Abstract and Applied Analysis

Volume 2013 (2013), Article ID 631639, 9 pages

http://dx.doi.org/10.1155/2013/631639

Research Article

A Note on the Observability of Temporal Boolean Control Network

Wenping Shi,¹ Bo Wu,1,2 and Jing Han³

¹College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China

²Academic Affairs Division, Zhejiang Normal University, Jinhua 321004, China

³Department of Mathematics, Tongji University, Shanghai 200092, China

Received 2 December 2012; Accepted 24 February 2013

Academic Editor: Qi Luo

Copyright © 2013 Wenping Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Temporal Boolean network is a generalization of the Boolean network model that takes into account the time series nature of the data and tries to incorporate into the model the possible existence of delayed regulatory interactions among genes. This paper investigates the observability problem of temporal Boolean control networks. Using the semi tensor product of matrices, the temporal Boolean networks can be converted into discrete time linear dynamic systems with time delays. Then, necessary and sufficient conditions on the observability via two kinds of inputs are obtained. An example is given to illustrate the effectiveness of the obtained results.

1. Introduction

Boolean network (BN) is the simplest logical dynamic system. It was proposed by Kauffman for modeling complex and nonlinear biological systems; see [1–3]. Since then, it has been a powerful tool in describing, analyzing, and simulating the cell networks. In this model, gene state is quantized to only two levels: true and false. Then, the state of each gene is determined by the states of its neighborhood genes, using logical rules.

The control of BN is a challenging problem. So far, there are only few results on it because of the shortage of systematic tools to deal with logical dynamic systems; see [4, 5]. Recently, a new matrix product, which was called the semitensor product (STP) [4], was provided to convert a logical function into an algebraic function, and the logical dynamics of BNs could be converted into standard discrete-time dynamics. Based on this, a new technique has been developed for analyzing and synthesizing Boolean (control) networks (BCNs); see [4, 6–9]. Furthermore, [10] have presented some simple criteria to judge the controllability with respect to input-state incidence matrices of BCNs. A Mayer-type optimal control problem for BCNs with multi-input and single input has been studied in [11, 12].

Systematic analysis of biological systems is an important topic in systems biology, and the observability is a structural property of systems. There have been many results on the controllability and observability of dynamic systems; see [13–18]. When it comes to the observability problem of BNs, Cheng and Qi have obtained necessary and sufficient conditions for the observability of BCNs in [8]. However, simple Boolean method cannot be used to study the kinetic properties of networks because it does not have time components, and time delay behaviors happen frequently in biological and physiological systems. In [19], the observability problem for a class of Boolean control systems with time delay is investigated.

It is well known that time delay phenomenon is very common in the real world [20, 21] and very important in analysis and control for dynamic systems. Since many experiments involve obtaining gene expression data by monitoring the expression of genes involved in some biological process (e.g., neural development) over a period of time, the resulting data is in the form of a time series [22]. It is interesting to understand how the expression of a gene at some stage in the process is influenced by the expression levels of other genes during the stages of the process preceding it. Temporal Boolean networks (TBNs) are developed to help model the temporal dependencies that span several time steps and model regulatory delays, which may come about due to missing intermediary genes and spatial or biochemical delays between transcription and regulation; see [23–25].

It should be noticed that TBCN is similar with higher-order BCN from Chapter 5 of [26] in which the higher-order BCN can be rewritten by a BCN by using the first algebraic form of the network. Hence, the observability analysis for higher-order BCNs can be obtained from [26]. However, if the first algebraic form is used, the dimension of network transition matrix depending on the number of logical variables will be much larger which would make computation cost much higher [27]. Motivated by the above analysis, the purpose of this paper is to use STP developed in [4, 6–9, 28] to analyze the observability problem of TBCN without changing it into BCN, which generalizes the BN model to cope with dependencies that span over more than one unit of time.

The rest of this paper is organized as follows. Section 2 provides a brief review for the STP of matrices and the matrix expression of logical function. In Section 3, we convert TBCN into discrete time delay systems. In Section 4, necessary and sufficient conditions for the observability of the temporal BCNs are obtained. An example is given to illustrate the efficiency of the proposed results in Section 5. Finally, a brief conclusion is presented.

2. Preliminaries

For simplicity, we first give some notations as in [4]. Denote as the set of all matrices. The delta set , where is the th column of identity matrix with degree . A matrix is called a logical matrix if the columns set of , denoted by , satisfies . The set of all logical matrices is denoted by . Assuming , we denote it as .

We recall the concept of STP. Let be a row vector of dimension and a column vector of dimension . Then, we split into equal-sized blocks as , which are rows. Define the STP, denoted by , as

In this paper, “” is omitted, and throughout this paper the matrix product is assumed to be the semi-tensor product as in [9].

The swap matrix is an matrix. Label its columns by and its rows by . Then, its element in the position is assigned as

When , we briefly denote . Furthermore, for and ,   and   .

A logical domain, denoted by , is defined as . To use matrix expression, we identify each element in with a vector as and and denote . Using STP of matrices, a logical function with arguments can be expressed in the algebraic form as follows.

Lemma 1 (see [9]). Any logical function with logical arguments can be expressed in a multi-linear form as

where is unique which is called the structure matrix of L.

Lemma 2 (see [9]). Assume that with logical arguments , then

where ,  .

3. Algebraic Form of Temporal Boolean Networks

We consider the temporal Boolean network [25] of a set of nodes as follows:

where ,   are logical functions, , and is a positive integer delay.

Using Lemma 1, for each logical function ,  , we can find its structure matrix . Let . Then, the system (5) can be converted into an algebraic form as

From Lemma 2, multiplying all systems in (6) together yields

Denote . Then (7) can be expressed as

and is called the network transition matrix of (5).

Next, we consider temporal Boolean control network with outputs as follows:

where ,   are inputs (or controls); ,   are outputs; ,  ;   are logical functions.

In this paper, two kinds of inputs (or controls) are considered for (9).

(A) The controls satisfying certain logical rules are called input networks such as

where ,   are logical functions, and the initial states ,  , can be arbitrarily given.

(B) The controls are free Boolean sequences, which means that the controls do not satisfy any logical rule.

Let ,  . From Lemma 1, for every logical function ,  ,  , we can find its structure matrix ,  ,  ,  ,  ,  , respectively. Then from (9) and (10), we can obtain

Similar with (7), multiplying (11) yields

And, multiplying (12), it leads to

Multiplying (13) yields , where . From the above conclusion, in an algebraic form, a BCN (9) and (10) can be expressed as

where are the network transition matrices of two kinds of equations in (9), respectively, and is the network transition matrix of (10).

Remark 3. It should be noticed that by using the first algebraic form of the network from Chapter 5 of [26], TBCN can be rewritten by a BCN with no delay. Hence, it can be a good idea to study the observability of TBCNs by using the corresponding BCNs from the results in [10]. However, if the first algebraic form is used, the dimension of network transition matrix of corresponding BCNs will be much bigger which would make computation cost much higher. From (16), it is easy to calculate that the dimension of is . However, if the TBCNs are rewritten by BCNs using the first algebraic form, then the dimension of the corresponding network transition matrix of the BCNs would be , which is much bigger if or is a large number. Furthermore, considering the TBCNs directly, we can find the relationship between the network transition matrix (or the Boolean functions) of the TBCN and the state clearly. However, if the BCN is used, the relationship would not be so clear.

4. Observability of Temporal Boolean Control Networks

In this section, we consider the observability problem of temporal Boolean control network (9), equivalently (16), and the analysis is given via two kinds of controls (A) and (B), respectively.

Definition 4 (see [19]). The temporal Boolean network (16) is observable if for the initial state sequence ,  , there exists a finite time , such that the initial state sequence can be uniquely determined by the input controls and the outputs .

For simplicity, we denote the vector ,  .

Definition 5 (see [19]). For temporal Boolean network (16) and control (17) with fixed , the input-state transfer matrix ,  , is defined as follows: for any and any ,  , we have

Now we need a dummy operator to add some fabricated variables when these variables do not appear. Define

A straightforward computation shows the following.

Lemma 6. Consider the temporal Boolean network (16),

Proof. Since , from the definition of , we have

Hence,

4.1. Observability of Input Boolean Networks

We first consider the case that controls satisfy certain logical rules as (17). Define a sequence of matrices as (23):

where and ,  , and the transition matrices , , and are defined in (16) and (17). Furthermore, we split , , into equal blocks as with each , , .

Theorem 7. Consider the temporal Boolean network (16) with control (17). Assume that ,  . Then, (16) and (17) are observable if and only if there exists a finite time such that , where

Proof. Firstly, from Lemma 6 and (16),

Since , we have from (18) that

For , we can obtain that

From the above analysis, and definition of in (24), we can see that

Since ,  . It implies that is determined uniquely by the outputs if and only if there exist no similar elements in , or equivalently, there are no equal columns in , that is, rank. The proof is completed.

Corollary 8. Consider the temporal Boolean network (16) with control (17). Equations (16) and (17) are observable if and only if there exist a finite time and such that .

Remark 9. When the time delay , then the temporal Boolean control network (16) and (17) become a Boolean control network. In this case, it can be induced from (23) that

Then, the observability of the BCN with input Boolean network controls can be deduced from Theorem 7 and Corollary 8.

4.2. Control via Free Boolean Sequence

In the following, the case where the controls are free Boolean sequences is considered. We split given in (16) into equal blocks as

with each ,  . Define a sequence of matrices ,  , as (31):

where , the transition matrices , , and are defined in (16) and (17).

Theorem 10. Consider the temporal Boolean network (16). Assume that the controls are free Boolean sequences with . Then, (16) is observable if and only if there exists a finite time such that , where

Proof. Since the controls are free Boolean sequences with , , , from (16) we have

For , we can obtain that

Thus, from (25) and the definition of in (32), we can see that

Similar with the proof of Theorem 7, we conclude that can be determined uniquely by the outputs if and only if rank. The proof is completed.

Corollary 11. Consider the temporal Boolean network (16). The system (16) is observable if and only if there exists a finite time and a sequence such that .

Remark 12. As a special case, when , then from the proof of Theorem 10, we have , and

Then, Corollary 11 is equivalent with Theorem  26 in [8] for the observability of BCNs.

Remark 13. For Theorems 7 and 10, when , the third explicit expressions of in (23) and in (31) for should be omitted.

5. An Example

Given logical arguments , we have the following structure matrices for the fundamental logical functions: , , , , , where , , , , .

Example 14. Consider the following temporal Boolean network:

Let , it is easy to get , , and .

(A) When the controls satisfy the logical rule

then the transition matrix . Now, assume that , by calculation, we have

Hence, for any , there are only linearly independent columns, which means that rank for any , and the system is not observable from Theorem 7. Similarly, if , we have the same conclusion.

(B) When controls are free sequences with , , . By calculation, it leads to

and hence,

When , it is enough to see that there are no equal columns in . So, the system is observable by Theorem 10.

From cases (A) and (B), it is easy to notice that the selection of controls is very important for the observability of the temporal Boolean control network.

6. Conclusion

In this brief paper, necessary and sufficient conditions for the observability of temporal Boolean control networks have been derived. By using semi-tensor product of matrices and the matrix expression of logic, we have converted the temporal Boolean control networks into discrete systems with time delays. Moreover, the observability has been investigated via two different kinds of controls. Finally, an example has been given to show the efficiency of the proposed results.

Acknowledgments

The authors would like to take this opportunity to thank the reviewers for their constructive comments and useful suggestions. This work was partially supported by the NNSF of China (Grants nos. 11101373, 11271333, and 61074011).

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