞMr ðI 2  Mr Þ ^ 3 uðtÞ~ x 1 ðtÞ~ x 2 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ¢ L x ðtÞ:  ðI4  Mn Þ ðI 4  Mr ÞEd W½4;2 Ed u1 ðtÞ~ Case 4: at time t, when node 4 is selected for update, ^ 4 uðtÞ~ x~ðt þ 1Þ ¼ x~1 ðtÞ~ x 2 ðtÞ~ x 3 ðtÞMd x~24 ðtÞu1 ðtÞ ¼ ðI8  Md ÞW½2;16 u1 ðtÞ~ x 1 ðtÞ~ x 2 ðtÞ~ x 3 ðtÞ~ x 4 ðtÞ¢ L x ðtÞ: Therefore, all of the control-depending network transition matrices can be calculated as follows: 8 ^ 1 ¼ d16 ½9; 10; 3; 4; 13; 14; 7; 8; 9; 10; 3; 4; 13; 14; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 9; 10; 11; 12; 13; 14; 15; 16 > L > > > L > > > :^ L 4 ¼ d16 ½1; 1; 3; 3; 5; 5; 7; 7; 9; 9; 11; 11; 13; 13; 15; 15; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16: 2 3 3 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 2 3 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 60 0 0 0 3 2 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6 60 0 0 0 0 0 2 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 60 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 60 0 0 0 0 0 0 0 0 0 1 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 60 0 0 0 0 0 0 0 1 1 0 0 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 4 6 X 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ~¼ ^ i W½24 ;21  ¼ 6 L L 6 4 i¼1 46 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 1 0 0 0 0 0 1 1 0 0 0 0 0 6 60 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 4 0 0 1 1 0 0 1 1 0 0 0 6 60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 3 4 1 0 0 0 0 0 1 1 0 6 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 1 6 60 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 1 0 1 1 0 6 60 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 0 1 6 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 0 07 7 07 7 07 7 07 7 07 7 7 07 7 07 7: 07 7 07 7 07 7 17 7 07 7 17 7 05 2

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Randomly choose the initial state x~ð0Þ ¼ d616 ð1; 0; 1; 0Þ and time step s ¼ 3. Next, we find all of reachable state set of system (16) from the initial states ~x 1 ð0Þ ¼ 1, ~x 2 ð0Þ ¼ 0, ~x 3 ð0Þ ¼ 1, ~x 4 ð0Þ ¼ 0 at time step s ¼ 3. One can obtain 2

~ 3 3 d616 ~ s x0 ¼ L L

11 6 6 7 6 6 1 6 6 0 6 6 6 3 6 6 1 6 6 6 5 6 1 6 0 ¼ 6 64 6 6 6 6 6 12 6 6 0 6 6 0 6 6 6 11 6 6 7 6 6 0 4 0

8 10 1 0 2 2 4 0 4 14 0 0 8 10 1 0

8 10 1 0 3 2 4 0 4 14 0 0 8 10 0 0

4 14 1 0 1 4 3 0 2 16 0 0 4 14 1 0

9 10 0 0 4 2 2 0 4 14 0 0 9 10 0 0

5 14 0 0 2 4 2 0 2 16 0 0 5 14 0 0

5 14 0 0 4 4 0 0 2 16 0 0 5 14 0 0

3 0 7 19 7 7 0 7 7 0 7 7 7 0 7 7 8 7 7 7 0 7 7 0 7 7: 0 7 7 7 18 7 7 0 7 7 0 7 7 7 0 7 7 19 7 7 0 7 5 0

(17)

Note that the row vectors in 4th, 8th, 11th, 12th, and 16th rows of the matrix (17) are zero, that implies states 12 16 fd416 ; d816 ; d11 16 ; d16 ; d16 g cannot be reached from the given initial states at time step 3 no matter how to choose control sequences. By means of Lemma 2, one can get the reachable set from state (1, 0, 1, 0) with probability at time step 3 is 13 14 15 fd116 ; d216 ; d316 ; d516 ; d616 ; d716 ; d916 ; d10 16 ; d16 ; d16 ; d16 g shown in Fig. 3. Generally, under different control sequences, there will be more than one way to reach a destination state from a given initial state at a certain time step, and we are always interested in the choice of controls which can get the maximum probabilities. Using the matrix (17), it is convenient to calculate the expected control sequence. Assume xd ¼ d216 ð1; 1; 1; 0Þ, the maximum probability 19=64 at (2, 8) of matrix (17), that means control d88 can steer the initial state (1, 0, 1, 0) to destination state (1, 1, 1, 0) at time step 3 with probability 19=64. Correspondingly, we can calculate uð3Þ ¼ u1 ð0Þu1 ð1Þu1 ð2Þ¼d88 ð0; 0; 0Þ, i.e., u1 ð0Þ¼0, u1 ð1Þ¼0, u1 ð2Þ¼0. And, the overall reachable set Rðx0 Þ can be found by using Algorithm 1. Based on Eq. (13), the condensed version of control-depending network transition matrix L0 can be calculated as follows: 2

1

6 60 6 60 6 6 60 6 60 6 6 60 6 60 6 6 60 L0 ¼ 6 61 6 6 60 6 60 6 6 60 6 60 6 6 60 6 60 4 0

1 1

0

1 0

0

0 0

0

0 0

0

0 0

1 0 0 1

1 1

0 1 0 0

0 1

0 0 0 0

0 0

0 0 1 0

0 0

0 0 0 0

0 0

1

0 0

0

1 0

0

0 1

0

0 0

0 0 0 0

0 0

1 1 0 1

0 0

0 0 1 0

0 0

0 0 0 0

0 0

0 0 0 0

0 0 0 0

0 0

1 0 0 0

1 0

1 0 1 0

0 0

0 0 0 0

0 0

0 1 0 0

0 0 1 0

0 0

0 0 0 0

0 0

0 1 0 0

1 1

0 0 0 1

1 0

0 0 1 0

0 1

0

0 0

0

0 1

0

1 1

0

0 1

0 0 0 0

1 0

0 0 1 0

0 0

0 0 0 0

0 0

0 1 0 0

0 1

0 0 1 1

0 0 0 0

0 0

0 1 0 0

0 1

0 0 0 0

0 0

0 0 0 0

0 0

1 0 0 1

0 0

0

0 0

0

1 0

0

0 0

0

0 0

0

3

7 07 7 07 7 7 07 7 07 7 7 07 7 07 7 7 07 7: 17 7 7 07 7 07 7 7 17 7 07 7 7 17 7 17 5 1

When the initial state is x~ð0Þ ¼ d616 ð1; 0; 1; 0Þ, based on Algorithm 1, we can calculate

(18)

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outwardly under any controls. The above result also can be checked by the method of Lemma 2. Example B The onset of M (mitosis) and S (DNA replication) phases of the cell cycle are controlled by the periodic activation of cyclin-dependent kinases (cdks), the differential equations model of the above process and its corresponding Boolean model were proposed by Romond et al..44 and Heidel et al.,4 respectively. Based on the above studies, the Boolean model with input controls was extended into multiplex architecture as follows: 8 > A1 ðt þ 1Þ ¼ 1 þ D1 ðtÞ þ B1 ðtÞ þ D1 ðtÞ B1 ðtÞ þ u1 ðtÞ > < B1 ðtþ1Þ ¼ A1 ðtÞ þ u1 ðtÞ > > : 1 D ðtþ1Þ ¼ B1 ðtÞ; (19) 8 02 2 2 2 2 > A ðt þ 1Þ ¼ D ðtÞ þ B ðtÞ þ D ðtÞ

B ðtÞ þ u ðtÞ 2 > < B2 ðtþ1Þ ¼ A0 2 ðtÞ > > : 2 D ðtþ1Þ ¼ B2 ðtÞ u2 ðtÞ;

FIG. 3. The state transfer graph of system (16) from initial state (1010) in 3 steps.

C1 ¼ XðColðL0 Þ6 Þ ¼ f2; 5; 6; 14g;

 C2 ¼ [i2C1 XðColðL0 Þi Þ [ C1 ¼ f1; 2; 5; 6; 7; 10; 13; 14; 15g;

 C3 ¼ [i2C2 XðColðL0 Þi Þ [ C2 ¼ f1; 2; 3; 5; 6; 7; 9; 10; 13; 14; 15g;

 C4 ¼ [i2C3 XðColðL0 Þi Þ [ C3 ¼ f1; 2; 3; 5; 6; 7; 9; 10; 11; 13; 14; 15g:

where þ Mp ¼ d2 ½2; 1; 1; 2 and Mc ¼ d2 ½1; 2; 2; 2. The controls in system (19) are produced by input control network as follows:  u1 ðt þ 1Þ ¼ :u2 ðtÞ (20) u2 ðt þ 1Þ ¼ u1 ðtÞ: Based on linear representation, one can obtain uðt þ 1Þ ¼ u1 ðtÞu2 ðtÞ ¼ Mn u2 ðtÞu1 ðtÞ ¼ Mn W ½2 uðtÞ and G ¼ Mn W ½2 ¼ d4 ½3; 1; 4; 2. ~ is selected for update, Case 1: at time t, when node A ~ þ 1Þ ¼ Mp Mp Mp Mn D1 ðtÞB1 ðtÞMc D1 ðtÞB1 ðtÞu1 ðtÞ Xðt ~ DðtÞ ~ A~0 ðtÞ  BðtÞ ¼ Mp Mp Mp Mn ðI4  Mc ÞðI16  W½8;2 Þ  ðI4  W ½4;2 ÞW½8;2 Mr Mr ðI 2  Mr Mr ÞEd ~ BðtÞ ~ DðtÞ ~ A~0 ðtÞ W Ed W u1 ðtÞu2 ðtÞAðtÞ ½2;16

It can be verified that C4 ¼ C5 , hence, after 4 steps, all 16 of states except d416 ; d816 ; d12 16 ; d16 can be reachable with probability from the initial state x~ð0Þ ¼ d616 . Furthermore, we can get that, in state space of system (16), four states 16 d416 ; d816 ; d12 16 ; d16 constitute a local cluster shown in Fig. 4, which can be seen as a whole only with state transition

½2

~ ^ 1 uðtÞXðtÞ: ¢L ~ is selected for update, Case 2: at time t, when node B 1 02 ~ ~ ~0 ~ þ 1Þ ¼ AðtÞM Xðt d Mp A ðtÞu1 ðtÞA ðtÞDðtÞA ðtÞ

¼ ðI2  Md Mp ÞðI 2  W½8;2 ÞðI4  W ½2 ÞW½2  ðI 8  Mr Mr ÞEd W ½4;2 Ed W ½2 u1 ðtÞu2 ðtÞ ~ BðtÞ ~ DðtÞ ~ A~0 ðtÞ  AðtÞ ~ ^ 2 uðtÞXðtÞ: ¢L ~ is selected for update, Case 3: at time t, when node D 1 2 ~0 ~ BðtÞM ~ ~ Xðtþ1Þ ¼ AðtÞ d B ðtÞMc B ðtÞu2 ðtÞA ðtÞ

¼ ðI 4 Md ÞðI8 Mc ÞðI16 W ½2 ÞðI 2 Mr Mr Þ ~ BðtÞ ~ DðtÞ ~ A~0 ðtÞ Ed W ½4;2W½2;16 Ed u1 ðtÞu2 ðtÞAðtÞ ~ ^ 3 uðtÞXðtÞ: ¢L FIG. 4. A local cluster consisting of four states.

Case 4: at time t, when node A~0 is selected for update,

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2 2 ~ BðtÞ ~ þ 1Þ ¼ AðtÞ ~ DðtÞM ~ Xðt p Mp Mp D ðtÞB ðtÞ

 Mc D2 ðtÞB2 ðtÞu2 ðtÞ ¼ ðI 8  Mp Mp Mp ÞðI 32  Mc ÞðI4  W ½2;4 Þ  ðI 8  W½2;8 ÞðI2  Mr Mr ÞðI 4  Mr Mr ÞEd ~ BðtÞ ~ DðtÞ ~ A~0 ðtÞ W W Ed u1 ðtÞu2 ðtÞAðtÞ ½8;2

½2;16

~ ^ 4 uðtÞXðtÞ: ¢L ~ ~ ~ Assume initial state ðAð0Þ; Bð0Þ; Dð0Þ; A~0 ð0ÞÞ ¼ ð1; 1; 0; 1Þ, by means of algebraic representation, one can ~ obtain Xð0Þ ¼ d316 . When time step is s ¼ 4, according to Theorem 3, we can calculate that 8  3 ðI22  ðLG  2 ÞÞðI 24  ðLGÞÞðI   > HG ð4Þ ¼ LG > 26  LÞ > > > ðI24  U2 ÞðI22  U2 ÞU2 > < U2 ¼ ðI2  W½2 ÞMr ðI2  Mr Þ¼d16 ½1; 6; 11; 16 > 4 > 1X > > ^i  > L L¼ > : 4 i¼1 2

0:3359 6 6 0:0781 6 6 0:1172 6 6 6 0:0430 6 6 0:0078 6 6 6 0:0078 6 6 0 6 6 6 0:0625 G ~ H ð4ÞW ½24 ;22  X 0 ¼ 6 6 0:1641 6 6 6 0:0391 6 6 0:0859 6 6 6 0:0195 6 6 0 6 6 6 0 6 6 0 4 0:0391

0:2773 0:3477 0:0781 0:1445 0:0820 0:1367 0 0:0859 0:0039

0

0:0469 0 0 0:0078 0:0859 0:0078 0:1914 0:1133 0:0625 0:0547 0:1055 0:0352 0:0469 0:0469 0 0:0078

0 0

0 0:0039 0:0117 0:0156

3 0:2656 7 0:1758 7 7 0:0625 7 7 7 0:0898 7 7 0 7 7 7 0:0234 7 7 0 7 7 7 0:0273 7 7: 0:1563 7 7 7 0:0586 7 7 0:0781 7 7 7 0:0508 7 7 0 7 7 7 0:0039 7 7 0 7 5 0:0078 (21)

Based on Lemma 4, the reachable set from the initial ~ state Xð0Þ ¼ d316 at time s ¼ 4 includes all of states except 13 d16 . Moreover, when an initial control u0 is given, the specific reachable set can also be found. For example, assume ðu1 ð0Þ; u2 ð0ÞÞ ¼ ð1; 1Þ d14 , the corresponding reachable set is G  KðXðH ð4ÞW½24 ;22  X 0 u0 ÞÞ2n n o 11 12 16 : ¼ d18 ; d28 ; d38 ; d48 ; d58 ; d68 ; d88 ; d98 ; d10 8 ; d8 ; d8 ; d8

Except states d716 ð1; 0; 0; 1Þ, d13 16 ð0; 0; 1; 1Þ, ð0; 0; 1; 0Þ, and d15 ð0; 0; 0; 1Þ, the rest states have 16 the possibility to be reached at time step 4 from initial state ð1; 1; 0; 1Þ under initial control ð1; 1Þ. Furthermore, we can get that the destination state d116 ð1; 1; 1; 1Þ has the maximum probability 0.3359 to be reached under the above d14 16

conditions and the destination state d516 ð1; 0; 1; 1Þ and d616 ð1; 0; 1; 0Þ have the minimum probability 0.0078. In many cases, for instance in the therapeutic intervention, a destination state of system is already given and a control sequence which can drive the system from the initial state to a given destination with maximum probability is required. Based on the above discussion, we can get a solution. Assume a required destination state at time 4 is ~ d ¼ d9 ð0; 1; 1; 1Þ. By means of matrix (21), it is easy to X 16 get the maximum probability is 0.1914 at the Row 9 and Column 2. According to Theorem 3 and Lemma 5, we can calculate the initial control u0 ¼ d24 ð1; 0Þ, i.e., u1 ð0Þ ¼ 1 and u2 ð0Þ ¼ 0. V. CONCLUSIONS

In this article, based on the previous studies, the model of ABMCNs is first proposed. By using STP technique, the linear representation of ABMCNs is achieved. Considering asynchronous BNs are non-deterministic, we proved that only control-depending fixed points can be controlled with probability one, that showed the reasonableness of discussing the controllability of ABMCNs based on probability. Consequently, respectively, for two kinds of controls, the controllability of ABMCNs is studied by revealing the reachable sets at time s under a given control sequence or arbitrary controls, providing the algorithm to find the overall reachable sets of an initial state and calculating the probability of a specified destination state under certain control sequence. Furthermore, we also showed the procedure to calculate a precise control sequence steering the system between two states with maximum probability. ACKNOWLEDGMENTS

This research was supported by the National Natural Science Foundation of China (Nos.: 61370145, 61173183, 60973152, 60970004, and 61272094), the Superior University Doctor Subject Special Scientific Research Foundation of China (No.: 20070141014), Program for Liaoning Excellent Talents in University (No.: LR2012003), the National Natural Science Foundation of Liaoning province (No.: 20082165) and the Fundamental Research Funds for the Central Universities (No.: DUT12JB06). 1

S. A. Kauffman, “Metabolic stability and epigenesis in randomly constructed genetic nets,” J. Theor. Biol. 22(3), 437–467 (1969). 2 B. Drossel, “Random Boolean networks,” in Reviews of Nonlinear Dynamics and Complexity, edited by H. G. Schuster (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008), Vol. 1, p. 69. 3 M. Sun, X. Cheng, and J. E. S. Socolar, “Causal structure of oscillations in gene regulatory networks: Boolean analysis of ordinary differential equation attractors,” Chaos 23(2), 025104 (2013). 4 J. Heidel, J. Maloney, C. Farrow et al., “Finding cycles in synchronous Boolean networks with applications to biochemical systems,” Int. J. Bifurcation Chaos 13(03), 535–552 (2003). 5 N. Berntenis and M. Ebeling, “Detection of attractors of large Boolean networks via exhaustive enumeration of appropriate subspaces of the state space,” BMC bioinf. 14(1), 361 (2013). 6 D. S. Zheng, G. W. Yang, X. Y. Li, Z. C. Wang, F. Liu, and L. He, “An efficient algorithm for computing attractors of synchronous and asynchronous Boolean networks,” PLoS One 8(4), e60593 (2013).

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