Observability of Boolean multiplex control networks

Boolean multiplex (multilevel) networks (BMNs) are currently receiving considerable attention as theoretical arguments for modeling of biological systems and system level analysis. Studying control-related problems in BMNs may not only provide new views into the intrinsic control in complex biological systems, but also enable us to develop a method for manipulating biological systems using exogenous inputs. In this article, the observability of the Boolean multiplex control networks (BMCNs) are studied. First, the dynamical model and structure of BMCNs with control inputs and outputs are constructed. By using of Semi-Tensor Product (STP) approach, the logical dynamics of BMCNs is converted into an equivalent algebraic representation. Then, the observability of the BMCNs with two different kinds of control inputs is investigated by giving necessary and sufficient conditions. Finally, examples are given to illustrate the efficiency of the obtained theoretical results.

Human Genome Project, which is an international scientific research project with the goal of determining the sequence of nucleotide base pairs 1 , inspired a new view of biology called the systems biology. Instead of investigating individual genes, proteins or cells, systems biology studies the behavior and relationships of all cells, proteins, DNAs and RNAs in the same biological system, called a cellular network 2 . The Boolean Networks (BNs) as a powerful tool in describing, analyzing, and simulating the cellular networks, has been most widely used [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] .
From decades ago, Kauffman put forward the theory which can describe the net of cell and gene using BNs 4 . And he made expatiation about the relationship between BNs and gene as well as life 5,6 . Because the construction and evolutionary process of cell and gene can be revealed very well by BNs, BNs turn into a hot topic concerned by biologists, physicists and neuroscientists. Huang In recent years accompany with the development of biology, control of biological system becomes into a hot topic [19][20][21][22][23][24][25][26][27][28][29] . As to the research of genetic regulatory networks (GRNs), one of the major goals is to carry out the therapeutic intervention strategies for diseased targets 30,31 . 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. So the interests to the BCNs are increasingly going up. The application of BCNs includes not only GRNs 32 , but also other various fields, such as man-machine dynamic game 33 and internal combustion engines 34,35 . Recently, based on semi-tensor product (STP) proposed by Cheng, D. et al. 36 , many basic problems for BCNs have been investigated, for example, realization 23 , controllability 24,26 , optimal control 15,33 , etc. Observability of a system is a structural property. It is also a fundamental concept in control theory and systematic science and, not surprisingly, it has found many applications in systems biology. As early as 1976, Cobelli et al. studied controllability, observability and structural identifiability of biological compartmental systems of any structure 37 . In evolutionary dynamics, observability is the key to study whether the genetic process itself can be recovered from measurements of phenotypic characteristics 38 . Observability analysis is a necessary preliminary step to the design of observers, that is, systems that provide an estimate of the complete internal state based on measurements of the inputs and outputs 39  From the view of systems biology, the analysis in system-level of biological regulation needs to consider the interactions of genes on a holistic level, rather than the independent characteristics of isolated parts of an organism 41 . To understand the intricate variability of biological systems, where many hierarchical levels and interactions coexist, a new level of description is required. Thereupon, multiplex networks as an extension of complex networks were firstly proposed by Mucha in 2010 42 , which is composed of several layered networks interrelated with each other shown in Fig. 1. The previous description implicitly assumes that all biochemical signals are equivalent and then collapses information from different pathways. Actually, in cellular biochemical networks, many different signaling channels do work in parallel 43 . Not only in cellular biochemical, multiplex networks have been applied to the natural, social, and information sciences 42 . As an old concept, multi-layer social networks have been studied from 1975 44 . Transportation systems are natural candidates for a multi-layer network representation. In a recent paper, a two-layer structure has been created by merging the world wide port and airport networks 45 . In multiplex networks, each layer could have particular features and dynamical processes. Between layers, interconnections are represented by some special nodes on behalf of different roles participating in multiple layers of interactions. The final states of those common nodes at each time are determined by all involved layers, which is different from the traditional sense of coupling.
Recent years more and more researchers studied the BMNs. For example, Xu, M. et al. investigated the synchronizability of two-layer networks 46 47 . But when it comes to the observability problem of BMCNs, to our best knowledge, there have been even no results, because there are many differences between BMNs and single-layer BNs. Even for the degenerated BMNs, their observability are different from the single-layer BNs' , for example the BCNs studied by Cheng, D. et al. 24 and Li, F. et al. 25 . Because even when the number of layer is one, our system still has holistic states, which have logical relationship with the states in basic layers. From above discussions, we can know that a study of the observability of the BMNs is meaningful and challenging.
In this article, by following this stream of research, we first address and characterize observability of BCNs defined on multiple topological layers. Based on the model of Boolean multiplex networks presented by Coozo et al. 43 , we introduce the input controls and the outputs. The model of BMCNs are changed into algebraic representation using STP tools. We consider the observability of BMCNs, following the standard formulation of the observability problem in systems theory, namely, we assume that the BN structure is known and that the goal is to infer the initial condition based on an output sequence. To clearly show the results of our research, we gave observable and unobservable examples in the final part of our essay.
The rest of this article is organized as follows. In Section II, we introduce the dynamic structure of BMCNs. In Section III, some concepts and properties of the STP are introduced, and we change our network into discrete form using STP tool. In Section IV, necessary and sufficient conditions for the observability of the BMCNs are obtained. Examples are given to show the effectiveness of the obtained results in Section V. Finally, a brief summary is given in Section VI.

Model of BMCNs
In this section, we introduce the model of BMCNs. For multiplex networks, different from the single-layer model, some nodes exist in multiple layers, the states which on different layers evolve independently of each other. The multiplex network we defined has N nodes per layer and K layers, and the number of total different nodes is n (N ≤ n ≤ NK). For example in Fig. 1, we have that N = 4, K = 2, n = 5. For statement ease, we define some related notions.
•  is the set of {0, 1}. which refers the set of j which has γ i,j,l = 1. And we set In Fig. 1, we have that the layers set of node 1 is  = (1) {1,2}, and a 1,1 = 1 and a 1,2 = 1, the layers set of node 2 , and a 2,1 = 1 and a 2,2 = 0. The incoming neighbors set of node 1 in layer 1 is Γ in (1) (1) = {1,4}, and . In each layer, for the specific ∈ … l K {1,2, , }, if a i,l = 1, we assume x t ( ) i l represents the state of node i on layer l at time t, then the update dynamics of state x i l can be described as , s e e F i g . 2 .
i i s i n f l u e n c e d b y , so we describe it as ), 1,2, , , When considering control-related problems for BMNs, based on above system structure, we introduce the m-dimensional control … ∈ u t u t u t ( ), ( ), , ( ) m 1 2  as the inputs of the system, correspondingly, then we have the outputs  … ∈ y t y t y t ( ), ( ), , ( ) p 1 2 , then the BMCNs can be described as where  f i is the canalizing function of node i with the controls … u t u t u t ( ), ( ), , ( ) m 1 2 , see Fig. 3. In finally, the output dynamics of the BMCNs are given by the following equation where h j is the output function. i , which determine the value of the holistic states. Even when the number of layer is one, our system still has holistic states, which have logical relationship with the states in basic layers through the canalizing functions. So it is still different from the one layer BNs.

Algebraic representation of BMCNs
In this section, we introduce some concepts and properties, changing our BMCNs into algebraic representation.
Concepts and properties of the semi-tensor product. In this subsection, some concepts and properties of the STP will be briefly introduced 36 .

Definition 1. 36
• Let X be a row vector of dimension np, and Y = [y 1 , y 2 ,… , y p ] T be a column vector of p dimension. Then we split X into p equal-size blocks as X 1 , X 2 , … , X p , which are 1 × n rows. Define the STP, denoted by , as A B and m n p q . If either n is a factor of p, say nt = p and denote it as  A B t , or p is a factor of n, say n = pt and denote it as  A B t , then we define the STP of A and B, denoted by  = C A B, as the following: C consists of m × q blocks as C = (C ij ) and each block is where A i is the i-th row of A and B j is the j-th column of B. And here we give some fundamental properties of the STP in the following 36 : 36 Assume A ∈ M m×n is given.
• Let Z ∈ R t be a row vector. Then It is easy to find out that STP of matrix can be seen as a generalization of conventional matrix product. All the fundamental properties of conventional matrix product, such as distributive rule, associative rule, remain true. So we can omit the symbol .
are positive integer constants. We can briefly denoted it as And the set of m × n logic matrices is denoted by  × m n .
Then we define a swap matrix ∈ × W m n mn mn [ , ]  , which is constructed in the following way: label its columns by (11,12, … , 1n, … , m1, m2, … , mn) and its rows by (11,21, … , m1, … , 1n, 2n, … , mn). And its element in the position ((I, J), (i, j)) is assigned as , , [ , ] For the logical function with n arguments → f : n  , we can convert it into an algebraic function using the STP of matrices. A logical domain, denoted by , is defined as . Based on this, we have Lemma 4. 36 Any logical function f(x 1 , x 2 , … , x n ) with logical arguments x 1 , x 2 , … , x n ∈ Δ , can be expressed in a multi-linear form as is unique, which is called the structure matrix of logical function f. And here we give another lemma: Here M r = δ 4 [1,4], which is power-reducing matrix and it can be verified that P 2 = M r P, ∀ P ∈ Δ . Based on the above properties of STP, we put forward an obvious proposition.
, we can find a matrix R i such that Algebraic structure of the BMCNs. In this subsection, we change our BMCNs into discrete version using STP tool. To express it more clearly, here we give some description of variables.
means the state of layer l.
means the state of all layers.
In the following step we will change the given BMCNs (3)-(4) into algebraic representation, as we will find out the algebraic relation between x(t + 1) and x(t) as well as the algebraic relation between +  x t ( 1) and x(t). At the first place, we will find out the algebraic relation between x(t + 1) and x(t). Using lemma 4 and proposition 1, for each logical rule f i l , we can find its structure matrix M i l , so we obtain that where Scientific RepoRts | 7:46495 | DOI: 10.1038/srep46495 , so we obtain that  . Subsequently, we will find out the algebraic relation between +  x t ( 1) and x(t). Using the similar steps above, the algebraic representation of (4) can be obtained as Scientific RepoRts | 7:46495 | DOI: 10.1038/srep46495 . So we obtain that Similarly, by letting = … y t y t y t y t ( ) ( ) ( ) ( ) , we obtain the algebraic expression of the output dynamics (5) as follows:  , here H j s are the structure matrixes of h j , j = 1, 2, … , p. Here we give an example to illustrate this process.
where ¬ ∨ ∧ → ↔ , , , and represent the logical functions of negation, disjunction, conjunction, implication, and equivalence, respectively. Based on Lemma 4, we obtain the corresponding structure matrices of those logical operators, as given in Table 1.
. Then we calculate the control-depending network transition matrix of system.

Observability of BMCNs
In this section, we will analyze and characterize the observability of the BMCNs, with two different types of controls. We first provide some definitions as follows. Consider the BMCN (3)-(4) with output dynamics (5). For any initial state ∈ ∆ x(0) , 2 NK and control input sequence =  u u u { (0), (1), }, the holistic trajectory at time t is denoted by  x t x u ( ; , (0)). Output trajectory at time t denote by y (; u, x(0)).

Definition 2. The BMCN (3)-(5) is observable if there exists a finite control sequence
In other words, there exists a control input sequence for which the initial state can be uniquely determined from the knowledge of the output sequence.

Remark 2.
Our definition is motivated by the definition of observability for BCNs proposed in Laschov, et al. 39 , which is different from the one proposed by Cheng, D. et al. 24 . In Cheng, D. et al. 24 , a BCN is said to be observable if the initial state can be uniquely determined from the knowledge of the control inputs (which may depend on the initial state) and the outputs.
We consider two kinds of controls. The first is that the controls are determined by certain logical rules, which called the input networks. , , , are logical function. According Lemma 4, we know that the input network (11) can be expressed as 1, , j are the structure matrix of logical function g i , respectively. Then, Theorem 1. Consider (3)-(5) (or equivalently (8)-(10)) with input network control (11). The system is observable if and only if there exists finite time s, s > 0, such that  Proof. By considering the input network, put together (8)-(9) with (12), we can obtain the system Lu t x t y t Hx t A straightforward computation shows, we calculate the output  (1) ( 1) (0) (0), Hence, in the matrix form, we obtain T

T T T c
From the solution structure of the system of linear algebraic equations, we know that for some initial control input δ = u (0) This completes the proof.

Remark 3.
From the proof of above theorem, we obtain that for some δ ∈ ∆ , , then the initial state x(0) can be reconstructed by the left inverse of δ ( ) In the following, we consider the case when the controls are free Boolean sequence. Precisely, m controls are described as  Proof. If the controls come from a free Boolean sequence, the system is that If free control inputs which is the all initial states of all layers in the initial time. Boolean control network (3)-(4) is observable if for the initial state , there exists finite time ∈ s , such that the initial state can be uniquely determined from the knowledge of the controls ... u u us (0), (1), , ( ) and the outputs ... y y y s (0), (1), , ( ). Based on the initial state ∈ ∆ x (0) 2 NK , we can easily obtain the holistic states are also observable.

Examples
In this section, we will give some examples to illustrate our results. Example 2 is a observable case and Example 3 is an unobservable case.

Example 2.
(Continue to Example 1) Consider the two-layer BMCN given in Example 1. Assume that the control inputs are determined by the following input network Then we obtain that δ = G [16,15,14,13,12,11,10,9,8,7,6,5,4, 3, 2, 1], 16 If we take δ = . u (0) 16 16  Then we have that And we can obtain that  δ = = rank( ) 2 16 c NK 16 16 . Then from Theorem 1, we know that the system is observable under the input network (18). Example 3. Consider following two-layer BMCN, with N = 2, K = 2, n = 3 and m = 1 x t u t x t 1: (   Then we calculate the control-depending network transition matrix of system.  .  x t x t x t x t x t x t x t

M u t x t M u t x t M u t x t M u t x t u t x t
Lu t x t x t x t x t x t

M u t M M u t x t M u t x t M u t M u t x t M u t M u t x t u t x t
Lu t x t Then, according to properties of STP, we obtain the matrix expression of output, as follows Now, we analyze the observability of this system, based on Theorem 2. We can calculate that while δ = x(0) 16   And while δ = x(0) 16 2 , we have that  Then, by induction, we easy obtain that, for any s > 0, and free control input =  Then we obtain that for arbitrary s > 0, we still have that  < rank ( ) 2 f NK , by Theorem 2, the system is unobservable.

Conclusions
In this paper, input controls were introduced into BMNs. By means of STP approach, the above logical dynamics has been converted into an algebraic form and the observability of dynamics is discussed. Firstly, we gave the theorem about the observability of whole dynamic system. Subsequently, the observability of each node in the special layer has been proved. Finally, we put forward some examples to illustrate our results.