Dynamics of Random Boolean Networks under Fully Asynchronous Stochastic Update Based on Linear Representation
Chao Luo, Xingyuan Wang* Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, China
Abstract 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 lengths 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.
Citation: Luo C, Wang X (2013) Dynamics of Random Boolean Networks under Fully Asynchronous Stochastic Update Based on Linear Representation. PLoS ONE 8(6): e66491. doi:10.1371/journal.pone.0066491 Editor: Peter Csermely, Semmelweis University, Hungary Received March 18, 2013; Accepted May 6, 2013; Published June 13, 2013 Copyright: ß 2013 Luo and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This research is supported by the National Natural Science Foundation of China (Nos: 61173183, 60973152, and 60573172), 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). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: [email protected]
Introduction Ref. [22], ‘‘The simplest class of Boolean network is synchronous, which means that all elements update their activities at the same The gene regulatory networks (GRNs) as complex systems moment’’. All of elements in RBNs evolve according to a global present diverse dynamical behaviors. In 1969, random Boolean synchronized clock, which is based on a tacit assumption that networks (RBNs) were first introduced by Stuart Kauffman to update scheme would not affect the essential properties of model gene regulatory networks [1]. Each gene is represented by a dynamics. However, some doubts have been cast on the validity node in RBNs with two possible states, i.e. the logical 0 and 1. of the above assumption when more and more evidences showed Generally, the ‘‘1’’ value represents the state ‘‘on’’ corresponding the dynamic behaviors of systems under asynchronous update to a gene that is being transcribed and the ‘‘0’’ value represents presented considerable divergence comparing with the counter- ‘‘off’’ corresponding to a gene that is not being transcribed [2]. A parts under synchronous update [29–30]. ARBNs was firstly directed edge from one node to another stands for the interaction proposed in 1997 by Harvey et al. [31] where only one node was between genes, the mutual regulation of which is described by an randomly chosen for update at each time step. Since then, many assigned Boolean function. The values of all nodes in a network results on ARBNs have been presented and some modified are updated in parallel and all of them together constitute the state versions of ARBNs were also proposed [32–38]. We think, if of the whole network. Since dynamics is deterministic and the state ‘‘synchronous update’’ is considered as ‘‘idealization’’ [31], space is finite, the states of a particular RBN eventually converge Harvey’s model as the extreme opposite of synchronous models into a series of periodically recurring states, which are called should be another form of ‘‘idealization’’. Motivated biologically, attractors including the fixed points and cycles, starting from an since the expression of genes is not an instantaneous process, but arbitrary initial condition. Attractors and their transient states the transcription of DNA and the transport of enzymes may take compose the basins of attraction, which present the dynamical from milliseconds up to a few seconds [36]. So when modeling the behaviors of systems. RBNs provide a way to formalize large scale complexity in a realistic manner by describing the gene expression above dynamics by means of a discrete form, at each time step, the with ON/OFF and reflect the statistical features of real living cells condition to strictly limit only one node for update is too strong to by adjusting the parameters of networks [3]. Recent years, along describe the real situation. with the research of networks in molecular biology, chemistry, Although RBNs have been much analyzed, but as logical neurobiology, and economy etc [4–16], RBNs have been systems, they are difficult to be investigated analytically. Recently, extensively investigated and a wealth of results have been achieved a new matrix theory called the semi-tensor product (STP) was [17–28]. proposed by Cheng et al. [39], whereby logical equations can be As abstract models of physical and biological phenomena, a represented as a linear discrete-time dynamic equation certain degree of simplification is necessary. Usually, synchrony idealization is adopted as update scheme in the classical RBNs, as x(tz1)~L|D x(t) ð1Þ
PLOS ONE | www.plosone.org 1 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
~|D n where x(t) i~1xi is the semi-tensor product of logical variables. Methods In contrast with usual transition matrix expression, Eq. (1) contains complete information of the logical equations of RBNs, based on In this section, the semi-tensor product of matrices is briefly which dynamics of systems, such as attractors and basins, can be introduced. Some concepts and properties related to this article analyzed. Furthermore, this form can easily be extended to the are presented. Boolean control networks. Certain control problems, such as controllability [40,41], observability [42], and realization [43], Definition 1 ([39]) etc., can be investigated. So far, the previous studies mainly focus 1) Let X be a row vector of dimension np, and Y be a column on models under synchronous update. However, ARBNs as vector of dimension p. Then we split X into p equal-size nondeterministic systems are considerable different from RBNs. blocks as X 1, ...,X p, which are 1|n rows. Define the STP, For instance, in Ref. [39], Cheng et al. proposed a method by STP denoted by , as technique to find attractors of RBNs, but, which is based on a fact 8 that RBNs are deterministic systems and attractors are circular. Pp > i n Obviously, this prerequisite can’t be satisfied for ARBNs. So, we <> X|D Y~ X yi[R , think it’s interesting and meaningful to extend the related research i~1 > Pp into the field of asynchronous random Boolean networks. > T T i T n : Y |D X ~ yi(X ) [R : Based on the above discussion, firstly, from the aspect of model i~1 selection, most of the previous results on ARBNs are based on the Harvey’s model. As mentioned above, we think Boolean networks with a more general asynchronous update schedule are worth 2) Let A[Mm|n and B[Mp|q. If either n is a factor of p, say investigating. Secondly, from the aspect of methodology selection, nt~p and denote it as A[tB,orp is a factor of n, say n~pt despite of the overload of matrix operations, STP technique and denote it as A]tB, then we define the STP of A and B, provides a new perspective to make the analytical analysis of denoted by C~A|D B, as the following: C consists of m|q logical dynamics. From the viewpoint of theoretical part, it’s a blocks as C~(Cij) and each block is valuable technique to be considered. Therefore, in this article, we focus on dynamics of ARBNs where a random number of nodes ij i ~ |D ~ ~ can be updated at each time step. Mainly, two parts of work are C A Bj, i 1, ...,m, j 1, ...,q involved in this article. First of all, we complete an algebraic where Ai is the i-th row of A and B is the j-th column of B. representation of the studied logical models. At the best of our j knowledge, the related result is firstly presented. Network It’s easy to verify that for two column vectors X[Rm and Y[Rn, transition matrix takes the essential role for the linear represen- mn X|D Y[R . tation of logical dynamics based on STP technique. Actually, in Note that STP of matrix can be seen as a generalization of the the previous works, Cheng et al. [39] have achieved results of conventional matrix product, so the properties of matrix product, RBNs. Here, we engage in a particular class of ARBNs, from the such as distributive rule, associative rule, etc, still hold. perspective of update schedule, this special update scheme can Some notations in this article are defined as follows involve the case of synchronous update, which allow all of nodes are updated at a certain time step. As a result, a general formula of r | 1) dn denotes the r-th column of the n n identity matrix In and network transition matrices is achieved, based on which one can ~ r Dn : dnj1ƒrƒn , which is the set of all n columns of In. calculate all of network transition matrices of a particular ARBN r ~ An operation is defined as H (dn) fgr , furthermore, when including the counterparts under synchronous update and C~ di1 ,di2 , ...,dip , H(C)~ i ,i , ...,i . Harvey’s model. Secondly, a necessary and sufficient algebraic n n n 1 2 p criterion is discussed, which determine whether a given group of s 2) A matrix L[Mn|m can be called a logical matrix if ~ i1 i2 im states could compose a loose attractor of length s in a specific L dn ,dn , ...,dn , which is briefly denoted by ARBN. Consequently, a novel approach to detect the attractors L~dn½ i1,i2, ...,im . And the set of n|m logical matrices is and basins of ARBNs is designed, which is completely based on set denoted by Ln|m. operations and different from the previous studies. 3) Let matrix A[Mn|m, all of elements in i-th column of matrix As we know, to search attractors in RBNs is a NP-hard problem. A compose a set denotedS by C(A)i,andwhenaset ~ Recent years, many powerful algorithms [3,17,20,35,45] have S[r(fg 1,2, ...,m ), C(A)S C(A)i, where rðÞ. represents been presented to improve the search efficiency and reduce the i[S time complexity. Yet, it should be noted that our main interest in the power set. this article is to provide a complete quantitative method and a Next, we define the swap matrix W , let X[Rm and Y[Rn be novel perspective to analyze the dynamics of ARBNs from the ½m,n view point of theoretical part. And results would be taken as two column vectors preparations for the further research, e.g. Boolean control network under asynchronous update. W½m,n XY~YX, This article is organized as follows. In the section of Methods, some concepts and properties of STP are introduced. In Results where W½m,n is a mn|mn matrix labeled columns by and discussion, the linear representation of ARBNs is discussed, (11,12, ...,1n, ...,m1,m2, ...,mn) and rows by based on which the dynamic properties of ARBNs are studied and (11,21, ...,m1, ...,1n,2n, ...,mn), the elements in position algorithms to find attractors and basins of ARBNs are presented. ((I,J),(i,j)) is Some examples are shown to illustrate the main results. Finally, a concluding remark is given.
PLOS ONE | www.plosone.org 2 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Mr~d4½1,4 , which is power-reducing matrix and it can be 2~ 1, I~i and J~j verified that P PMr, VP[D2. w(I,J),(i,j)~ : Then, the following result can be achieved. 0, otherwise Theorem 1. Equation (4) can be represented as
W½m,m is briefly denoted by W½m . z ~ = In order to get the matrix expression of logic, the Boolean values x(t 1) LHt x(t), Ht[rðÞfg1,2, ...N ,Ht W ð5Þ ~ 1 ~ 2 should be denoted as vector forms T 1*d2 and F 0*d2. For a mt 1 logical function L(P1, ...,Pr) with arguments P1, ...,Pr, it has ~ 6 where LHt P (I iu{uz(u{1)N Miu ) P (I (iv{1)z(v{1)(N{1) u~1 2 v~m 2 been proved in Ref. [39] that L(P1, ...,Pr) can be represented as t mt{1 mt L(P1, ...,Pr)~MLP1P2 ...Pr, where matrix ML[M2|2r is 6 6 mt W N{iv{(m {v) N ) P (I2w Ed ) P W (m {y)z(iy{1) WN . unique, which is called the structure matrix of logical function L. ½2 t ,2 w~0 y~1 ½2 t ,2 More details on STP can be found in Ref. [39]. In this article, the matrix products are assumed to be STP and the symbol is Proof omitted. x(tz1)~x1(t), ...,xi1{1(t)Mi1 x(t)xi1z1(t), ..., ximt {1(t)Mimt x(t)ximt z1(t), ...,xN (t) ~ 6 Results and Discussion (I2i1{1 Mi1 )x1(t), ...,xi2{1(t)Mi2 x(t)xi2z1(t), ..., ximt {1(t)Mimt x(t)ximt z1(t), ...,xN (t) Model mt
In this section, the algebraic representation of asynchronous ~ P (I iu{uz(u{1)N 6Miu )x1(t), ..., ~ 2 random Boolean networks is discussed. Without special note, u 1 xi1{1(t)x(t)xi1z1(t), ..., ‘‘asynchrony’’ in this article implies mt (mt[fg1, ...,N ) nodes are ximt {1(t)x(t)ximt z1(t), ...,xN (t) randomly selected to be updated at time step t. Here, we don’t mt ~ consider the case mt 0 i.e. none of nodes are updated. ~ u 6 m 6 P (I2i {uz(u{1)N Miu )(I2(i t {1)z(mt{1)(N{1) ARBNs with N nodes can be described as follows: u~1 W½2N{imt ,2N )x1(t), ...,xi1{1(t)x(t)xi1z1(t), ...,ximt {1 z ~ (t)ximt z1(t), ...,xN (t)x(t) xi(t 1) fi(xi (t), ...,xip (t)), i[Ht 1 ð2Þ mt 1 z ~ { u 6 v xj(t 1) xj(t), j[fg1,2, ...N Ht ~ P (I2i {uz(u{1)N Miu ) P (I2(i {1)z(v{1)(N{1) u~1 v~mt 6W N{iv{(m {v) N x t ... Where xi(t) represents the state of node i at time t, which receives ½2 t ,2 ) 1( ), , mt x 1{ (t)x 1z (t), ...,ximt {1(t)ximt z1(t), ...,xN (t)x (t) inputs from p distinct neighbors and xij’ (t) is the j’ th input. The i 1 i 1 m 1 nodes in ARBNs take values from the set fg0,1 corresponding to t u 6 v 6 ~ P (I2i {uz(u{1)N Miu ) P (I2(i {1)z(v{1)(N{1) two levels of gene expressions. fi is a Boolean logic from u~1 v~mt p v 6 fg0,1 ?fg0,1 , which is assigned to node i. At time t, there are W½2N{i {(mt{v),2N )Ed (I2 Ed ), ..., 6 mt (mt[fg1, ...,N ) nodes which are selected at random for (I2mt{1 Ed )xi1 (t)xi2 (t), ...,ximt (t)x1(t), ..., mt update, and all of updated elements are included in set xi1{1(t)xi1z1(t), ...,ximt {1(t)ximt z1(t), ...,xN (t)x (t) 1 mt m Ht~ i , ...,i , without the loss of generality, we assume that t 1 a a ~ u 6 v 6 i 1 vi 2 when a1va2. The updated value of selected node i is P (I2i {uz(u{1)N Miu ) P (I2(i {1)z(v{1)(N{1) u~1 v~mt x (t), j’[ 1,2, ...p { determined by all of the input values of ij’ fgand mt 1 v 6 the logic fi. At the same time, the remaining nodes keep the values W½2N{i {(mt{v),2N ) P (I2w Ed ) w~0 at time step t. m z1 mt { t W½2(mt{1)z(i1{1),2 W½2(mt{2)z(i2{1),2 , ...,W½2i 1,2 x (t) Firstly, with the structure matrix Mi of each Boolean logic fi, Eq. (2) can be converted into mt 1 u 6 v 6 ~ P (I2i {uz(u{1)N Miu ) P (I2(i {1)z(v{1)(N{1) u~1 v~mt mt{1 xi(tz1)~Mix(t), i[Ht v 6 ð3Þ W½2N{i {(mt{v),2N ) P (I2w Ed ) xj(tz1)~xj(t), j[fg1,2, ...N {Ht w~0 mt m y t When the in-degree of node i is less than N, a dummy matrix P W½2(mt{y)z(i {1),2 WN x(t) y~1 Ed ~d2½ 1,2,1,2 should be applied. It’s easy to verify that for any Boolean variable u, one can get Ed u~I2. Assume there was no input from node q to i, xq(t) can be substituted by Ed xq(t). This completes the proof. N Multiplying all the equations in (3) as |D N One can prove that the mapping i~1 : D2 ?D2n is a bijective mapping. So Eq. (5) can adequately describe the dynamics of Eq. z ~ (3) at time t, say, LH involves all of transition information of x(t 1) x1(t), ...,xi1{1(t)Mi1 x(t)xi1z1(t), ..., t ð4Þ system from time t to tz1 when set Ht is chosen for update, which m m m xi t {1(t)Mi t x(t)xi t z1(t), ...,xN (t), is called network transition matrix under update elements Ht. Whether RBNs or ARBNs, Network transition matrix is an ~|D N According to Ref [39], when XN (t) i~1xi(t) and xi(t)[D2, one essential concept for linear representation of logical dynamics N 2 ~ ~ 6 6 based on STP technique. As deterministic system, there is only one can obtain XN (t) WN XN (t), where WN P I i{1 ½(I2 i~1 2 network transition matrix for a particular RBN. However, { 6 W½2,2N i )Mr and refers to the Kronecker product. Here, multiple matrices exist in ARBNs depending on different asynchronous update scheme. For models discussed in this article,
PLOS ONE | www.plosone.org 3 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks there are total 2N {1 network transition matrices. By means of And all of network transition matrices compose a set Theorem 1, when getting the structure matrix Mi of each Boolean T~fgL1,L2,L3,L1,2,L1,3,L2,3,L1,2,3 . logic fi for any specific ARBN, all of network transition matrices It should be noted that matrix operations are involved in could be calculated. quantity during the above calculations. Fortunately, the related Example 1. A logical dynamics is described as matrices are typically sparse and many efficient algorithms can be 8 applied to optimize the calculation procedure. <> x1(tz1)~x2(t) _ x3(t) The asynchronous network transition table x (tz1)~x (t) ^ x (t) ð6Þ :> 2 1 3 In order to present the algorithm to find the attractors and z ~ x3(t 1) x2(t) basins of ARBNs, we first provide the following definition. Definition 2. For a random Boolean network with N nodes where xi(t) represents the state of node i at time t, and the symbols ~ under asynchronous stochastic update, T fgA1,A2, ... ,A2N {1 is ‘‘^’’, ‘‘_’’ represent conjunction and disjunction, respectively. By ~ i i the set of network transition matrices, where Ai d2N a1,a2, ..., truth tables, one can get _*Md ~d ½ 1,1,1,2 and 2 i N a N , i[ 1, ...,2 {1 . The asynchronous network transition ^*Mc~d2½ 1,2,2,2 . 2 The algebraic form of Eq. (6) as table, briefly called transition table, is defined as a1 a1 a1 8 1 2 2N > z ~ < x1(t 1) Md x2(t)x3(t) a2 a2 a2 1 2 2N x (tz1)~M x (t)x (t) ð7Þ I~ : ð8Þ :> 2 c 1 3 . . . . . P . x3(tz1)~x2(t) . . . N { N { N { a2 1 a2 1 a2 1 1 2 2N Case 1: at time t, when only node 1 is selected for update, Example 2. Recall Example 1. The transition table of system (7) is as follows z ~ z 12381238 x(t 1) x1(t 1)x2(t)x3(t) ~ 14147878 Md x2(t)x3(t)x2(t)x3(t) 14145858 ~ Md W½2,4 x2(t)x2(t)x3(t)x3(t) ~ I 14183438 ð9Þ ~ 6 Md W½2,4 Mr(I2 Mr)x2(t)x3(t) 11481148 6 13247788 ~Md W½2,4 Mr(I2 Mr)Ed x1(t)x2(t)x3(t) 13283348
Case 2: at time t, when all of three nodes are selected for It should be noted that the asynchronous network transition table update, is specially designed for ARBNs. As mentioned above, more than one network transition matrix exists in ARBNs, which is different from RBNs. Therefore, there is a question: how to organize these x(tz1)~x1(tz1)x2(tz1)x3(tz1) network transition matrices can provide an efficient way for ~Md x2(t)x3(t)Mcx1(t)x3(t)x2(t) applications. For the above consideration, the asynchronous network transition table is presented, by means of which ~Md (I46Mc)x2(t)x3(t)x1(t)x3(t)x2(t) algorithms to find attractors and basins of ARBNs can be achieved ~Md (I46Mc)W½2,4 x1(t)x2(t)Mrx3(t)x2(t) completely based on set operations, which are shown later in Algorithms 1 and 2. ~Md (I46Mc)W½2,4 (I46Mr)x1(t)x2(t)x3(t)x2(t)
~Md (I46Mc)W½2,4 (I46Mr)(I46W½2 )(I26Mr) Fixed Points (point attractors) Proposition 1. For a given ARBN with N nodes, when x1(t)x2(t)x3(t) ~ ~ i C(I)i fgi , the state x d2N is a fixed point, where I is the asynchronous network transition table. Here, we just show the calculations of network transition matrices Proof. Assume T is the set of network transition matrices. under the above two cases. Similarly, all of the network transition ~ ~ When C(I)i fgi , we can obtain H(Col(L)i) fgi ,VL[T, i.e. the matrices can be calculated and results are as follows. i i-th column of L is equal to d2N , which holds for any network i i 8 transition matrices in T. And, it’s easily verified that Ld N ~d N , ~ 6 ~ 2 2 > L1 Md W½2,4 Mr(I2 Mr)Ed d8½ 1,2,3,8,1,2,3,8 > VL[T, which completes the proof. > 6 6 > L2~(I2 Mc)Mr(I2 Mr)Ed W½2 ~d8½ 1,4,1,4,7,8,7,8 Note that Col(.) represents the i-th column of a matrix. > i > ~ 6 ~ Example 3. Recall Example 1. We get C(I) ~fg1 and > L3 (I2 Mr)Ed W½4,2 d8½ 1,4,1,4,5,8,5,8 1 > ~ 1 8 < 2 C(I)8 fg8 . So, two fixed points are d8*111 and d8*000, which L1,2~Md (I46Mc)W (I46M )~d8½ 1,4,1,8,3,4,3,8 ½2,4 r can be checked in the state space graph of Fig. 1. > 2 > L1,3~Md W½4 M Ed ~d8½ 1,1,4,8,1,1,4,8 > r > 6 6 > L2,3~(I2 Mc)Mr(I2 W½2 )~d8½ 1,3,2,4,7,7,8,8 Attractors (Loose attractors) > > ~ 6 6 6 6 > L1,2,3 Md (I4 Mc)W½2,4 (I4 Mr)(I4 W½2 )(I2 Mr) Fixed points can be seen as point attractors. In the following, we : focus on loose attractors in ARBNs. ~d8½ 1,3,2,8,3,3,4,8
PLOS ONE | www.plosone.org 4 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Figure 1. An asynchronous Boolean network with 3 nodes and its state space graph. doi:10.1371/journal.pone.0066491.g001 S {1 {1 Theorem 2. For a given ARBN with N nodes and I is the set S(D2N , L (S)~ L (x). For simplicity, asynchronous network transition table, a state set of x[S {1 {1 {1 {t ~ i1 i2 is § L|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}L ...L %hbrace(x) is denoted by L (x). C d2N ,d2N , ...,d2N (s 2) is an attractor of length s, iff t ~ For nondeterministic systems, a partition can’t be found to H(C) C(I)H(C), ð10Þ and separate the whole state space of ARBNs into several disjointed sets. As mentioned in Ref. [31], basins of attractors in = asynchronous random Boolean networks involve two kinds: the C(I)H(C{C’)\H(C’) W, VC’5C, ð11Þ definite basins and possible basins. Then, the overlapping parts of basins of different attractors as common phenomena exist the state Proof. (Necessity) Assume T is the set of network transition i1 i2 is space of ARBNs, which mean certain transient states can jump matrices. When C~ d N ,d N , ...,d N is an attractor of length s, 2 2 2 from one basin of attractors into another. More studies related to Ldij [C, VL[T, Vj[ 1,2, ...,s one can obtain 2N fg, which implies basins in ARBNs can refer to Refs. [32–34]. C(I)H(C)(H(C). Since all of states in C can be reached, that is, for Generally, by a backstepping manner, one can find the basin of each element p in set H(C), one can find q[H(C) and L’[T to a given attractor C. Starting from the state set of attractor C,we q ~ p satisfy L’d2N d2N , i.e. p[C(I)H(C), which implies H(C)(C(I)H(C). can determine parent states of each state in C, which compose a ~ So, one can obtain H(C) C(I)H(C) and Eq. (10) holds. new set C’. Repeat to find the parent states of each state in C’ and Assume there exists a set C’5C, which satisfy C(I)H(C{C’)\ add them into C’ until there aren’t any new states involved. The H(C’)~W. For Vx[C’ and Vx’[C{C’, we can’t find a path from set C’|C is the basin of attractor C. The above procedure is x’ to x, which is contradiction to the definition of attractor. So Eq. shown in Algorithm 2. (11) holds. Example 4. Recall Examples 2. The attractors of system (7) 1 (Sufficiency) Firstly, we should prove that each state in set C can include the fixed points 000 and 111. For the fixed point d8,by find a way to reach all of the other states. Assume there exists a state checking the transition table I in (9), we can get L{1(d1)~ p p 8 d [C d 1 2 3 5 6 {1 1 {2 1 1 2 3 5 6 7 2N and all of states that 2N can’t reach compose a non-empty set d ,d ,d ,d ,d , L (d )|L (d )~ d ,d ,d ,d ,d ,d . Since p { 8 8 8 8 8 8 8 8 8 8 8 8 8 C’.Sinced N can reach each state in set C C’, so any states in set 2 3 2 {i 1 ~ {i 1 1 C{C’ also can’t reach states in C’,i.e.C(I) { \H(C’)~W, | L (d8) | L (d8), the basin of the fixed point d8 is H(C C’) i~1 i~1 which is a contradiction to Eq. (11). So, each state in C can composed of six states d1*111,d2*110,d3*101,d5*011, reach the other states. 8 8 8 8 d6*010,d7*001. Similarly, for the fixed point d8, one can get C 8 8 8 Secondly, we should prove each state in set can’t reach any 2 3 4 5 6 7 8 states out of set C. the basin d8,d8,d8,d8,d8,d8,d8 . The above results can be verified q q ~ in Fig. 1. We assume there exist d2N [C and L’[T which satisfy L’d2N a In the above example, except state d4 is the definite basin of d2N 6[C. Then a[C(I)H(C) and a6[H(C), which is a contradiction to 8 8 Eq. (10). fixed point d8, all the rest transient states are the overlapping part The proof is complete. between the basins of two fixed points. As a consequence, we develop a procedure to find all of attrac- tors of ARBNs based on the above result, that is shown later in Algorithms for detecting the attractors and basins of Algorithm 1. ARBNs Given a specific logical dynamics under asynchronous update, Basins of attractors all of the network transition matrices could be calculated based on Definition 3. For ARBNs with N nodes, T and I are the set of Theorem 1. Consequently, the asynchronous network transition network transition matrices and transition table, respectively. Let table can also be achieved. By means of the transition table, we x,x’[D2N and L[T, when Lx’~x, x’ is called the parent state of x. formulate the following algorithms to find attractors and basins of All of parent states of x are denoted by the set L{1(x). And, for a ARBNs.
PLOS ONE | www.plosone.org 5 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Algorithm 1: An algorithm for finding attractors in ARBNs with as follows, which is a model of phosphorylation/dephosphoryla- n nodes and the asynchronous network transition table I. tion cycles considered by Gonze et. al [44] and re-investigated in [45]. Begin: Algorithm 1 Example 5. A logical dynamics is described as ~ 1 2n ~ F d2n , ...,d2n ,s 1 /* F is a feasible state set and s is the 8 length of attractor. */ > x (tz1)~x (t) > 1 2 for (i~1; iv~2n; izz) do /*find the fixed points in ARBNs */ < x2(tz1)~x3(t) i C(I) ~ i d n ð12Þ if i fgthen /* 2 is a fixed point */ > x (tz1)~x (t) i :> 3 4 Find the basin of state d n ; /* the procedure to find basin of 2 x (tz1)~x (t) a particular attractor refers to Algorithm 2*/ 4 1 i Remove the state d2n and its basin from set F; where xi(t) represents the state of node i at time t. System (12) end if presents a model of protein–protein interactions. As shown in [45], end for there are three cycles of length four, one cycle of length two, and while Nonempty (F) do two fixed points under synchronous update. Here, we study the attractors and basins of system under asynchronous stochastic s~sz1 update. The algebraic form of Eq. (12) as ~ i v~ i F’ d2n card(C(I)i) s and d2n [F /* card(.) is the num- 8 ber of elements in set. */ > x1(tz1)~Ed (I26Ed )(I46Ed )(I26W )x(t) > ½2,4 Update F’ by removing states whose successors aren’t in < x2(tz1)~Ed (I26Ed )(I46Ed )(I46W½2 )x(t) F’. ð13Þ > x (tz1)~E (I 6E )(I 6E )x(t) while Nonempty (F’) do :> 3 d 2 d 4 d i x4(tz1)~Ed (I26Ed )(I46Ed )W½2,8 x(t) Randomly pick up one state d2n from F’ and all reachable i states from d2n compose a set C. 4 where x(t)~|D xi(t). if card(C)~s and H(C)~C(I) then /* C is an i~1 H(C) Then, the structure matrices of logical functions are as follows attractor of length s */ Find the basin of attractor C and remove the 8 ~ 6 6 6 ~ corresponding states from sets F and F’. > M1 Ed (I2 Ed )(I4 Ed )(I2 W½2,4 ) > i > d else remove state d n from F’ > 2½1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2 2 > end if > M ~E (I 6E )(I 6E )(I 6W )~ > 2 d 2 d 4 d 4 ½2 end while < d2½1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2 end while > M ~E (I 6E )(I 6E )~ End: Algorithm 1 > 3 d 2 d 4 d > Since, according to the length, the attractors are found > d2½1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2 > successively from small to large, so it isn’t necessary to check the > ~ 6 6 ~ :> M4 Ed (I2 Ed )(I4 Ed )W½2,8 ) condition of Eq. (11) to determine attractors. d ½1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2 Algorithm 2: An algorithm for finding basin of a particular 2 attractor C in ARBNs with n nodes.
Begin: Algorithm 2 By Theorem 1, all of network transition matrices can be C’~C /* C is the set of a particular attractor. */ calculated and the asynchronous network transition table is while C’=L{1(C’) do constructed as follows. 1 2 3 4 13 14 15 16 1 2 3 4 13 14 15 16 {1 1 2 7 8 1 2 7 8 9 10 15 16 9 10 15 16 C’~C’|L (C’) 1 4 1 4 5 8 5 8 9 12 9 12 13 16 13 16 1 1 3 3 5 5 7 7 10 10 12 12 14 14 16 16 end while 1 2 7 8 9 10 15 16 1 2 7 8 9 10 15 16 End: Algorithm 2 1 4 1 4 13 16 13 16 1 4 1 4 13 16 13 16 In Ref. [39], Cheng’s approach to find the attractors of length s 1 1 3 3 13 13 15 15 2 2 4 4 14 14 16 16 s in RBNs needs the computation of L , where L is the network ~ I 1 4 5 8 1 4 5 8 9 12 13 16 9 12 13 16 transition matrix. Moreover, Cheng’s algorithm is based on a 1 1 7 7 1 1 7 7 10 10 16 16 10 10 16 16 prerequisite that RBNs are deterministic systems and a state on a 1 3 1 3 5 7 5 7 10 12 10 12 14 16 14 16 cycle of length s will be back afters steps. However, this 1 4 5 8 9 12 13 16 1 4 5 8 9 12 13 16 prerequisite can’t hold for ARBNs. Therefore, we propose a novel 1 1 7 7 9 9 15 15 2 2 8 8 10 10 16 16 way to detect attractors and basins of ARBNs by means of the 1 3 1 3 13 15 13 15 2 4 2 4 14 16 14 16 transition table. 1 3 5 7 1 3 5 7 10 12 14 16 10 12 14 16 Examples 1357 9 1113152 4 6 8 10121416 In this section, some examples are presented to illustrate the main results of this article. Firstly, an idealized example is shown
PLOS ONE | www.plosone.org 6 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
1 ~ ~ d16 come from this former cycle. The same case occurs between By means of Algorithm 1, since C(I)1 fg1 and C(I)16 fg16 in the former cycle d8 ,d15,d14,d12 andthefixedpointd16.As the transition table I, it’s easy to get two fixed points fd1 *1111g 16 16 16 16 16 16 to the third former cycle d4 ,d7 ,d13,d10 , it plays a role of and fd16*0000g. Before finding the attractors of system, we first 16 16 16 16 16 bridge between the rest former cycles of length four, but it determine the basins of fixed points by Algorithm 2 to reduce the hasn’t any direct links pointing to either of fixed points. The ~ 1 search scope in state space. For the fixed point C fd16g, one can related state transfers are depicted in Fig. 2. Former cycle { { get L 1(C)~ d1 ,d2 ,d3 ,d5 ,d6 ,d9 ,d11 and L 1(C)| 6 11 16 16 16 16 16 16 16 d16,d16 is more special, each state of which has links pointing {2 ~ i ~ L (C) d16ji 1, ...,15 . So far, we can know the basin of to all of states in state space except for itself, which is shown in 1 the fixed point d16 includes all of the states in state space except for Fig. 3. 16 Under synchronous update, all of attractors present similar the other fixed point d16. Therefore, we can conclude that there aren’t any attractors in this system. Similarly, the basin of the fixed dynamics, e.g. stability, and it’s difficult to determine whether 16 1 there is difference among them. However, some divergences point d involves all of the states except for d . A big overlapping 16 16 among these attractors can be observed under asynchronous part exists between the basins of two fixed points. update. Such as the above example in Fig. 2, two former cycles of One can verify, under synchronous update, System (12) has two length four are close to the fixed points, which have greater d1 d16 fixed points as f 16g and f 16g, three cycles of length four as possibility to flow into the corresponding fixed points. So, we can 2 3 5 9 8 15 14 12 4 7 13 10 d16,d16,d16,d16 , d16,d16 ,d16,d16 and d16,d16,d16,d16 , one reasonably consider that the states of the two cycles would be faster 6 11 cycle of length two as d16,d16 . Without ambiguity, we call into stable states. Yet, the remaining period-4 cycle is in the these cycles under synchronous update as ‘‘former cycles’’. When intermediate position, statistically, whose location determines it system (12) is under asynchronous update, all of former cycles turn would take more time to converge into stability. As to the former into the overlapping part of the basins of two fixed points, say, they period-2 cycle in Fig. 3, to some extent, its characteristic is more can jump from one basin of attractor into another. Furthermore, unstable under asynchronous update. ~ ~ we also can observe some interesting phenomena. Each state in Example 6. Consider an ARBN with N 5,K 2, where a 2 3 5 9 random number of nodes can be updated at each time step former cycle d16,d16,d16,d16 has a direct link pointing to the 1 described as fixed point d16, at the same time, all of in-edges of the fixed point
Figure 2. State transfers (Including two fixed points and the former cycles of length four). doi:10.1371/journal.pone.0066491.g002
PLOS ONE | www.plosone.org 7 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Figure 3. State transfers (The basin of the former cycle of length two). doi:10.1371/journal.pone.0066491.g003
8 > x1(tz1)~x1(t) ^ x3(t) 8 > > z ~ > M1~McEd W½4,2 Ed W½4,2 Ed W½2 ~d2½1,1,1,1,2,2,2,2,1,1,1,1, < x2(t 1) x2(t)?x4(t) > > 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 x3(tz1)~:(x2(t) ^ x5(t)) ð14Þ > > > > z ~ > M2~MiEd W Ed W Ed ~d2½1,1,2,2,1,1,2,2,1,1,1,1,1,1,1, > x4(t 1) x3(t)?x4(t) > ½4,2 ½2 : > x (tz1)~x (t)
x3(tz1)~MnMcEd W 2 Ed W 2 Ed x(t) ð15Þ > ½ ½ > z ~ > x4(t 1) MiEd W½4,2 Ed Ed x(t) : By Theorem 1, all of network transition networks can be x5(tz1)~MiEd W½2 Ed W½2 Ed W½2 x(t) calculated and the asynchronous network transition table is as ~5 where x(t) i~1xi(t). follows. Then, the structure matrices of logical functions are as follows.
PLOS ONE | www.plosone.org 8 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
as definite basin and four states fd6 ,d8 ,d14,d16g as possible basin. Construct the feasible set F~fdi ji~1,2, ...,32g. According to 32 32 32 32 32 For attractor fd4 ,d12g, there are total two states fd8 ,d16g in its ~ 2 32 32 32 32 Algorithm 1, since C(I)2 fg2 , the fixed point d32 can firstly be 17 18 21 22 basin. Similarly, as to attractor fd32,d32,d32,d32g, one can find a found. Consequently, the basin of this fixed point can be detected 1 5 9 6 8 10 14 16 definite basin fd32,d32,d32g and the rest of transient states in its as d32,d32,d32,d32,d32 by Algorithm 2. Remove the fixed point basin form the possible basin. For simplicity, here, we just show and its basin from feasible set F. Next, to find the attractors of part of state transfers of states in attractors’ possible basins. ~ ~ 1 4 5 9 11 12 length s 2, one could obtain F’ fd32,d32,d32,d32,d32,d32, Example 7. A Boolean model of cAMP signaling in 18 21 11 3 18 17 9 1 5 21 22 d32,d32g. Since d32?d32,d32?d32 and d32?d32?d32?d32?d32, Dictyostelium [45–47] as follows these states should be removed from F’ because their successors 8 ’ ’~ d4 ,d12 z ~ z aren’t involved in set F . After updating, set F f 32 32g. Pick > A(t 1) 1 C(t) 4 ~ 4 12 > up state d32 and form the reachable set C2 fd32,d32g. It’s easy > B(tz1)~1zA(t) ~ ~ ~ ~ > to check card(C2) 2 and H(C2) C(I)H(C ) f4,12g,soC2 > : 2 > C(tz1)~H(t)zB(t) H(t) 4 12 <> fd32,d32g is an attractor of length 2. And, we can find the basin of D(tz1)~C(t) 8 16 ð16Þ C2 is fd ,d g. Remove C2 and its basins from F,F’ and > 32 32 > E(tz1)~D(t) continue to the above procedure. Similarly, we can find a loose > > F(tz1)~1zE(t) s~4 C ~ d17,d18,d21,d22 > attractor of length as 4 f 32 32 32 32g and its basin is > i > G(tz1)~D(t)zE(t)zD(t):E(t) fd ji[f1,3,5,6,7,8,9,11,13,14,15,16,19,20,23,24,25,26,27,28,29, :> 32 z ~ z : 30,31,32gg. After removing the states C4 and its basin from F, the H(t 1) G(t) F(t) G(t) feasible set F is empty and the procedure is terminated. The state Where the symbols A,B, ...,H represent PDI, PDE, cAMP(out- 1 2 3 4 21 22 23 24 9 10 11 12 29 30 31 32 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1211125615161234567817 18 27 28 21 22 31 32 17 18 19 20 21 22 23 24 527452749101112910111221 18 23 20 21 18 23 20 25 26 27 28 25 26 27 28 12 3 4 5 6 5 6 910111213141314 17 18 19 20 21 22 21 22 25 26 27 28 29 30 29 30 12 3 4 5 6 7 8 910111213141516 18 17 20 19 22 21 24 23 26 25 28 27 30 29 32 31 1211122122313212 3 4 21222324 17 18 27 28 21 22 31 32 17 18 19 20 21 22 23 24 5 2 7 4 21 18 23 20 9 10 11 12 25 26 27 28 21 18 23 20 21 18 23 20 25 26 27 28 25 26 27 28 1 2 3 4 21 22 21 22 9 10 11 12 29 30 29 30 17 18 19 20 21 22 21 22 25 26 27 28 29 30 29 30 1 2 3 4 21 22 23 24 9 10 11 12 29 30 31 32 18 17 20 19 22 21 24 23 26 25 28 27 30 29 32 31 5215125215121234123421 18 31 28 21 18 31 28 17 18 19 20 17 18 19 20 1211125613141234565617 18 27 28 21 22 29 30 17 18 19 20 21 22 21 22 1211125615161234567818 17 28 27 22 21 32 31 18 17 20 19 22 21 24 23 52745252910111291091021 18 23 20 21 18 21 18 25 26 27 28 25 26 25 26 527452749101112910111222 17 24 19 22 17 24 19 26 25 28 27 26 25 28 27 12 3 4 5 6 5 6 910111213141314 18 17 20 19 22 21 22 21 26 25 28 27 30 29 30 29 ~ I 5215122118312812 3 4 17181920 21 18 31 28 21 18 31 28 17 18 19 20 17 18 19 20 1211122122293012 3 4 21222122 17 18 27 28 21 22 29 30 17 18 19 20 21 22 21 22 1211122122313212 3 4 21222324 18 17 28 27 22 21 32 31 18 17 20 19 22 21 24 23 5 2 7 4 21 18 21 18 9 10 11 12 25 26 25 26 21 18 23 20 21 18 21 18 25 26 27 28 25 26 25 26 5 2 7 4 21 18 23 20 9 10 11 12 25 26 27 28 22 17 24 19 22 17 24 19 26 25 28 27 26 25 28 27 1 2 3 4 21 22 21 22 9 10 11 12 29 30 29 30 18 17 20 19 22 21 22 21 26 25 28 27 30 29 30 29 5215125213101234121221 18 31 28 21 18 29 26 17 18 19 20 17 18 17 18 5215125215121234123422 17 32 27 22 17 32 27 18 17 20 19 18 17 20 19 1211125613141234565618 17 28 27 22 21 30 29 18 17 20 19 22 21 22 21 52745252910111291091022 17 24 19 22 17 22 17 26 25 28 27 26 25 26 25 5215122128292612 3 4 17181718 21 18 31 28 21 18 29 26 17 18 19 20 17 18 17 18 5215122118312812 3 4 17181920 22 17 32 27 22 17 32 27 18 17 20 19 18 17 20 19 1211122122293012 3 4 21222122 18 17 28 27 22 21 30 29 18 17 20 19 22 21 22 21 5 2 7 4 21 18 21 18 9 10 11 12 25 26 25 26 22 17 24 19 22 17 22 17 26 25 28 27 26 25 26 25 5215125213101234121222 17 32 27 22 17 30 25 18 17 20 19 18 17 18 17 5215122118292612 3 4 17181718 22 17 32 27 22 17 30 25 18 17 20 19 18 17 18 17 transfer graph is shown in Fig. 4. side the cell), cAR1/3, Erk-2, RegA, ACA and cAMP (inside the 2 cell), respectively. z*M ~d ½ 2,1,1,2 and :*M ~d ½ 1,2,2,2 . In Fig. 4, we can find one fixed point d32, one attractor of length p 2 c 2 ~ 4 12 The above biochemical system is known for producing cyclic two C2 fd32,d32g and one attractor of length four ~ 17 18 21 22 aggregation signals. In [45], researchers studied the attractors of C4 fd32,d32,d32,d32g. Correspondingly, the basin of each attrac- 2 10 system under synchronous update and found one fixed point and tor is also depicted. As to fixed point d32, there exist one state d32 two cycles of length five. Here, we re-investigate this well-studied
PLOS ONE | www.plosone.org 9 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Figure 4. Attractors and basins of system (14). doi:10.1371/journal.pone.0066491.g004 system under asynchronous update. Considering the interactions Based on the above discussions, one can calculate the of the elements in the actual model, one element is randomly asynchronous network transition table of system. The detailed chosen for update at each time step. computations are skipped and the results are presented as follows. 124 The algebraic form of Eq. (16) is represented as It can be verified that only one fixed point d256*(10000100) exists in system (16) under asynchronous update, which is the same 8 > A(tz1)~L (I 6M )(I 6W )K(t) as the one under synchronous update. Fig. 5 and Fig. 6 show the > 6 128 n 4 ½2,32 > basins of the fixed point under asynchronous and synchronous > z ~ 6 > B(t 1) L6(I128 Mn)W½2,128 K(t) update, respectively. > > z ~ 6 6 6 6 Comparing Fig. 5 with Fig. 6, under asynchronous update, the > C(t 1) L5(I64 Mp)(I128 Mc)(I2 W½2,64 )(I128 Mr)K(t) > basin of the fixed point contains fewer states but has more complex > > D(tz1)~L (I 6W )K(t) interconnections. There are 11 transient states in Fig. 5 and the > 6 4 ½2,32 > farthest distance from one transient state to the fixed point is 3 < z ~ 6 E(t 1) L6(I8 W½2,16 )K(t) steps, for instance, d186(01000110)?d58 ð17Þ 256 256 > z ~ 6 6 60 124 > F(t 1) L6(I128 Mn)(I16 W½2,8 )K(t) (11000110)?d256(11000100)?d256(10000100). However, in > Fig. 6, there are more states directly flowing to the fixed point, > 6 6 6 6 > G(tz1)~L5(I64 Mp)(I128 Mp)(I256 Mc)(I128 W½2 ) > furthermore, any states can reach the fixed point though at most 2 > > (I 6M )(I 6M )(I 6W )K(t) steps. There are no attractors in system (16) under asynchronous > 64 r 128 r 8 ½4,8 > update, or from another perspective, the remaining states together > z ~ 6 6 6 6 > H(t 1) L5(I64 Mp)(I128 Mc)(I32 W½2,4 )(I128 Mr) construct a complicated and big loose attractor. :> (I 6W )K(t) 64 ½2 Conclusions In this article, we have presented a new approach to study the a dynamics of random Boolean networks under asynchronous where La~ P (I2i 6Ed ) and K(t)~A(t)B(t)C(t)D(t)E(t)F(t) i~0 stochastic update. By semi-tensor product of matrix, the logic of G(t)H(t). ARBN is converted into linear representation. A general formula
PLOS ONE | www.plosone.org 10 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks
Figure 5. The basin of the fixed point under asynchronous update. doi:10.1371/journal.pone.0066491.g005 of network transition matrices is presented, by which one can been proved to determine whether a given group of states compose obtain all of network transition matrices of a given logical attractor of length s on a particular ARBN. Asynchronous network dynamics whether under synchronous update or under asynchro- transition table as a transformed set of network transition matrices nous update. A necessary and sufficient algebraic criterion has is presented. Based on the above results, algorithms to detect the
Figure 6. The basin of the fixed point under synchronous update. doi:10.1371/journal.pone.0066491.g006
PLOS ONE | www.plosone.org 11 June 2013 | Volume 8 | Issue 6 | e66491 Dynamics of Asynchronous Random Boolean Networks attractors and basins of ARBNs are achieved. Examples are shown Author Contributions to demonstrate the validity of results. As a preparation, results in Conceived and designed the experiments: CL XYW. Performed the this article can be applied in the investigation of Boolean control experiments: CL. Analyzed the data: CL XYW. Contributed reagents/ problems under asynchronous update, which would be the work in materials/analysis tools: CL XYW. Wrote the paper: CL XYW. the future.
References 1. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed 23. Klemm K, Bornholdt S (2005) Stable and unstable attractors in Boolean genetic nets. J Theor Biol 22:437–467. networks. Physical Review E 72:055101. 2. Drossel B (2008) Random Boolean Networks. In Annual review of nonlinear 24. Aracena J, Goles E, Moreira A (2009) On the robustness of update schedules in dynamics and complexity, vol. 1, New York: Wiley. Boolean networks. Biosystems 97:1–8. 3. Farrow C, Heidel J, Maloney J, Rogers J (2004) Scalar equations for 25. Bassler KE, Lee C, Lee Y (2004) Evolution of Developmental Canalization in synchronous Boolean networks with biological applications. IEEE Trans. Neural Networks of Competing Boolean Nodes. Phys Rev Lett 93:038101. Netw 15:348–354. 26. Aldana M (2003) Boolean dynamics of networks with scale-free topology. 4. Steuer R (2007) Computational approaches to the topology, stability and Physica D 185:45–66. dynamics of metabolic networks. Phytochemistry 68:2139–2151. 27. Paul U, Kaufman V, Drossel B (2006) Properties of attractors of canalyzing 5. Liu YJ, Chen CLP, Wen GX, Tong SC (2011) Adaptive neural output feedback random Boolean networks. Physical Review E 73: 026118. tracking control for a class of uncertain discrete-time nonlinear systems. IEEE 28. Krawitz P, Shmulevich I (2007) Basin Entropy in Boolean Network Ensembles. Transactions on Neural Networks 22:1162–1167. Physical Review Letters 98:158701. 6. Zhang HG, Luo YH, Liu DR (2009) Neural-network-based near-optimal control 29. Hopfield JJ (1982) Neural networks and physical systems with emergent for a class of discrete-time affine nonlinear systems with control constraints.IEEE collective computational abilities. Proceedings of the National Academy of Transactions on Neural Networks 20:1490–1503. Sciences 79:2554–2558. 7. Steuer R, Gross T, Selbig J, Blasius B (2006) Structural kinetic modeling of 30. Dellaert F, Beer R (1994) Toward an evolvable model of development for metabolic networks. Proc Natl Acad Sci 103:11868–11873. autonomous agent synthesis. Proceedings of Alife 4, pp.246–257, Cambridge 8. Fu J, Zhang HG, Ma TD, Zhang QL (2010) On passivity analysis for stochastic MA, MIT Press. neural networks with interval time-varying delay. Neurocomputing 73:795–801. 31. Harvey I, Bossomaier T (1997) Time out of joint: Attractors in asynchronous 9. Sporns O, Chialvo DR, Kaiser M, Hilgetag CC (2004) Organization, random boolean networks, in Proceedings of the Fourth European Conference development and function of complex brain networks. Trends in Cognitive on Artificial Life (ECAL97, MIT Press), pp. 67–75. Sciences 8:418–425. 32. Shreim A, Berdahl A, Greil F, Davidsen J, Paczuski M (2010) Attractor and basin entropies of random Boolean networks under asynchronous stochastic 10. Liu YJ, Tong SC, Wang D, Li TS, Chen CLP (2011) Adaptive neural output update. Physical Review E 82: 035102. feedback controller design with reduced-order observer for a class of uncertain 33. Greil F, Drossel B (2005) The dynamics of critical Kauffman networks under nonlinear SISO systems. IEEE Transactions on Neural Networks 22:1328–1334. asynchronous stochastic update. Physical Review Letters 95:048701. 11. Chmiel AM, Sienkiewicz J, Suchecki K, Holyst JA (2007) Networks of 34. Deng XT, Geng HM, Matache MT (2007) Dynamics of asynchronous random companies and branches in Poland. Physica A 383: 134–138. Boolean networks with asynchrony generated by stochastic processes. Biosystems 12. Zhang HG, Wang YC (2008) Stability analysis of Markovian jumping stochastic 88:16–34. Cohen-Grossberg neural networks with mixed time delays. IEEE Transactions 35. Skodawessely T, Klemm K (2011) Finding attractors in asynchronous Boolean on Neural Networks 19: 366–370. dynamics. Advances in Complex Systems 14:439–449. 13. Liu YJ, Tong SC and Chen CLP (2013) Adaptive fuzzy control via observer 36. Greil F, Drossel B, Sattler J (2007) Critical Kauffman networks under design for uncertain nonlinear systems with unmodeled dynamics. IEEE deterministic asynchronous update. New Journal of Physics 9:373. Transactions on Fuzzy Systems 21:275–288. 37. Di Paolo EA (2001) Rhythmic and non-rhythmic attractors in asynchronous 14. Colizza V, Barrat A, Barthelemy M, Vespignani A (2006) The role of the airline random Boolean networks. BioSystems, 59:185–195. transportation network in the prediction and predictability of global epidemics. 38. Hallinan J, Wiles J (2004) Asynchronous dynamics of an artificial genetic Proc Natl Acad Sci 103:2015–2020. regulatory network. In Ninth International Conference on the Simulation and 15. Zhang HG, Ma TD, Huang GB, Wang ZL (2010) Robust global exponential Synthesis of Living Systems. ALife9, Boston. synchronization of uncertain chaotic delayed neural networks via dual-stage 39. Cheng D, Qi H, Li Z (2011) Analysis and Control of Boolean Networks: A Semi- impulsive control. IEEE Transactions on System Man and Cybernetics Part B- Tensor Product Approach. London, U.K.: Springer-Verlag. Cybernetics 40:831–844. 40. Cheng D, Qi H (2009) Controllability and observability of Boolean control 16. Uchida S (2008) Stability and Structure of Model Food-webs with Adaptive networks. Automatica 45:1659–1667. Behavior. Ph.D. thesis, Technische Universita¨t Darmstadt Dissertation, D17. 41. Li F, Sun J (2011) Controllability of probabilistic Boolean control networks. 17. Dubrova E, Teslenko M (2011) A SAT-Based Algorithm for Finding Attractors Automatica 47:2765–2771. in Synchronous Boolean Networks. IEEE-ACM Transactions on Computational 42. Li F, Sun J, Wu Q (2011) Observability of Boolean control networks with state Biology and Bioinformatics 8:1393–1399. time delays. IEEE Trans. Neural Networks 22: 948–954. 18. Dubrova E, Teslenko M (2005) Compositional properties of random Boolean 43. Cheng D, Li ZQ, Qi H (2010) Realization of Boolean control networks. networks. Physical Review E 71:056116. Automatica 46:62–69. 19. Lloyd-Price J, Gupta A, Ribeiro AS (2012) Robustness and information 44. Gonze D, Goldbeter A (2000) A model for a network of phosphorylation– propagation in attractors of Random Boolean Networks. PLOS ONE 7:e42018. dephosphorylation cycles displaying the dynamics of dominoes and clocks. 20. Akutsu T, Kosub S, Melkman AA (2012) Tamura T. Finding a Periodic J Theor Biol 210:167–186. Attractor of a Boolean Network. IEEE-ACM Transactions on Computational 45. Heidel J, Maloney J, Farrow C, Rogers JA (2003) Finding cycles in synchronous Biology and Bioinformatics 9:1410–1421. Boolean networks with applications to biochemical systems. International 21. Yang LJ, Meng Y, Bao C, Liu WH, Ma CZ, et al. (2013) Robustness and Journal of Bifurcation and Chaos 13:535–552. Backbone Motif of a Cancer Network Regulated by miR-17-92 Cluster during 46. Goldbeter A (1996) Biochemical Oscillations and Cellular Rhythms. Cambridge the G1/S Transition. PLOS ONE 8:e57009. Univ. Press, Cambridge. 22. Kauffman SA (1993) The Origins of Order: Self-Organization and Selection of 47. Aubry L, Firtel R (1999) Integration of signaling networks that regulate Evolution. Oxford: Oxford University Press. Dictyostelium differentiation. Ann Rev Cell Dev Biol 15:469–517.
PLOS ONE | www.plosone.org 12 June 2013 | Volume 8 | Issue 6 | e66491