Introduction

Human Genome Project, which is an international scientific research project with the goal of determining the sequence of nucleotide base pairs1, 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 network2. The Boolean Networks (BNs) as a powerful tool in describing, analyzing, and simulating the cellular networks, has been most widely used3,4,5,6,7,8,9,10,11,12,13,14,15,16,17.

From decades ago, Kauffman put forward the theory which can describe the net of cell and gene using BNs4. And he made expatiation about the relationship between BNs and gene as well as life5,6. 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, S. et al. talked about the Boolean modeling and analyzing of biological system10,11. Aldana, M. et al. studied the topological structure of BNs7. Akutsu, T. et al. and Albert, R. et al. considered the dynamic features of BNs12,13. More detailedly and recently, Lu, J. et al. analyzed the synchronization problem of master-slave probabilistic BNs18.

In recent years accompany with the development of biology, control of biological system becomes into a hot topic19,20,21,22,23,24,25,26,27,28,29. As to the research of genetic regulatory networks (GRNs), one of the major goals is to carry out the therapeutic intervention strategies for diseased targets30,31. 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 GRNs32, but also other various fields, such as man-machine dynamic game33 and internal combustion engines34,35. Recently, based on semi-tensor product (STP) proposed by Cheng, D. et al.36, many basic problems for BCNs have been investigated, for example, realization23, controllability24,26, optimal control15,33, 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 structure37. In evolutionary dynamics, observability is the key to study whether the genetic process itself can be recovered from measurements of phenotypic characteristics38. 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 outputs39. There have been many studies on the observability of BCNs in recent years. Cheng, D. et al. have investigated the controllability and observability of BCNs24. Li, F. et al. have studied the observability of time-delayed BCNs25. Laschov, D. et al. have considered the observability of BNs through a graph-theoretic approach39. Zhang, K. et al. have proposed a unified approach to determine all the four types of observability of BCNs in the literature40.

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 organism41. 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 201042, 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 parallel43. Not only in cellular biochemical, multiplex networks have been applied to the natural, social, and information sciences42. As an old concept, multi-layer social networks have been studied from 197544. 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 networks45. 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.

Figure 1: Schematic of multiple networks with two layers.
figure 1

Here K = 2 means that the system is a two layers network. N = 4 means that there are four nodes in each layer. And n = 5 means that there are five total different nodes in the system. are the states in layer 1, and are the states in layer 2.

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Recent years more and more researchers studied the BMNs. For example, Xu, M. et al. investigated the synchronizability of two-layer networks46. Cozzo talked about the stability of BMNs43. Luo et al. studied the controllability of BMCNs27. Zhong, J. et al. studied controllability problem for multi-level Boolean control networks47. 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}.

  • , and ai,l = 1 if node i in the l layer and 0 otherwise. The layers set of node i is which refers the set of l which has ai,l = 1.

  • , and γi,j,l = 1 if node j is the incoming neighbors of node i in the l layer. The incoming neighbors set of node i in the l layer is which refers the set of j which has γi,j,l = 1. And we set .

In Fig. 1, we have that the layers set of node 1 is , and a1,1 = 1 and a1,2 = 1, the layers set of node 2 is , and a2,1 = 1 and a2,2 = 0. The incoming neighbors set of node 1 in layer 1 is Γin(1)(1) = {1,4}, and .

In each layer, for the specific , if ai,l = 1, we assume represents the state of node i on layer l at time t, then the update dynamics of state can be described as

where , is the update rule of node i on layer l.

Furthermore, assume represents the holistic states of node i at time t, which means the global states of , see Fig. 2. is influenced by , so we describe it as

Figure 2: Schematic illustration of the relationship of node states in the fixed layers with the holistic states in BMNs.
figure 2

Where, represent the holistic states of BMNs. For example is the holistic state of and . It is affected by and through canalizing function . The second node is only existed in layer one. So holistic state is only affected by .

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where.. is the canalizing function.

When considering control-related problems for BMNs, based on above system structure, we introduce the m-dimensional control as the inputs of the system, correspondingly, then we have the outputs , then the BMCNs can be described as

and

where is the canalizing function of node i with the controls , see Fig. 3.

Figure 3: Schematic illustration of BNs with control and output.
figure 3

The inputs m dimension control have been introduced. denote outputs. From the figure, we see that inputs affect the node states in each layers as well as the abstract holistic states. And outputs are affected by holistic states .

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In finally, the output dynamics of the BMCNs are given by the following equation

where hj is the output function.

Remark 1. The BMCNs are not the simple superposition of K single-layer BCNs. Because we have the holistic states which are affected by corresponding states in each layers. Between the holistic states with the states in basic layers, we have the canalizing functions, 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 introduced36.

Definition 1. 36

  • Let X be a row vector of dimension np, and Y = [y1, y2,…, yp]T be a column vector of p dimension. Then we split X into p equal-size blocks as X1, X2, …, Xp, which are 1 × n rows. Define the STP, denoted by , as

  • Let . If either n is a factor of p, say nt = p and denote it as , or p is a factor of n, say n = pt and denote it as , then we define the STP of A and B, denoted by , as the following: C consists of m × q blocks as C = (Cij) and each block is

where Ai is the i–th row of A and Bj is the j–th column of B.

And here we give some fundamental properties of the STP in the following36:

Lemma 1.36 Assume , then (where ⊗ is the Kronecker product, It is the identity matrix).

Assume , then

Lemma 2.36 Assume A ∈ Mm×n is given.

  • Let Z ∈ Rt be a row vector. Then

  • Let Z ∈ Rt be a column 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 .

Here we defined some notions for statement ease.

  • , where denotes the i–th column of the identify matrix In.

  • Assume a matrix , where are positive integer constants. We can briefly denoted it as .

  • A matrix is called a logical matrix if the columns of A, denoted by Col (A), satisfy . And the set of m × n logic matrices is denoted by .

Then we define a swap matrix , 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

And we denote W[n] = W[n, n] when m = n.

Lemma 3.36 Let and be two columns. Then

For the logical function with n arguments , we can convert it into an algebraic function using the STP of matrices. A logical domain, denoted by , is defined as . We identify each element in with a vector as , and . Based on this, we have

Lemma 4.36 Any logical function f(x1, x2, …, xn) with logical arguments x1, x2, …, xn ∈ Δ, can be expressed in a multi-linear form as

where is unique, which is called the structure matrix of logical function f.

And here we give another lemma:

Lemma 5.36 Assume , then

where

Here Mr = δ4[1, 4], which is power-reducing matrix and it can be verified that P2 = MrP, ∀P ∈ Δ.

Based on the above properties of STP, we put forward an obvious proposition.

Proposition 1. For each , if , we can find a matrix Ri 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 control.

  • 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 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 , we can find its structure matrix , so we obtain that

where , and . Hence

with . And we have defined that , so we obtain that

where .

Subsequently, we will find out the algebraic relation between and x(t). Using the similar steps above, the algebraic representation of (4) can be obtained as

where is the structure matrix of logical function . And we have that . So we obtain that

where .

Means that

Similarly, by letting , we obtain the algebraic expression of the output dynamics (5) as follows:

where , here Hjs are the structure matrixes of hj, j = 1, 2, …, p.

Here we give an example to illustrate this process.

Example 1. Consider following two-layer BMCN, with N = 2, K = 2, n = 4 and m = 4

and we have that

where 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.

Table 1 Structure matrices of some basic logical functions.
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Define . Then we calculate the control-depending network transition matrix of system.

and we have that

Then

Here we have found out the algebraic relation between x(t + 1) and x(t) as well as the algebraic relation between and x(t). Furthermore, we assume that

Then, according to properties of STP, we obtain the matrix expression of output, as follows

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 and control input sequence , the holistic trajectory at time t is denoted by . 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 , with s > 0, such that for any with i ≠ j, we have for some .

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.

where , are logical function.

According Lemma 4, we know that the input network (11) can be expressed as

where are the structure matrix of logical function gi, respectively. Then,

with .

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 , for some , where

Proof. By considering the input network, put together (8)-(9) with (12), we can obtain the system

A straightforward computation shows, we calculate the output as follows

Hence, in the matrix form, we obtain

From the solution structure of the system of linear algebraic equations, we know that for some initial control input , the system of linear equations (14) with 2NK-dimension unknown vector x(0) has a unique solution if and only if the system matrix has rank 2NK. That is, for some initial control input , the initial state x(0) is uniquely determined by the knowledge of the output sequence if and only if

This completes the proof.

Remark 3. From the proof of above theorem, we obtain that for some if the matrix has full column rank, means that , then the initial state x(0) can be reconstructed by the left inverse of operation on output sequence ,

In the following, we consider the case when the controls are free Boolean sequence. Precisely, m controls are described as and freely designed.

Theorem 2. Consider (3)-(4) and (5) (or equivalently (8), (9) and (10)), with a free Boolean sequence control. The system is observable if and only if there exists a finite control sequence with such that , where

Proof. If the controls come from a free Boolean sequence, the system is that

If free control inputs are given, then a straightforward computation shows the following:

Hence, in the matrix form, we obtain

As a similar analysis discussed in the proof of Theorem 1, we know that for a given free control inputs , the system of linear equations (17) with 2NK-dimension unknown vector x(0) has a unique solution if and only if the system matrix has rank 2NK. That is, for a given free control inputs , the initial state x(0) is uniquely determined by the knowledge of the output sequence if and only if . Furthermore, as mentioned in Remark 3, the initial state x(0) can exactly calculated as This completes the proof.

Remark 4. The observability in our paper is the observability of 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 , such that the initial state can be uniquely determined from the knowledge of the controls and the outputs . Based on the initial state , we can easily obtain the holistic states through the canalizing function . So 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

If we take By calculation, we have

Then we have that

And we can obtain that . 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

and we have that

Then we calculate the control-depending network transition matrix of system.

and we have that

Then

Furthermore, we assume that

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 ,

And while , we have that

Then, by induction, we easy obtain that, for any s > 0, and free control input , , and furthermore, . That implies, for any s > 0, and free control input

So the linear homogeneous equation

has the non-zero solution. Then we obtain that for arbitrary s > 0, we still have that , 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.

Additional Information

How to cite this article: Wu, Y. et al. Observability of Boolean multiplex control networks. Sci. Rep. 7, 46495; doi: 10.1038/srep46495 (2017).

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