This paper investigates the delay synchronization between two temporal Boolean networks base on semi-tensor product method, which improve complete synchronization. Necessary and sufficient conditions for delay synchronization are drawn base on algebraic expression of temporal Boolean networks. A example is presented to show the effectiveness of theoretical analysis.

Topics
Algebraic logic, Matrix methods, Mathematical modeling, Dynamical systems, Genomics, Human genome, Proteins, Gene network, Gene regulation, Systems biology

In 1990, the completion of the human genome project marked the beginning of life science into the era of systems biology. From then on, people study the running mechanism of biological systems focus on the level of the genome structure and function.

Systems biology research biological system, among which contain many different levels and different forms of organization of networks, including the micro protein molecules and signaling transduction network, macro ecological network and gene regulatory network, etc.1–4 Research of gene regulatory network starting from the interaction among genes, which reveal the essence of complex life phenomenon of organism. Gene regulatory network is the important content of functional genomics.

Gene regulatory networks research not only involves the knowledge of biology but also covers mathematics, control theory, computer science and other disciplines. Due to gene regulatory network of various physiological activities of organisms is very complex. At present, the structure of regulatory network can not be established by biological experiment. However, gene regulatory network topology can be built by mathematic analytical method,5–7 thus simulation real gene regulatory network. The establishment of mathematical models of gene regulatory network is an important mission for gene regulatory network. At present, there are various ways to establish gene regulatory network mathematical model, such as Boolean networks,8,9 bayesian networks10 and differential equation,11 piecewise linear differential equation,12 qualitative networks,13 etc. These methods all describe gene regulatory networks different perspectives.

Boolean networks (BNs) is the fundamental method which can systematically describe gene networks. In 1969, Kauffman firstly brought forward the concepts of Boolean networks.14 The abstract model of Boolean networks is built by Boolean logic and binary “0,1”, which study the biological activities. Boolean networks is an effective tool to describe the complex structure of gene regulatory network and analysis of the network dynamics. In the Kauffman proposed model, gene expression are abstracted as two states “0” and “1”: logic value“1” represents that gene is in the activated state, logic value “0” represents that gene is in the inhibitory state. The state of every gene in networks is in “0” or “1” at arbitrary moment. The state of gene is decided by gene regulation rules, which is run by Boolean function.14,15

In recent years, Boolean networks is researched widely, including time complexities for Boolean networks,16 symbolic dynamics of Boolean networks,17 controllability and observability of Boolean networks, etc. Synchronization is a typical collective behavior,18,19 which exist in the natural world, has pointed that physiological rhythms are central to life and synchronization of networks is essential for biological rhythms and information processing in biological organism. In the past decades, synchronization of dynamic system has been developed.20,21 The study of synchronization of Boolean networks is meaningful too, which can provide useful information on the coevolution of several biological species whose genetic dynamics influence each other.22 Besides, the synchronization of two coupled Boolean networks can be applied to the synchronization of two lasers.23 

Research on synchronization of Boolean networks have acquired some achievements. Li and Chu propose complete synchronization between two Boolean networks.24 Li etc. propose synchronization of Boolean networks with delay.25 Li etc. propose complete synchronization between two Boolean networks base on semi-tensor product.26 Li etc. propose complete synchronization between two Boolean networks with time delay.27 Li study complete synchronization between two large-scale Boolean networks.28 Li etc. study complete synchronization between two Boolean networks with delays.29 Lu etc. present complete synchronization between two output-coupled temporal Boolean networks.30 Zhong etc. present complete synchronization in array output-couple Boolean networks.31 Although there are some achievements on synchronization of Boolean networks, synchronization of Boolean networks is focused mainly on complete synchronization. According to biological knowledge, some genes will happen interaction in specific time and specific conditions. The internal and external of the networks are perturbation. So the delay synchronization is unavoidable in Boolean networks. There are few paper research on delay synchronization between two Boolean networks. So, we propose definition of delay synchronization and achieve delay synchronization of temporal Boolean networks.

The rest of this paper is organized as follows: In Section II, the semi-tensor product and the algorithm are briefly introduced. In Section III, the delay synchronization of temporal Boolean networks is defined and necessary and sufficient conditions are drawn. An illustrative example is given to illustrate correctness of theoretical analysis in Section IV. The conclusion is finally drawn in Section V.

The property and computations of the semi-tensor product is adopted in this paper, the semi-tensor product of matrices are introduced in the following sections.

Definition 2.1
Ref. 29 
The following is definition of semi-tensor product,
D ⋉ E ≜ ( D ⊗ I α / n ) ( E ⊗ I α / p ) ,
where matrices D ∈ Rm×n and E ∈ Rp×q, ⋉ symbols represents semi-tensor product of matrices operations, ⊗ symbols represents the Kronecker product, α is the least common multiple of n and p.

When n = p, the semi-tensor product operations become common matrix operations. In the following research, we omit the symbols of the semi-tensor product.

We need to know the following basic symbol.

  1. D ≜ 1 , 0 ⁠.

  2. Δ n ≜ δ n i i = 1 , 2 , … , n , where δ n i is i the column of the identity matrix Ik.

  3. Mm×n is the set of m × n real matrixes.

  4. 1m is 1 × k row vector which each element is 1, 1 m = [ 1 , 1 , … , 1 ] ︸ m ⁠.

  5. Coli(E)[Rowi(E)] is the i the column [line] of matrix E, the set of column of matrix E are expressed as Col(E).

  6. Let T be a matrix, T = [ δ n i 1 , δ n i 2 , … , δ n i r ] ∈ M m × n ⁠, T can be simply as T = δn[i1, i2, …, ir].

  7. Classical logic theory field: true (T ∼ 1) and false (F ∼ 0), which can be expressed as D = 0 , 1 ⁠. Logical values true and false can be expressed as T ∼ δ 2 1 and F ∼ δ 2 2 ⁠, respectively. So logic theory field can be expressed as D ⁠.

We need to know the following common methods for structuring structure matrix of the general logic operator.

  1. Dummy matrix : Given:c ∈ Δp, d ∈ Δq, then Dp,qW[p,q]cd = d, Dp,qW[p,q]cd = c, where D p , q = 1 p T ⊗ I q ⁠, 1 p = [ 1 , 1 , … , 1 ] ︸ p T , W [ p , q ] is swap matrix.

  2. Descending power matrix : Given: x ∈ Δk, then there are x2 = Mr,kx, where M r , k = diag ( δ k 1 , δ k 1 , … , δ k k ) ⁠.

Some necessary and sufficient conditions are summarized to achieve delay synchronization between two temporal Boolean networks.

The Boolean networks with n nodes can be showed as follows,

(1)
(2)
i = 1 , 2 , 3 , … , n ,

where Ci and Di represent nodes of drive networks (1) and response networks (2), respectively. h i : D n ( τ 1 + 1 ) → D , l i : D n ( τ 1 + τ 2 + 2 ) → D are Boolean functions, τ1 ≥ 0 and τ2 ≥ 0 are time delays.

To convert (1) and (2) into algebraic manipulation, we define C ( t ) = ⋉ i = 1 n C i ( t ) , x ( t ) = ⋉ i = 0 τ 1 C ( t − i ) ⁠. Let the structure matrix of hi and li be Mi and Ni, respectively. Then (1) can be expressed as follows,

(3)

Suppose

(4)

then

(5)

where Col i ( L 1 ) =   ⋉ j = 1 n Col i ( M j ) ⁠. Hence, the drive temporal Boolean networks (1) can be expressed as,

(6)
x ( t + 1 ) = ⋉ i = 0 τ 1 C ( t − i ) = C ( t + 1 ) C ( t ) … C ( t − τ 1 + 1 )
= L 1 x ( t ) C ( t ) … C ( t − τ 1 + 1 )
= L 1 x ( t ) C ( t ) … C ( t − τ 1 + 1 )
= L 1 W [ 2 n τ 1 , 2 n ( τ 1 + 1 ) ] Φ n τ 1 x ( t )
≜ F x ( t ) .

Defining D ( t ) = ⋉ i = 1 n D i ( t ) , y ( t ) = ⋉ i = 0 τ 2 D ( t − i ) ⁠, so

(7)
D i ( t + 1 ) = N i C ( t ) C ( t − 1 ) … C ( t − τ 1 ) D ( t ) D ( t − 1 ) … D ( t − τ 2 )
= N i x ( t ) y ( t ) .

So

(8)

where C o l i ( L 2 ) = ⋉ j = 1 n C o l i ( L j ) ⁠. Hence, the response temporal Boolean networks (2) can be expressed as,

(9)
y ( t + 1 ) = ⋉ i = 0 τ 2 D ( t + 1 − i )
= D ( t + 1 ) D ( t ) … D ( t − τ 2 + 1 )
= L 2 x ( t ) y ( t ) D ( t ) … D ( t − τ 2 + 1 )
= L 2 x ( t ) W [ 2 n τ 2 , 2 n ( τ 2 + 1 ) ] Φ n τ 2 y ( t )
= L 2 ( I 2 n ( τ 1 + 1 ) ⊗ W [ 2 n τ 2 , 2 n ( τ 2 + 1 ) ] Φ n τ 1 ) x ( t ) y ( t )
≜ G x ( t ) y ( t ) .

So Eqs.(6)- (9) can be summarized as follows

(10)
(11)

where F = L 1 W [ 2 n τ 1 , 2 n ( τ 1 + 1 ) ] Φ n τ 1 ∈ L 2 n ( τ 1 + 1 ) × 2 n ( τ 1 + 1 ) ⁠, G = L 2 ( I 2 n ( τ 1 + 1 ) ⊗ W [ 2 n τ 2 , 2 n ( τ 2 + 1 ) ] Φ n τ 1 ) ∈ L 2 n ( τ 2 + 1 ) × 2 n ( τ 1 + τ 2 + 2 ) ⁠.

In this section, some necessary and sufficient conditions are drawn for achieving the delay synchronization between networks (1) and (2).

Definition 3.1.

Delay synchronization can be achieved between temporal Boolean networks (1) and (2), for any initial value x ( 0 ) = ⋉ i = 0 τ 1 C ( − i ) ⁠, there is a positive integer s and s ≤ t − 1 such that C(t) = D(t − s) for all y ( 0 ) = ⋉ i = 0 τ 2 D ( − i ) ⁠.

Note: Mn stand for the operation of negation. C ( t )   =   ⋉ i = 1 n C i ( t ) ,   D ( t ) = ⋉ i = 1 n D i ( t ) , x ( t )   =   ⋉ i = 0 τ 1 C ( t − i ) , y ( t ) = ⋉ i = 0 τ 2 D ( t − i ) ⁠, where ⋉ i = 1 n : Δ 2 → Δ 2 n , ⋉ i = 1 τ 1 : Δ 2 n → Δ 2 n ( τ 1 + 1 ) , ⋉ i = 1 τ 2 : Δ 2 n → Δ 2 n ( τ 2 + 1 ) are bijective mappings.26 

Lemma 3.1.
(a) Assuming the initial state of Eq.(10) is x(0), therefore x(t) = Fx(t − 1) = ⋯ = Ftx(0).
(12)

Proof.

According to Eq. (10)-(11), the following Eq. (13) can be gotten

(13)
x ( t + 1 ) ⋉ y ( t + 1 ) = [ F ⋉ x ( t ) ] ⋉ [ G x ( t ) ⋉ y ( t ) ]
= F ⋉ ( I 2 n ( τ 1 + 1 ) ⊗ G ) ⋉ Φ n ( τ 1 + 1 ) ⋉ x ( t ) ⋉ y ( t )
= ( F ⊗ G ) ⋉ ( Φ n ( τ 1 + 1 ) ⊗ I 2 n ( τ 2 + 1 ) ) ⋉ ( x ( t ) ⋉ y ( t ) )
= ε ⋉ ( x ( t ) ⋉ y ( t ) ) .

So x(t)⋉y(t) = ε⋉(x(t − 1)⋉y(t − 1)) and y(t) = ε⋉ϑt−1⋉(x(0)⋉y(0)).

The necessary and sufficient conditions are established for achieving delay synchronization between Boolean networks (1) and (2).

Theorem 3.1.
Delay synchronization of Boolean networks (1) and (2) occurs if and only if there is an positive integer k such that
(14)
where ε is defined in Eq.(12).

Proof
Sufficiency

(15)
x ( k ) = F k x ( 0 ) = F k x ( 0 ) ( 1 2 n ( τ 2 + 1 ) y ( 0 ) ) = F k ( 1 2 n ( τ 1 + 1 ) ⊗ 1 2 n ( τ 2 + 1 ) ) ( x ( 0 ) y ( 0 ) )
= ( F k ⊗ 1 ) ( 1 2 n ( τ 1 + 1 ) ⊗ 1 2 n ( τ 2 + 1 ) ) ( x ( 0 ) y ( 0 ) )
= ( F k ⊗ 1 2 n ( τ 2 + 1 ) ) ( x ( 0 ) y ( 0 ) ) .

According to Eq.(15) and Lemma 3.1, the following formulas can be derived,
(16)
 
(17)
So, if Eq.(14) holds with a positive integer s( s ≤ k − 1 ), C(k) = D(k − s) for all x ( 0 ) = ⋉ i = 0 τ 1 C ( − i ) ∈ Δ 2 n ( τ 1 + 1 ) , y ( 0 ) = ⋉ i = 0 τ 2 D ( − i ) ∈ Δ 2 n ( τ 2 + 1 ) ⁠.
Assumptions about Fk+1 = δ2n(τ1+1)(α1, α2, …, α2n(τ1+1)), the following can be verified
(18)

The above Eq.(18) prove the Eq.(14) is true for t ≥ k. The networks (1) and (2) achieve delay synchronization.

(Necessity): Base on Definition 3.1, if the networks (1) and networks (2) achieve delay synchronization, then C(k) = D(k − s) for any x(0) ∈ Δ2n(τ1+1), y(0) ∈ Δ2n(τ2+1). k is a positive integer. These derivation demonstrate that
(19)
for any x(0) ∈ Δ2n(τ1+1), y(0) ∈ Δ2n(τ2+1). These prove the necessity.

The minimum of k is determined in the following section at last.

The following recursive relation can be established:
(20)
 
(21)
x ( t ) y ( t ) = ε t x ( 0 ) y ( 0 )
= ( F t x ( 0 ) ) [ G ε t − 1 x ( 0 ) y ( 0 ) ]
= ( F t ⊗ G ε t − 1 ) ( Φ n ( τ 1 + 1 ) ⊗ I 2 n ( τ 2 + 1 ) ) x ( 0 ) y ( 0 ) .

Analysis Eq. (21) can gotten εt = (Ft ⊗ Gεt−1)(Φn(τ1+1) ⊗ I2n(τ2+1)). Then

(22)
[ ( I 2 n ⊗ 1 2 n τ 1 ) ⊗ ( I 2 n ⊗ 1 2 n τ 2 ) ] ε t = [ ( I 2 n ⊗ 1 2 n τ 1 ) ⊗ ( I 2 n ⊗ 1 2 n τ 2 ) ] ( F t ⊗ G ε t − 1 ) ( Φ n ( τ 1 + 1 ) ⊗ I 2 n ( τ 2 + 1 ) )
= [ ( I 2 n ⊗ 1 2 n τ 1 ) F t ⊗ ( I 2 n ⊗ 1 2 n τ 2 ) G ε t − 1 ] ( Φ n ( τ 1 + 1 ) ⊗ I 2 n ( τ 2 + 1 ) ) .

Let T1, T2 and T3 be T1 = I2n ⊗ 12nτ1, T2 = I2n ⊗ 12nτ2, T = T1 ⊗ T2. Assuming that Fk = δ2n(τ1+1)(β1, …, β2n(τ1+1)), then
(23)
 
(24)
Base on Eq. (23) and Eq. (24), the following Eq.(25) can be gotten,
(25)

If there are k0 and s0, then

(26)
( I 2 n ⊗ 1 2 n τ 1 ) ( F k 0 ⊗ 1 2 n ( τ 2 + 1 ) ) = ( I 2 n ⊗ 1 2 n τ 2 ) ( Gε k 0 − s 0 − 1 )
= ( ( I 2 n ⊗ 1 2 n τ 1 ) F k 0 ) ⊗ 1 2 n ( τ 2 + 1 )
= δ 2 n ( ρ 1 ( k 0 ) , … , ρ 1 ( k 0 ) , ︸ 2 n ( τ 2 + 1 ) … , ρ 2 n ( τ 1 + 1 ) ( k 0 ) , … , ρ 2 n ( τ 1 + 1 ) ( k 0 ) ︸ 2 n ( τ 2 + 1 ) ) ,
where ρ 1 ( k 0 ) , … ρ 2 n ( τ 1 + 1 ) ( k 0 ) ∈ 1 , 2 , … , 2 n , ( ( I 2 n ⊗ 1 2 n τ 1 ) F k 0 ) ⊗ 1 2 n ( τ 2 + 1 ) = δ 2 n ( ρ 1 ( k 0 ) , … ρ 2 n ( τ 1 + 1 ) ( k 0 ) ) ⁠, for every t ≥ 1.

So T ε k 0   =   δ 2 2 n ( ( β 1 ( k 0 ) − 1 ) 2 n   +   β 1 ( k 0 ) , ( β 2 ( k 0 ) − 1 ) 2 n   +   β 2 ( k 0 ) , … , ( β 2 n ( τ 1 + 1 ) ( k 0 ) − 1 ) 2 n + β 2 n ( τ 1 + 1 ) ( k 0 ) ) ⊗ 1 2 n ( τ 2 + 1 ) ⁠. Supposing that k0 > b, Let h be h = min i ≥ 0 : ε b + i + 1 = ε b ⁠. There is a integer z and satisfy b ≤ z ≤ b + h, εz = εk0. Col(εb) = Col(εb+h+1) ⊆ Col(εz) ⊆ Col(εb), therefore
C o l ( ε b ) = C o l ( ε b + h + 1 ) = C o l ( ε k 0 ) .

Supposing that ε b = δ 2 n ( τ 1 + τ 2 + 2 ) ( β 1 ′ , … , β 2 n ( τ 1 + τ 2 + 2 ) ′ ) ⁠, then T ε b   =   ( C o l β ′ ( T ) , … , C o l β 2 n ( τ 1 + τ 2 + 2 ) ′ ( T ) ) and Col(Tεb) = Col(Tεk0). Let μ1, …, μ2n(τ1+τ2+2) be μ 1 , … , μ 2 n ( τ 1 + τ 2 + 2 ) ∈ 1 , … , 2 n ⁠, therefore (I2n ⊗ 12nτ2) Gεb−s′−1 = δ2n(μ1, …, μ2n(τ1+τ2+2)).

According to Eq.(22),
T ε b = δ 2 2 n ( ( ρ 1 ( b ) − 1 ) 2 n + μ 1 , … , ( ρ 1 ( b ) − 1 ) 2 n + μ 2 n ( τ 2 + 1 ) , … , ( ρ 2 ( b ) − 1 ) 2 n + μ 2 n ( τ 2 + 1 ) + 1 , … , ( ρ 2 ( b ) − 1 ) 2 n + μ 2 n ( τ 2 + 1 ) + 1 , … , ( ρ 2 n ( τ 1 + 1 ) ( b ) − 1 ) 2 n + μ ( 2 n ( τ 2 + 1 ) − 1 ) 2 n ( τ 2 + 1 ) , … , ( ρ 2 n ( τ 1 + 1 ) ( b ) − 1 ) 2 n + μ 2 n ( τ 1 + τ 2 + 2 ) ) .
So,
ρ i ( b ) = μ ( i − 1 ) 2 n ( τ 2 + 1 ) + j , i = 1 , 2 , … , 2 n ( τ 1 + 1 ) , j = 1 , 2 , … , 2 n ( τ 2 + 1 ) ,
 
( I 2 n ⊗ 1 2 n τ 2 ) Gε b − s ′ − 1 = δ 2 n ( μ 1 , … , μ 2 n ( τ 1 + τ 2 + 2 ) ) = δ 2 n ( ρ 1 ( b ) , … , ρ 1 ( b ) , ︸ 2 n ( τ 2 + 1 ) … , ρ 2 n ( τ 1 + 1 ) ( b ) , … , ρ 2 n ( τ 1 + 1 ) ( b ) ︸ 2 n ( τ 2 + 1 ) ) = δ 2 n ( ρ 1 ( b ) , … , ρ 2 n ( τ 1 + 1 ) ( b ) ) ⊗ 1 2 n ( τ 2 + 1 ) = ( ( I 2 n ⊗ 1 2 n τ 2 ) F b ) ⊗ 1 2 n ( τ 2 + 1 ) = ( I 2 n ⊗ 1 2 n τ 2 ) ( F b ⊗ 1 2 n ( τ 2 + 1 ) ) ,
these can illustrate contradicting the minimality of k0. The proof is complete.

Let s be s = 1, then C(t) = D(t − 1). Let the drive Boolean networks (1) be the following equation,

(27)

Let C(t) be C ( t ) = ⋉ i = 1 2 C i ( t ) ⁠. Let x(t) be x ( t ) = ⋉ i = 0 1 C ( t − i ) ⁠. So

(28)

where Mc is the structure matrice of logical function “∧″ and Md is the structure matrice of logical function “∨″. So

(29)

where C o l i ( L 1 ) = ⋉ j = 1 2 C o l i ( M i ) ⁠. Then

(30)

Hence, Boolean networks (27) can be expressed the following the algebraic format,

(31)

with F = δ16[1, 1, 1, 1, 2, 2, 10, 10, 7, 3, 7, 3, 8, 4, 16, 12].

Let the response Boolean networks be the following equation,

(32)

Let D(t) and y(t) be D ( t ) = ⋉ i = 1 2 D i ( t ) and y ( t ) = ⋉ i = 0 1 D ( t − i ) ⁠, so

(33)

Thus D(t + 1) = D1(t + 1) D2(t + 1) = L2x(t) y(t), where C o l i ( L 2 ) = ⋉ j = 1 2 C o l i ( N j ) ⁠. Then

y ( t + 1 ) = D ( t + 1 ) D ( t ) = L 2 x ( t ) y ( t ) B ( t ) = L 2 ( I 16 ⊗ W [ 4 , 16 ] ) x ( t ) y ( t ) ≜ G x ( t ) y ( t ) ,

Therefore, Boolean networks (32) can be expressed the following the algebraic format,

y ( t + 1 ) = G x ( t ) y ( t ) ,
G = δ 16 [ 2 , 1 , 2 , 3 , 1 , 1 , 3 , 3 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 1 , 2 , 5 , 5 , 5 , 8 , 8 , 8 , 7 , 7 , 10 , 9 , 9 , 9 , 11 , 11 , 12 , 12 , 12 , 14 , 14 , 15 , 15 , 13 , 13 , 7 , 7 ] .

So

(34)

These illustrative example prove the BN (32) and BN (27) can delay synchronization according to Lemma 3.1.

Remark.

According to Eq.(34), the BN (32) and BN (27) can enter a fixed point δ 4 1 ⁠.

The delay synchronization for temporal Boolean networks is investigated in this paper. Delay synchronization has been achieved and sufficiency and necessity conditions are given based on semi-tensor product. Compared with complete synchronization, delay synchronization is more common and more general. An example has been worked out to prove the correctness of the theoretical analyses.

Synchronization of Boolean networks need be further investigated, such as feedback synchronization, limited number of iterations synchronization, etc.

The first author would like to thank the support of Doctor Start-up capital of the Beihua University (Nos: 199500096). The second author would like to thank the support of the National Natural Science Foundation of China (No: 61363082). The third author would like to thank the support of the Natural Science Foundation of the Xinjiang Uygur Autonomous Region (Nos:201442137-26). The fourth author would like to thank the support of the Special Fundation for Outstanding Young Doctor of Dalian University (No. 2014YL07). The fifth author would like to thank the support of the Natural Science Foundation of Sichuan Province (No. 2016JY0179), the Innovation Group Build Plan for the Universities in Sichuan (No. 15TD0024), the Youth Science and Technology Innovation Group of Sichuan Provincial (No.2015TD0022), the High-level Innovative Talents Plan of Sichuan University of Science and Engineering (2014), and the Talents Project of Sichuan University of Science and Engineering (No. 2015RC50).

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2016
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