# Attractor Transformation by Impulsive Control in Boolean Control Network

Authors: Bo Gao; Haipeng Peng; Dawei Zhao; Wenguang Zhang; Yixian Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Boolean control networks have recently been attracting considerable interests as computational models for genetic regulatory networks. In this paper, we present an approach of impulsive control for attractor transitions in Boolean control networks based on the recent developed matrix semitensor product theory. The reachability of attractors is estimated, and the controller is also obtained. The general derivation proposed here is exemplified with a kind of gene model, which is the protein-nucleic acid interactions network, on numerical simulations.


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## Body

## 1. Introduction

Gene regulatory networks (GRNs) have been offering plenty of holistic approaches to biological processes. They can explicitly represent the causality of developmental processes and exactly describe the state set of biological systems [1]. Waddington and Kacser proposed a metaphor that the developmental process of GRNs can be represented by a ball rolling down along a landscape with peaks and valleys, and the steady states, which were called attractors, were found at the bottom of the basins [2]. In cell model, there is a one-to-one relationship between the attractors and the observed phenotypes. This means that different cell types can be characterized by different attractors [3]. The states of a GRN will stay in an attractor, unless it is perturbed by an outside impact [4].In several studies on GRNs such as genetic organogenesis and diseases, researchers have considered to make the states of GRN transit from one attractor to another one by using control methods [5–7]. It was found that repression of a single RNA binding polypyrimidine tract-binding protein was sufficient to induce transdifferentiation of fibroblasts into functional neurons in [8]. An approach was presented to evaluate drug targets of GRN inference to ovarian cancer in [9]. The previous studies mainly focused on intervening the system to help it transit to the desirable attractors by controlling a (or some) valid genetic locus. Since most of the existing achievements in related fields were obtained based on the experiments, the actual impact of control on the same GRN is uncertain [10]. In brief, to estimate the effectiveness of the controller for the transformation of GRNs from one attractor to another one, still remains an open crucial theoretical problem [11–15].A Boolean network (BN) is often used as a model for gene regulation which treats genes as binary nodes that are either expressed or unexpressed [4]. In order to manipulate networks, the control of BNs is an important topic. A Boolean control network (BCN) can be considered as a BN with additional binary inputs. BCNs are attracting considerable interests as computational models for GRNs which use the exogenous inputs. BCN has been widely used in yeast cell-cycle [16], Drosophila melanogaster [17], and other kinds of cells.In this paper, we propose a theoretical method to estimate the effects of a certain impulsive controller in a BCN and solve the appropriate controller by using semitensor product. Compared with the existing methods, based on the results of the experiments, our mathematics-based approach is more accurate and simpler.The rest of this paper is organized as follows. Section2 reviews STP and the model of BN. In Section 3, the reachability of an attractor is realized and the controller is also obtained. Section 4 gives our main results of an example. Section 5 is the conclusion.


## 2. Preliminaries

### 2.1. Semitensor Product

Semitensor product (STP) of matrices was firstly proposed by Cheng and Dong. It is the algebraic form and the coordinate transformation of BN and BCN. Based on STP, BNs and BCNs can be converted into equivalent algebraic form of some standard discrete-time system [18]. In this paper, STP is denoted by “⋉”.Definition 1.
Assuming there are two matricesA∈ℝm×n and B∈ℝp×q, the STP of A and B is A⋉B=(A⊗Iα/n)(B⊗Iα/p), where α is the least common multiple of n and p, “⊗” is the Kronecker product, and Ik is the identity matrix.
The STP of matrices makes all the fundamental properties of the conventional matrix product remain true [19]. With STP, Boolean operation can be converted into matrix product. These two logical values, “true” and “false,” are expressed in vector forms as δ21 and δ22, where δnr denotes the rth column of the identity matrix Ik. Some fundamental logical functions are identified as M=[δni1,δni2,…,δnis], which is also briefly expressed as M=δn[i1,i2,…,is]. And the logic relationships are(1)
negation:Mn=δ2[2,1];(2)
disjunction:Md=δ2[1,1,1,2];(3)
conjunction:Mc=δ2[1,2,2,2];(4)
XOR:Mp=δ2[2,1,1,2].
The above matrices are called the structure matrices.


### 2.2. Attractor

A BN, which is typically formulated as a directed graph, composed ofn nodes, whose state indicates whether the gene is switched 1 (on) or 0 (off). The state of each node at time t+1 is determined by the state of its spatial neighbors at time t. The system can be described by
(1)x1(t+1)=f1(x1(t),x2(t),…,xn(t)),x2(t+1)=f2(x1(t),x2(t),…,xn(t)),⋮xn(t+1)=fn(x1(t),x2(t),…,xn(t)),
where fi(i=1,2,…,n) is an n-ary logical function.The BN is a globally convergence system. An attractor, called the stable state of system, is in the form of either a single state (fixed point) or a repeating set of states (cycle) [20]. Here, we consider how to find the attractors of (1). According to STP, we define
(2)A(t)=⋉i=1nxn(t).
Then
(3)A(t+1)=⋉i=1nMiA(t),
where Mi(i=1⋯n) is the structure matrix. Using the properties of STP, (3) can be converted into an algebraic form as
(4)A(t+1)=LA(t),
where L∈Δ2n×2n is called the transition matrix. The state of (1) is uniquely determined by the transition matrix. Each column of L, which is called state number, represents a state of the BN. The attractor has the following definitions:(1)
a statex(t)∈Δ2n is called a fixed point if Lx(t)=x(t);(2)
{x(t),Lx(t),…,Lkx(t)} is called a cycle with length k if Lkx(t)=x(t) and the elements in the set {x(t),Lx(t),…,Kk-1x(t)} are distinct.Theorem 2.
In system (1), the number of length s cycles, Ns, is inductively determined by
(5)N1=Trace(L),Nk=(Trace(Lk)-∑s∈𝒫(k)sNs)k,2≤k≤2n,
where 𝒫(k) is the set of proper factors of k. According to (5), one can find all the attractors in the state space of BN (1) [21].


## 2.1. Semitensor Product

Semitensor product (STP) of matrices was firstly proposed by Cheng and Dong. It is the algebraic form and the coordinate transformation of BN and BCN. Based on STP, BNs and BCNs can be converted into equivalent algebraic form of some standard discrete-time system [18]. In this paper, STP is denoted by “⋉”.Definition 1.
Assuming there are two matricesA∈ℝm×n and B∈ℝp×q, the STP of A and B is A⋉B=(A⊗Iα/n)(B⊗Iα/p), where α is the least common multiple of n and p, “⊗” is the Kronecker product, and Ik is the identity matrix.
The STP of matrices makes all the fundamental properties of the conventional matrix product remain true [19]. With STP, Boolean operation can be converted into matrix product. These two logical values, “true” and “false,” are expressed in vector forms as δ21 and δ22, where δnr denotes the rth column of the identity matrix Ik. Some fundamental logical functions are identified as M=[δni1,δni2,…,δnis], which is also briefly expressed as M=δn[i1,i2,…,is]. And the logic relationships are(1)
negation:Mn=δ2[2,1];(2)
disjunction:Md=δ2[1,1,1,2];(3)
conjunction:Mc=δ2[1,2,2,2];(4)
XOR:Mp=δ2[2,1,1,2].
The above matrices are called the structure matrices.


## 2.2. Attractor

A BN, which is typically formulated as a directed graph, composed ofn nodes, whose state indicates whether the gene is switched 1 (on) or 0 (off). The state of each node at time t+1 is determined by the state of its spatial neighbors at time t. The system can be described by
(1)x1(t+1)=f1(x1(t),x2(t),…,xn(t)),x2(t+1)=f2(x1(t),x2(t),…,xn(t)),⋮xn(t+1)=fn(x1(t),x2(t),…,xn(t)),
where fi(i=1,2,…,n) is an n-ary logical function.The BN is a globally convergence system. An attractor, called the stable state of system, is in the form of either a single state (fixed point) or a repeating set of states (cycle) [20]. Here, we consider how to find the attractors of (1). According to STP, we define
(2)A(t)=⋉i=1nxn(t).
Then
(3)A(t+1)=⋉i=1nMiA(t),
where Mi(i=1⋯n) is the structure matrix. Using the properties of STP, (3) can be converted into an algebraic form as
(4)A(t+1)=LA(t),
where L∈Δ2n×2n is called the transition matrix. The state of (1) is uniquely determined by the transition matrix. Each column of L, which is called state number, represents a state of the BN. The attractor has the following definitions:(1)
a statex(t)∈Δ2n is called a fixed point if Lx(t)=x(t);(2)
{x(t),Lx(t),…,Lkx(t)} is called a cycle with length k if Lkx(t)=x(t) and the elements in the set {x(t),Lx(t),…,Kk-1x(t)} are distinct.Theorem 2.
In system (1), the number of length s cycles, Ns, is inductively determined by
(5)N1=Trace(L),Nk=(Trace(Lk)-∑s∈𝒫(k)sNs)k,2≤k≤2n,
where 𝒫(k) is the set of proper factors of k. According to (5), one can find all the attractors in the state space of BN (1) [21].


## 3. Main Results

In GRNs, much attention focuses on making the whole system transit from one attractor to another by control methods [22, 23]. Since the impulsive controller could destroy the cycle structure of the biological system, it is widely used in GRNs [24]. First, we will judge the reachability of an attractor.Assume𝒜1={xh∣h=1⋯l1} and 𝒜2={yr∣r=1⋯l2} are two attractors in system (1). l1 and l2 are the lengths of 𝒜1 and 𝒜2, respectively. We will determine the reachability from 𝒜1 to 𝒜2.Here, we consider a BCN withm impulsive inputs at time t, and it is defined as
(6)x~1(t+1)=f~1(x1(t),…,xn(t),u1(t),…,um(t)),⋮x~n(t+1)=f~n(x1(t),…,xn(t),u1(t),…,um(t)),
where fi(i=1,2,…,n) is an n-ary logical function and uj(j=1⋯m) is the impulsive input. uj is described as
(7)uj(t)={input,t=tk,noinput,t≠tk,
where input is 0 or 1. When system (6) is at time t=tk, according to STP, we define
(8)A~(t)=(⋉i=1nxi(t))⋉(⋉j=1muj(t)),
then
(9)A~(t+1)=⋉i=1nMiA~(t),
where Mi(i=1⋯n) is the structure matrix. So (9) can be converted into an algebraic form as follows:
(10)A~(t+1)=L~A~(t),
where L~∈Δ2n×2n+m is called the state transition matrix of system (6).Theorem 3.
Consider (6). The transformation from attractor 𝒜1 to attractor 𝒜2 is reachable with controllers u1(t)⋯um(t) if and only if
(11)L~𝒜1∩𝒜2≠ϕ,
where ϕ is the null set.Proof.
SinceL~ is the linear representation of matrix L with the inputs u1(t)⋯um(t), L~𝒜1 is the reachable state set of 𝒜1. The intersection set of L~𝒜1 and 𝒜2 is the destination states from 𝒜1 to 𝒜2 with controllers u1(t)⋯um(t).Definition 4.
Assumeyr is the destination state for the transition from 𝒜1 to 𝒜2 if and only if
(12)yr∈L~𝒜1∩𝒜2,
then xh is the source state if and only if
(13)yr=L~xh,
where xh∈𝒜1 and yr∈𝒜2.
Next, we will find the existence of the controller for attractor transition.Theorem 5.
Consider system (6). Assume L~xh={ek∣k=1⋯2m}; if yr=ek, we have δ2mk. The impulsive controllers are obtained by
(14)δ2mk=u1(t)⋉u2(t)⋉⋯um(t),
where uj(t)(j=1⋯m) is the impulsive input at time t.Proof.
SinceL~xh is the destination state from xh with impulsive inputs u1(t)⋯um(t), each ek represents each state which is from xh by inputs u1(t)⋯um(t). Equation (14) is based on the properties of STP.
We can solve the input values for a deterministic target by using (14).


## 4. Example

In order to illustrate our approach, an example is given in this section. It is an idealized protein-nucleic acid interaction involved in gene regulation model in cells [25]. The type of unit component which we will study is shown as follows:
(15)x1(t+1)=1+x3(t)+x6(t)+x3(t)x6(t),x2(t+1)=x1(t),x3(t+1)=x2(t),x4(t+1)=1+x3(t)+x6(t)+x3(t)x6(t),x5(t+1)=x4(t),x6(t+1)=x5(t),
where A·B represents A Conjunction B and A+B represents the XOR operation between A and B. Based on (2)–(4), the L matrix of system (15) is L = δ64[373738383939404037138239340445454646474748484594610471148125353545455555656531754185519562061616262636364646125622663276428].Using (5), we obtain that there are the following two attractors in the state space:
(16)𝒜1=(19)⟶(46)⟶(19),𝒜2=(1)⟶(37)⟶(55)⟶(64)⟶(28)⟶(10)⟶(1).
The attractors of this system represent different quantities of the generation of a metabolic species.Next, assume the system is already in attractor𝒜1; we want to transit the whole system from 𝒜1 to 𝒜2 with some impulsive controllers.The BCN is expressed as(17)x1(t+1)=1+x3(t)+x6(t)+x3(t)x6(t),x2(t+1)=x1(t)+u1(t),x3(t+1)=x2(t),x4(t+1)=1+x3(t)+x6(t)+x3(t)x6(t),x5(t+1)=x4(t)+u2(t),x6(t+1)=x5(t),
where u1, u2 are controllers.Step 1.
Using (8)–(10), we obtain the matrix L~. Based on the computing of L~, we obtain L~𝒜1=δ64[64624846131719]. According to (11), the intersection of two sets is L~𝒜1∩𝒜2=δ64[641]≠ϕ. So, the transformation from 𝒜1 to 𝒜2 is reachable, and yr1=δ64[64] and yr2=δ64[1] are the destination states.Step 2.
Since𝒜1 is made up of two states, which are 𝒜11=δ64[19] and 𝒜12=δ64[46], we have
(18)L~𝒜11=δ64[64624846],L~𝒜12=δ64[131719],
then L~𝒜11∩yr1≠ϕ and L~𝒜12∩yr2≠ϕ.
Soxh1=𝒜11=δ64[19] and xh2=𝒜12=δ64[46] are source states.Step 3.
LettingL~𝒜11={ek∣k=1,…,4}, yr1=e1, we have δ41. Based on Theorem 5, the impulsive controllers are described as
(19)u1(t)={1,t=tk,noinput,t≠tk,u2(t)={1,t=tk,noinput,t≠tk.
Similarly, we can obtain other impulsive controllers which are described as
(20)u1(t)={1,t=tk,noinput,t≠tk,u2(t)={1,t=tk,noinput,t≠tk,
whose source state is 𝒜12=δ64[46].The conclusion of this example is that there are two kinds of controllers which can transform the state of system (17) from attractor 𝒜1 to attractor 𝒜2. They can be described as follows.The First. The source state number is 19, the destination state number is 64, and the impulsive controllers are
(21)u1(t)={1,t=tk,noinput,t≠tk,u2(t)={1,t=tk,noinput,t≠tk.The Second. The source state number is 46, destination state number is 1, and the impulsive controllers are
(22)u1(t)={1,t=tk,noinput,t≠tk,u2(t)={1,t=tk,noinput,t≠tk.


## 5. Conclusion

This paper explores the problem of attractor transformation by impulsive control in BCN. We propose an effective algorithm which allows us to realize the transformation among different attractors of the BCN. Although the protein-nucleic acid gene network has relatively simple structure compared with those exhibited by metazoans, the attractors transformation by impulsive control is impressively significant. Our findings open a new perspective in the attractor transformation by impulsive control which is of utmost importance in areas as diverse as drug target and gene regulation and so forth. Developing more effective algorithms or approximate techniques for the present approaches will be the future work.



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*Source: 674571-2013-07-28.xml*