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State analysis of Boolean control networks with impulsive and uncertain disturbances
Applied Mathematics and Computation 301 (2017) 187–192
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
State analysis of Boolean control networks with impulsive and uncertain disturbances Bo Gao a, Zheng-hong Deng b, Da-wei Zhao c,∗, Qun Song b a
School of Computer Information management, Inner Mongolia University of Finance and Economics, Hohhot 010051, China School of Automation, Northwestern Polytechnical University, Xi’an 710072, China c Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Jinan 250014, China b
a r t i c l e
i n f o
Keywords: Boolean Control Networks Semi-tensor Product Disturbance
a b s t r a c t In this paper, we put forward a method for the state analysis of Boolean control networks with impulsive and uncertain disturbances. In detail, we introduce the probability in the analysis of the state of Boolean networks. Mathematically, the STP as the analytical tool, has be used for the modeling of uncertainty. In addition, we give a mathematical proof for the model. Lastly, we conduct some simulations which validate the efficiency for this model. © 2016 Elsevier Inc. All rights reserved.
1. Introduction More and more attention has been paid to the Genetic regulatory network (GRN), a new research field in the biological science, during the last few years [1,2]. GRN, as a newly developing topic, has a close relationship with protein webs, neural networks and other biological systems [3–5]. Several types of genetic regulatory networks are proposed in recent years, including Markov-type genetic networks [6], probabilistic Boolean networks [7] and Boolean networks [8]. The Boolean network (BN), with a simple structure, was proposed initially for modeling and quantitatively describing the gene circuits. It has been widely applied in different fields, such as biology, physics, systems science and so on (see, e.g., [9–11]). There are mere two states of each node in a Boolean network: 1 (active) or 0 (inactive) at each discrete time, and each node evolves its state according to a logical Boolean function. As the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, Boolean network might be suitable for modeling GRN and biological systems, as well as other cellular processes. It is especially common in biological networks that many evolution processes may occur abrupt changes of states at certain time instants, which may be due to many factors, such as changes in the interconnections of subsystems, sudden changes of external environment. To describe the evolution of a process with a short-time perturbation, it is need to approximate the perturbation and neglect the duration as impulse disturbances, which may be caused by sudden noise or switching phenomenon. There are some literatures with impulsive disturbances [12,13]. However, there is a little research on studying the impulsive disturbances on Boolean control networks (BCN) [14]. In short, to investigate the Boolean control networks with impulsive disturbances still remains an open crucial theoretical problem In the present note, global stability and stabilization of switched Boolean network with impulsive effects are considered [15]. Ref. [16] investigates a Mayer-type optimal control and minimum-time control of a Boolean control network with ∗
Corresponding author. E-mail addresses: [email protected] (B. Gao), [email protected] (D.-w. Zhao).
http://dx.doi.org/10.1016/j.amc.2016.11.040 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
188
B. Gao et al. / Applied Mathematics and Computation 301 (2017) 187–192
impulsive disturbances. The stabilization, controllability and optimal control problem of Boolean networks with impulsive effects and state constraints are investigated [17]. These results are based on the known control impulse, however, there is a lack of research for uncertain control impulse [18–20]. Here, Semi-tensor Product is adopted as a mathematical tool to study the state analysis of BCN with impulsive and uncertain disturbances. We propose an algebraic approach to estimate the probability of each state in BCN with uncertain disturbances sequence or disturbances with the probability of each value. Unlike other existing methods, our approach, on the basis of clear and strict evidences, and the simulation results validation the efficiency for this model.
2. Preliminaries Semi-tensor Product (STP) is an extension of traditional matrix multiplication, which can make the multiplication between matrix with unequal rows and columns into reality. In this article, the symbol represents the calculation of STP [21]. Definition 2.1. Suppose that x is a lk dimensional row vector, y = {y1 , y2 ,…, yl } is a l dimensional column vector. The vector x is divided into l block, and each block has the same dimension, namely the size of each block is 1 × k. The calculation of STP as follow:
xy=
yT xT =
l l
i=1
i=1
xi yi ∈ Rn
(2.1)
yi ( xi ) ∈ Rn T
According to the mathematically meaning of STP, it is immediate that the matrix satisfied:
Ar×kl Bl×t = Cr×kt , Ar×l Bkl×t = Ckr×t .
(2.2)
Here, Ar×kl is a matrix with r row and kl column, and Br×kl is a matrix with l row and t column, Cr×kt is a matrix with r row and kt column. The Eq. (2.2) shows the variation of STP. Example 1. Consider two matrixes, where A = [15
AB = =
2 3 4 ] 6 7 8
and B = [10 20]. Then
(1 2 ) × 1 + (3 4 ) × 0 (1 2 ) × 0 + ( 3 4 ) × 2 ( 5 6 ) × 1 + (7 8 ) × 0 (5 6 ) × 0 + (7 8 ) × 2 1 5
2 6
6 14
8 . 16
Next, we introduce the definition of logical matrix, the logical calculation can transform into matrix multiplication according to the mathematical property of STP. Suppose that Ik is a unit matrix which can represented as Ik = {δki |i = 1, 2, . . . , k}, here δki is the ith column of unit matrix. So the logic number “0” and “1” can be denoted by δ22 = [10] and
δ21 = [10] respectively. i
i
in The logic calculation rule can be presented as matrix M = [δm1 , δm2 , . . . , δm ], which can be abbreviated as M = δ m [i1 , i2 ,…, in ]. The typical logic calculation are: logical “not” ¬ A, logical “and” A ∧ B, logical “or” A∨B, and logical “x or” AB. Accordingly, the logic calculation can be represented as:
M¬ = δ2 [2, 1]; M∧ = δ2 [1, 2, 2, 2]; M∨ = δ2 [1, 1, 1, 2]; M = δ2 [2, 1, 1, 2]. Theorem 2.1. A logical function f(x1 , x2 ,…, xn ), which have n dimensional variable x1 , x2 ,…, xn , can be represented as a linear mapping form:
f ( x1 , x2 , . . . , xn ) = M f x1 x2 · · · xn . Here, Mf is a 2 × 2n dimensional matrix.
(2.3)
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3. Main results 3.1. Algebraic form of BCNs with impulsive disturbances The model of Boolean networks with impulsive disturbances as follow:
⎧ x1 (t + 1 ) = f1 (x1 (t ), . . . , xn (t )), ⎪ ⎪ ⎨x2 (t + 1 ) = f2 (x1 (t ), . . . , xn (t )),
(1 ) . . ⎪ ⎪ ⎩. ⎧xn (t + 1 ) = x1 (t + 1 ) = ⎪ ⎪ ⎨x2 (t + 1 ) = (2 ) . . ⎪ ⎪ ⎩. xn (t + 1 ) =
t = k
fn (x1 (t ), . . . , xn (t )), f 1 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )), f 2 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )),
(3.1) t=k
f n (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t ))
This model contains two equations. When t = k, the system did not suffer the effect of impulsive disturbances, the function of each node is fi (i = 1, 2,…, n), which is a multivariate function with n variables, and x1 (t),…, xn (t) is stands for the state of node xi at the tth step. When t = k, the function of each node is f i (i = 1, 2,…, n), but the function is a multivariate function with n + m variables, and x1 (t),…, xn (t) is stands for the state of node xi at the tth step, and u1 (t),…, um (t) is impulsive disturbances at the tth step, so k is the step when the impulsive disturbances works. Suppose that the length of impulsive sequence equals to l, namely k = s, s + 1,…, s + l − 1, s ∈ Z + . According to the property of STP, the system (3.1) has the following conclusions: suppose that x(t ) = x1 (t )x2 (t ) . . . xn (t ) = ni=1 xi (t ), here xi (t) ∈ I2 ; and u(t ) = u1 (t )u2 (t ) . . . um (t ) = m u (t ), uj (t) ∈ I2 , therefore, system (3.1) can be transferred j=1 j into the following form [22]:
(1 )x(t + 1 ) = Lx(t ), t = k, (2 )x(t + 1 ) = L u(t )x(t ), t = k,
(3.2)
Here, L ∈ M2n ×2n , L ∈ M2n ×2n+m . In each column of matrix L, there exists only one entity “1”, which stands for the number of the following state. Also the matrix L is the transform matrix of system (3.1)(1). Likewise, In each column of matrix L , there exist only one entity “1”, which stands for the number of the following state. Matrix L is the transform matrix of system (3.1)(2). Matrixes L and L are the fundamental results of this paper. When the length of impulsive sequence equals tol = 1, the Eq. (3.1)(2) become a new form: x(t + 1) = L x(t)u(t). Therefore, we can have the following theorem.
v1 ], u (t ) = [ v2 ], …, um (t ) = [ vm ], here v (i = 1, 2,…, m) is the realization of input, Theorem 3.1. Suppose u1 (t ) = [1 − 2 i v 1−v 1−v 1
2
m
and the value of vi is either “0” or “1”, so the Eq. (3.1)(2) can obtained by the following equation:
L = L u1 (t ) u2 (t ) · · · um (t )
(3.3)
Here, L ∈ M2n ×2n .
Suppose that the Eq. (3.1)(2) has the initial state η0 , and η0 = ni=1 xi (0 ), the u(t ) = m u (t ), so the target state ηd can j=1 j be acquire by:
ηd = L u(t ) η0
(3.4)
Here, u(t ) ∈ I2m ×1 , ηd ∈ I2n ×1 . Proof. According to Eq. (3.2), we can conclude that matrix L is the linear expression of (3.1)(2). Therefore, compare with the property of STP, it can be concluded that when the initial state of system is η0 , the target state is ηd = L u(t)η0 when the input is u1 (t),…, um (t). 3.2. Uncertain disturbances sequence When we study the impulsive disturbances, it always can meet the situation that the disturbances cannot be well measured. So, in this section we study how to describe the state of the system when the input is an uncertain disturbances sequence. The Boolean network with disturbances sequence input can be linearized by (3.2)(2). When the length of the sequence equals to l, and k = 0, 1,…, l − 1, then the system (3.2)(2) can be described as:
x ( l ) = ( L )l u ( 0 ) u ( 1 ) · · · u ( l − 1 ) x ( 0 )
(3.5)
Here, (L )l ∈ M2n ×2n+m+l−1 , x(0) = x1 (0)x2 (0)xn (0) is the initial state of system with impulsive disturbances, u(0) = u1 (0)u2 (0)um (0), ……, u(l − 1) = u1 (l − 1)u2 (l − 1)um (l − 1).
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According to Eq. (3.5), we can see that as the increasing of length l, the increasing speed of matrix (L )l is very fast. Next, we put forward a method to reduction the scale of matrix (L )l . Based on this results, we analyze the effect of uncertain disturbances sequence foe the state of the system. Theorem 3.2. Suppose the input disturbances sequence of system (3.1)(2) equals to l, and the inputs u(0), u(1),…, u(l − 1) are uncertain. So the state that may occur of system (3.1)(2) can be described by state vector S,
S = (L )l 12n+m+l−1 Here, 12n+m+l−1
(3.6)
⎡ ⎤⎫ ⎡ ⎤ 1 ⎬ s1 s2 1 = ⎣ .. ⎦ 2n+m+l−1 is a column vector, which has 2n+m+l−1 row. S = ⎣ .. ⎦ is a column vector, which has . . ⎭ s2n
1 2n row., where si is the number of state i occurred in the system.
Proof. According to Eq. (3.5), (L )l is a matrix with 2n + m + l − 1 column, and each column has only one “1”. Suppose that ⎡ ⎤ ⎡ ⎤ a1 a1 a a2 2 l l l ⎣ ⎦ ⎣ . . ⎦1 n+m+l−1 , ai 1 n+m+l−1 = (L ) = .. , ai = [ωi1 , ωi2 , . . . , ωi2n+m+l−1 ] is the row of matrix (L ) . Thus, (L ) 12n+m+l−1 = .. 2 2 a2n a2n 2n+m+l−1 ωik . k=1 Therefore, si is the number of state i occurred in the system. According to the property of STP, when m and l are variable, the values of S are varied. In order to exclude the changing of m and l, we compute the occurrence value of each state when there exists uncertain disturbances in the system. Theorem 3.3. Suppose the input disturbances sequence of system (3.1)(2) equals to l, and the inputs u(0), u(1),…, u(l − 1) are uncertain. So the probability of each state that may occur of system (3.1)(2) can be described by state probability vector: Sp ,
Sp =
S 2n+m+l−1
(3.7)
⎤⎫ s p1 ⎪ ⎬ ⎢ s. ⎥ n Here, S p = ⎣ p2 .. ⎦⎪2 is a column vector, and spi is the probability that state i occurs. ⎭ s n ⎡
p2
Because Sp is a vector, and whose entities are probability, so we can derived that
2n
i=1 s pi
=1
3.3. Disturbances with the probability of each value In this section, we study another case for uncertain input disturbances. Suppose that the input of system (3.1)(2) are u1 (t), …, um (t), and the value of unknown input is either “0” or “1”, but we can compute the probability of the occurrence of “0” and “1”. In the section, we give a framework for the description of the type of system. p1 Theorem 3.4. Suppose the input of system (3.1)(2) are u1 (t),…, um (t), and u1 (t ) = [1−p ], here p1 is the probability of input 1
p2 pm value is “1”, so, 1 − p1 is the probability of input value is “0”, likewise, u2 (t ) = [1−p ], ……, um (t ) = [1−p ]. 2
m
Therefore, the system (3.1)(2) can be describe probability vector Vp :
Vp =
(L u1 (t ) u2 (t ) · · · um (t )) 12n 2n
(3.8)
⎤⎫ v p1 ⎪ ⎬ ⎢ v. ⎥ n Here, Va = ⎣ p2 .. ⎦⎪2 is a column vector, and vpi is the probability that state i occurs. v p2n ⎭ ⎡
Proof. According to Eq. (3.2)(2), (L )l is a matrix with dimensional 2n × 2n + m , and each column has only one “1”. ⎡ ⎤ b1 b2 (b u (t )u2 (t )···um (t ))12n Suppose that L = ⎣ .. ⎦, bi = [ωi1 , ωi2 , . . . , ωi2n+m ] is the row of matrix (L )l . Thus, v pi = i 1 , here 2n . b2n n+m (bi u1 (t ) u2 (t ) · · · um (t )) 12n = 2k=1 ωik is the sum of ith row of L , 2n is the number of occurrence. vpi is the probability that state i occurs. n Because Vp is a vector, and whose entities are probability, so we can derived that 2i=1 v pi = 1.
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4. Example In order to illustrate our main results, let us consider the Boolean control networks model, which is a reduced Boolean model for the lac operon in the bacterium Escherichia coli [23]. The model is given as follows:
x1 (t + 1 ) = x2 (t ) ∨ x3 (t ) (1 ) x2 (t + 1 ) = x1 (t ) t = k x ( t + 1 ) = x ( t ) 3 1 x1 (t + 1 ) = x2 (t ) ∨ x3 (t ) (2 ) x2 (t + 1 ) = x1 (t ) t=k x3 (t + 1 ) = x1 (t ) ∧ u(t )
(4.1)
Here, x1 (t) represent for lac mRNA, x2 (t) represent for lactose with higher concentration, x2 (t) represent for lactose with medium concentration, u(t) represent for lactose. According to the calculation of Eq. (3.2)(1), the matrixLcan be represented as L = δ 8 [11154448]. The sequence length of impulsive disturbances l = 1. So the Eq. (3.2)(2) can be represented as: L = δ 8 [1115444822264448]. When the input u(k) is 1, we have u(k ) = [10], so L u(k) = δ 8 [11154448]; likewise, when the input u(k) is 0, we have u(k ) = [10], so L u(k) = δ 8 [22264448].
⎡ ⎤ 3
When the input value is uncertain, S = (L )l 12n+m+l−1
⎢3⎥ ⎢0⎥ ⎢ ⎥ ⎢6⎥ ⎥ =⎢ ⎢1⎥. ⎢ ⎥ ⎢1⎥ ⎣ ⎦
0 2 ⎡ ⎤ 18.75% ⎢18.75%⎥ ⎢ 0% ⎥ ⎢ ⎥ ⎢ 37.5% ⎥ V ⎢ ⎥. The probability of each state equals to S p = n+m+l−1 = ⎢ ⎥ 2 ⎢ 6.25% ⎥ ⎢ 6.25% ⎥ ⎣ ⎦ 0% 12.5% Suppose that the input u(k) is a vector, whose value equals to 1 with probability 70%, and value equals to 0 with probability 30%, so the input u(k) can be presented as u(k ) = [00..73].
⎛⎡0.7
Vp =
L u ( k ) 1 2n 2n
⎜⎢ 0 ⎜⎢ 0 ⎜⎢ ⎜⎢0.3 = ⎜⎢ ⎜⎢ 0 ⎜⎢ 0 ⎝⎣ 0 0
⎡2.1⎤
⎡26.25%⎤
0 0.6
0% 7.5%
0 0.7 0 0.3 0 0 0 0
0.7 0 0 0.3 0 0 0 0
0 0.7 0 0.3 0 0 0 0
0.7 0 0 0.3 0 0 0 0
0 0.7 0 0.3 0 0 0 0
0 0 0 0 0.7 0 0 0.3
⎤
⎞
0 0 ⎥ ⎟ ⎟ 0 ⎥ ⎥ ⎟ 0 ⎥ ⎟ 18 ⎟/23 ⎥ 0 ⎥ ⎟ ⎟ 0.7⎥ ⎦ ⎠ 0 0.3
⎢2.1⎥ ⎢26.25%⎥ ⎢0⎥ ⎢ 0% ⎥ ⎢ ⎥ ⎢ ⎥ 1 . 8 ⎢ ⎥ ⎢ 22.5% ⎥ = ⎢ ⎥/ 8 = ⎢ ⎥ ⎢0.7⎥ ⎢ 8.75% ⎥ ⎢0.7⎥ ⎢ 8.75% ⎥ ⎣ ⎦ ⎣ ⎦ So, it can be concluded that the simulation results verify the efficiency of our methods. 5. Conclusion There is a lack of studies for the state analysis of BCN with impulsive and uncertain disturbances. However, for many evolution processes in the real world may experience abrupt changes of states at certain time instants. This is especially common in biological networks. These disturbances always cause the inaccuracy between the theoretical calculation and
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practical measurement. In this note, we introduce the probability in the analysis of the state of BCN, mathematically, the STP is a useful tool. The simulation results validation the efficiency for this model. So, it is can be concluded that the method in this note can resolve the concretizing problem of BCN, which can bridge the gap for the modeling of GRN based on BCN. Acknowledgments This paper is supported by the National Natural Science Foundation of China (Grant nos. 31560622, 31260538, 30960246, 61563038, and 71261015), Inner Mongolia Colleges and Universities Scientific and Technological Research Projects (Grant no. NJZY132), Inner Mongolia Talent Development Funded Project, Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (No. NMGIRT-A1609), Program for Science and Technology Innovation Guide Awards fund of Inner Mongolia Autonomous Region, Inner Mongolia University of Finance and Economics Institute of Big Data Research, the Shandong Province Outstanding Young Scientists Research Award Fund Project (Grant no. BS2015DX006), the Shandong Academy of Sciences Youth Fund Project (Grant no. 2016QN003) and Natural Science Foundation of Shandong Province (Grant no. ZR2016YL011). References [1] J M Bower, H Bolouri, Computational modeling of genetic and biochemical networks, Am. Math. Mon. (1) (2002) 58. [2] H Zang, T Zhang, Y Zhang, Bifurcation analysis of a mathematical model for genetic regulatory network with time delays, Appl. Math. Comput. 260 (C) (2015) 204–226. [3] H Huang, T Huang, X Chen, Global exponential estimates of delayed stochastic neural networks with Markovian switching, Neural Netw. Off. J. Int. Neural Netw. Soc. 36 (8) (2012) 136–145. [4] S Huang, D E Ingber, Shape-dependent control of cell growth, differentiation and apoptosis: switching between attractors in cell regulatory networks, Exp. Cell Res. 261 (1) (20 0 0) 91–103. [5] I Shmulevich, H LaHdesmaKi, E R Dougherty, et al., The role of certain Post classes in Boolean network models of genetic networks, Proc. Natl. Acad. Sci. U. S. A. 100 (19) (2003) 10734–10739. [6] Q Meng, H Jiang, Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays, Neurocomputing 74 (1-3) (2010) 362–368. [7] I Shmulevich, E R Dougherty, Z Wei, Gene perturbation and intervention in probabilistic Boolean networks, Bioinformatics 18 (10) (2002) 1319–1331. [8] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol. 22 (3) (1969) 437–467. [9] B Gao, L Li, H Peng, et al., Principle for performing attractor transits with single control in Boolean networks, Phys. Rev. E 88 (6) (2013). [10] B Drossel, T Mihaljev, F Greil, Number and length of attractors in a critical Kauffman model with connectivity one, Phys. Rev. Lett. 94 (8) (2004). [11] B Gao, Y Shi, C Yang, et al., STP-LWE: a variant of learning with error for a flexible encryption, Math. Probl. Eng. 2014 (2) (2014) 55–57. [12] K Mathiyalagan, H P Ju, R Sakthivel, Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities, Appl. Math. Comput. 259 (C) (2015) 967–979. [13] F Li, J Sun, Stability and stabilization of Boolean networks with impulsive effects, Syst. Control Lett. 61 (1) (2012) 1–5. [14] D Cheng, Disturbance decoupling of Boolean control networks, IEEE Trans. Autom. Control 56 (1) (2011) 2–10. [15] H Chen, J Sun, Global stability and stabilization of switched Boolean network with impulsive effects, Appl. Math. Comput. 224 (4) (2013) 625–634. [16] H Chen, B Wu, J Lu, A minimum-time control for Boolean control networks with impulsive disturbances, Appl. Math. Comput. 273 (C) (2016) 477–483. [17] C Hao, X Li, J Sun, Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, IEEE Trans. Autom. Control 60 (3) (2015) 806–811. [18] X Wang, X Li, J Lu, Control and flocking of networked systems via pinning, IEEE Circuits Syst. Mag. 10 (3) (2010) 83–91. [19] H. Li, L. Xie, Y. Wang, On robust control invariance of Boolean control networks, Automatica 68 (2016) 392–396. [20] B. Gao, Z. Deng, D. Zhao, Competing spreading processes and immunization in multiplex networks, Chaos Solitons Fractals 93 (2016) 175–181. [21] D Cheng, H Qi, A linear representation of dynamics of Boolean networks, IEEE Trans. Autom. Control 55 (10) (2010) 2251–2258. [22] D Cheng, H Qi, Z Li, Controllability and observability of Boolean control networks, Automatica 45 (7) (2011) 1659–1667. [23] L Rui, Y Meng, T Chu, State feedback stabilization for Boolean control networks, IEEE Trans. Autom. Control 58 (58) (2013) 1853–1857.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
State analysis of Boolean control networks with impulsive and uncertain disturbances Bo Gao a, Zheng-hong Deng b, Da-wei Zhao c,∗, Qun Song b a
School of Computer Information management, Inner Mongolia University of Finance and Economics, Hohhot 010051, China School of Automation, Northwestern Polytechnical University, Xi’an 710072, China c Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Jinan 250014, China b
a r t i c l e
i n f o
Keywords: Boolean Control Networks Semi-tensor Product Disturbance
a b s t r a c t In this paper, we put forward a method for the state analysis of Boolean control networks with impulsive and uncertain disturbances. In detail, we introduce the probability in the analysis of the state of Boolean networks. Mathematically, the STP as the analytical tool, has be used for the modeling of uncertainty. In addition, we give a mathematical proof for the model. Lastly, we conduct some simulations which validate the efficiency for this model. © 2016 Elsevier Inc. All rights reserved.
1. Introduction More and more attention has been paid to the Genetic regulatory network (GRN), a new research field in the biological science, during the last few years [1,2]. GRN, as a newly developing topic, has a close relationship with protein webs, neural networks and other biological systems [3–5]. Several types of genetic regulatory networks are proposed in recent years, including Markov-type genetic networks [6], probabilistic Boolean networks [7] and Boolean networks [8]. The Boolean network (BN), with a simple structure, was proposed initially for modeling and quantitatively describing the gene circuits. It has been widely applied in different fields, such as biology, physics, systems science and so on (see, e.g., [9–11]). There are mere two states of each node in a Boolean network: 1 (active) or 0 (inactive) at each discrete time, and each node evolves its state according to a logical Boolean function. As the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, Boolean network might be suitable for modeling GRN and biological systems, as well as other cellular processes. It is especially common in biological networks that many evolution processes may occur abrupt changes of states at certain time instants, which may be due to many factors, such as changes in the interconnections of subsystems, sudden changes of external environment. To describe the evolution of a process with a short-time perturbation, it is need to approximate the perturbation and neglect the duration as impulse disturbances, which may be caused by sudden noise or switching phenomenon. There are some literatures with impulsive disturbances [12,13]. However, there is a little research on studying the impulsive disturbances on Boolean control networks (BCN) [14]. In short, to investigate the Boolean control networks with impulsive disturbances still remains an open crucial theoretical problem In the present note, global stability and stabilization of switched Boolean network with impulsive effects are considered [15]. Ref. [16] investigates a Mayer-type optimal control and minimum-time control of a Boolean control network with ∗
Corresponding author. E-mail addresses: [email protected] (B. Gao), [email protected] (D.-w. Zhao).
http://dx.doi.org/10.1016/j.amc.2016.11.040 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
188
B. Gao et al. / Applied Mathematics and Computation 301 (2017) 187–192
impulsive disturbances. The stabilization, controllability and optimal control problem of Boolean networks with impulsive effects and state constraints are investigated [17]. These results are based on the known control impulse, however, there is a lack of research for uncertain control impulse [18–20]. Here, Semi-tensor Product is adopted as a mathematical tool to study the state analysis of BCN with impulsive and uncertain disturbances. We propose an algebraic approach to estimate the probability of each state in BCN with uncertain disturbances sequence or disturbances with the probability of each value. Unlike other existing methods, our approach, on the basis of clear and strict evidences, and the simulation results validation the efficiency for this model.
2. Preliminaries Semi-tensor Product (STP) is an extension of traditional matrix multiplication, which can make the multiplication between matrix with unequal rows and columns into reality. In this article, the symbol represents the calculation of STP [21]. Definition 2.1. Suppose that x is a lk dimensional row vector, y = {y1 , y2 ,…, yl } is a l dimensional column vector. The vector x is divided into l block, and each block has the same dimension, namely the size of each block is 1 × k. The calculation of STP as follow:
xy=
yT xT =
l l
i=1
i=1
xi yi ∈ Rn
(2.1)
yi ( xi ) ∈ Rn T
According to the mathematically meaning of STP, it is immediate that the matrix satisfied:
Ar×kl Bl×t = Cr×kt , Ar×l Bkl×t = Ckr×t .
(2.2)
Here, Ar×kl is a matrix with r row and kl column, and Br×kl is a matrix with l row and t column, Cr×kt is a matrix with r row and kt column. The Eq. (2.2) shows the variation of STP. Example 1. Consider two matrixes, where A = [15
AB = =
2 3 4 ] 6 7 8
and B = [10 20]. Then
(1 2 ) × 1 + (3 4 ) × 0 (1 2 ) × 0 + ( 3 4 ) × 2 ( 5 6 ) × 1 + (7 8 ) × 0 (5 6 ) × 0 + (7 8 ) × 2 1 5
2 6
6 14
8 . 16
Next, we introduce the definition of logical matrix, the logical calculation can transform into matrix multiplication according to the mathematical property of STP. Suppose that Ik is a unit matrix which can represented as Ik = {δki |i = 1, 2, . . . , k}, here δki is the ith column of unit matrix. So the logic number “0” and “1” can be denoted by δ22 = [10] and
δ21 = [10] respectively. i
i
in The logic calculation rule can be presented as matrix M = [δm1 , δm2 , . . . , δm ], which can be abbreviated as M = δ m [i1 , i2 ,…, in ]. The typical logic calculation are: logical “not” ¬ A, logical “and” A ∧ B, logical “or” A∨B, and logical “x or” AB. Accordingly, the logic calculation can be represented as:
M¬ = δ2 [2, 1]; M∧ = δ2 [1, 2, 2, 2]; M∨ = δ2 [1, 1, 1, 2]; M = δ2 [2, 1, 1, 2]. Theorem 2.1. A logical function f(x1 , x2 ,…, xn ), which have n dimensional variable x1 , x2 ,…, xn , can be represented as a linear mapping form:
f ( x1 , x2 , . . . , xn ) = M f x1 x2 · · · xn . Here, Mf is a 2 × 2n dimensional matrix.
(2.3)
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3. Main results 3.1. Algebraic form of BCNs with impulsive disturbances The model of Boolean networks with impulsive disturbances as follow:
⎧ x1 (t + 1 ) = f1 (x1 (t ), . . . , xn (t )), ⎪ ⎪ ⎨x2 (t + 1 ) = f2 (x1 (t ), . . . , xn (t )),
(1 ) . . ⎪ ⎪ ⎩. ⎧xn (t + 1 ) = x1 (t + 1 ) = ⎪ ⎪ ⎨x2 (t + 1 ) = (2 ) . . ⎪ ⎪ ⎩. xn (t + 1 ) =
t = k
fn (x1 (t ), . . . , xn (t )), f 1 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )), f 2 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )),
(3.1) t=k
f n (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t ))
This model contains two equations. When t = k, the system did not suffer the effect of impulsive disturbances, the function of each node is fi (i = 1, 2,…, n), which is a multivariate function with n variables, and x1 (t),…, xn (t) is stands for the state of node xi at the tth step. When t = k, the function of each node is f i (i = 1, 2,…, n), but the function is a multivariate function with n + m variables, and x1 (t),…, xn (t) is stands for the state of node xi at the tth step, and u1 (t),…, um (t) is impulsive disturbances at the tth step, so k is the step when the impulsive disturbances works. Suppose that the length of impulsive sequence equals to l, namely k = s, s + 1,…, s + l − 1, s ∈ Z + . According to the property of STP, the system (3.1) has the following conclusions: suppose that x(t ) = x1 (t )x2 (t ) . . . xn (t ) = ni=1 xi (t ), here xi (t) ∈ I2 ; and u(t ) = u1 (t )u2 (t ) . . . um (t ) = m u (t ), uj (t) ∈ I2 , therefore, system (3.1) can be transferred j=1 j into the following form [22]:
(1 )x(t + 1 ) = Lx(t ), t = k, (2 )x(t + 1 ) = L u(t )x(t ), t = k,
(3.2)
Here, L ∈ M2n ×2n , L ∈ M2n ×2n+m . In each column of matrix L, there exists only one entity “1”, which stands for the number of the following state. Also the matrix L is the transform matrix of system (3.1)(1). Likewise, In each column of matrix L , there exist only one entity “1”, which stands for the number of the following state. Matrix L is the transform matrix of system (3.1)(2). Matrixes L and L are the fundamental results of this paper. When the length of impulsive sequence equals tol = 1, the Eq. (3.1)(2) become a new form: x(t + 1) = L x(t)u(t). Therefore, we can have the following theorem.
v1 ], u (t ) = [ v2 ], …, um (t ) = [ vm ], here v (i = 1, 2,…, m) is the realization of input, Theorem 3.1. Suppose u1 (t ) = [1 − 2 i v 1−v 1−v 1
2
m
and the value of vi is either “0” or “1”, so the Eq. (3.1)(2) can obtained by the following equation:
L = L u1 (t ) u2 (t ) · · · um (t )
(3.3)
Here, L ∈ M2n ×2n .
Suppose that the Eq. (3.1)(2) has the initial state η0 , and η0 = ni=1 xi (0 ), the u(t ) = m u (t ), so the target state ηd can j=1 j be acquire by:
ηd = L u(t ) η0
(3.4)
Here, u(t ) ∈ I2m ×1 , ηd ∈ I2n ×1 . Proof. According to Eq. (3.2), we can conclude that matrix L is the linear expression of (3.1)(2). Therefore, compare with the property of STP, it can be concluded that when the initial state of system is η0 , the target state is ηd = L u(t)η0 when the input is u1 (t),…, um (t). 3.2. Uncertain disturbances sequence When we study the impulsive disturbances, it always can meet the situation that the disturbances cannot be well measured. So, in this section we study how to describe the state of the system when the input is an uncertain disturbances sequence. The Boolean network with disturbances sequence input can be linearized by (3.2)(2). When the length of the sequence equals to l, and k = 0, 1,…, l − 1, then the system (3.2)(2) can be described as:
x ( l ) = ( L )l u ( 0 ) u ( 1 ) · · · u ( l − 1 ) x ( 0 )
(3.5)
Here, (L )l ∈ M2n ×2n+m+l−1 , x(0) = x1 (0)x2 (0)xn (0) is the initial state of system with impulsive disturbances, u(0) = u1 (0)u2 (0)um (0), ……, u(l − 1) = u1 (l − 1)u2 (l − 1)um (l − 1).
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According to Eq. (3.5), we can see that as the increasing of length l, the increasing speed of matrix (L )l is very fast. Next, we put forward a method to reduction the scale of matrix (L )l . Based on this results, we analyze the effect of uncertain disturbances sequence foe the state of the system. Theorem 3.2. Suppose the input disturbances sequence of system (3.1)(2) equals to l, and the inputs u(0), u(1),…, u(l − 1) are uncertain. So the state that may occur of system (3.1)(2) can be described by state vector S,
S = (L )l 12n+m+l−1 Here, 12n+m+l−1
(3.6)
⎡ ⎤⎫ ⎡ ⎤ 1 ⎬ s1 s2 1 = ⎣ .. ⎦ 2n+m+l−1 is a column vector, which has 2n+m+l−1 row. S = ⎣ .. ⎦ is a column vector, which has . . ⎭ s2n
1 2n row., where si is the number of state i occurred in the system.
Proof. According to Eq. (3.5), (L )l is a matrix with 2n + m + l − 1 column, and each column has only one “1”. Suppose that ⎡ ⎤ ⎡ ⎤ a1 a1 a a2 2 l l l ⎣ ⎦ ⎣ . . ⎦1 n+m+l−1 , ai 1 n+m+l−1 = (L ) = .. , ai = [ωi1 , ωi2 , . . . , ωi2n+m+l−1 ] is the row of matrix (L ) . Thus, (L ) 12n+m+l−1 = .. 2 2 a2n a2n 2n+m+l−1 ωik . k=1 Therefore, si is the number of state i occurred in the system. According to the property of STP, when m and l are variable, the values of S are varied. In order to exclude the changing of m and l, we compute the occurrence value of each state when there exists uncertain disturbances in the system. Theorem 3.3. Suppose the input disturbances sequence of system (3.1)(2) equals to l, and the inputs u(0), u(1),…, u(l − 1) are uncertain. So the probability of each state that may occur of system (3.1)(2) can be described by state probability vector: Sp ,
Sp =
S 2n+m+l−1
(3.7)
⎤⎫ s p1 ⎪ ⎬ ⎢ s. ⎥ n Here, S p = ⎣ p2 .. ⎦⎪2 is a column vector, and spi is the probability that state i occurs. ⎭ s n ⎡
p2
Because Sp is a vector, and whose entities are probability, so we can derived that
2n
i=1 s pi
=1
3.3. Disturbances with the probability of each value In this section, we study another case for uncertain input disturbances. Suppose that the input of system (3.1)(2) are u1 (t), …, um (t), and the value of unknown input is either “0” or “1”, but we can compute the probability of the occurrence of “0” and “1”. In the section, we give a framework for the description of the type of system. p1 Theorem 3.4. Suppose the input of system (3.1)(2) are u1 (t),…, um (t), and u1 (t ) = [1−p ], here p1 is the probability of input 1
p2 pm value is “1”, so, 1 − p1 is the probability of input value is “0”, likewise, u2 (t ) = [1−p ], ……, um (t ) = [1−p ]. 2
m
Therefore, the system (3.1)(2) can be describe probability vector Vp :
Vp =
(L u1 (t ) u2 (t ) · · · um (t )) 12n 2n
(3.8)
⎤⎫ v p1 ⎪ ⎬ ⎢ v. ⎥ n Here, Va = ⎣ p2 .. ⎦⎪2 is a column vector, and vpi is the probability that state i occurs. v p2n ⎭ ⎡
Proof. According to Eq. (3.2)(2), (L )l is a matrix with dimensional 2n × 2n + m , and each column has only one “1”. ⎡ ⎤ b1 b2 (b u (t )u2 (t )···um (t ))12n Suppose that L = ⎣ .. ⎦, bi = [ωi1 , ωi2 , . . . , ωi2n+m ] is the row of matrix (L )l . Thus, v pi = i 1 , here 2n . b2n n+m (bi u1 (t ) u2 (t ) · · · um (t )) 12n = 2k=1 ωik is the sum of ith row of L , 2n is the number of occurrence. vpi is the probability that state i occurs. n Because Vp is a vector, and whose entities are probability, so we can derived that 2i=1 v pi = 1.
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4. Example In order to illustrate our main results, let us consider the Boolean control networks model, which is a reduced Boolean model for the lac operon in the bacterium Escherichia coli [23]. The model is given as follows:
x1 (t + 1 ) = x2 (t ) ∨ x3 (t ) (1 ) x2 (t + 1 ) = x1 (t ) t = k x ( t + 1 ) = x ( t ) 3 1 x1 (t + 1 ) = x2 (t ) ∨ x3 (t ) (2 ) x2 (t + 1 ) = x1 (t ) t=k x3 (t + 1 ) = x1 (t ) ∧ u(t )
(4.1)
Here, x1 (t) represent for lac mRNA, x2 (t) represent for lactose with higher concentration, x2 (t) represent for lactose with medium concentration, u(t) represent for lactose. According to the calculation of Eq. (3.2)(1), the matrixLcan be represented as L = δ 8 [11154448]. The sequence length of impulsive disturbances l = 1. So the Eq. (3.2)(2) can be represented as: L = δ 8 [1115444822264448]. When the input u(k) is 1, we have u(k ) = [10], so L u(k) = δ 8 [11154448]; likewise, when the input u(k) is 0, we have u(k ) = [10], so L u(k) = δ 8 [22264448].
⎡ ⎤ 3
When the input value is uncertain, S = (L )l 12n+m+l−1
⎢3⎥ ⎢0⎥ ⎢ ⎥ ⎢6⎥ ⎥ =⎢ ⎢1⎥. ⎢ ⎥ ⎢1⎥ ⎣ ⎦
0 2 ⎡ ⎤ 18.75% ⎢18.75%⎥ ⎢ 0% ⎥ ⎢ ⎥ ⎢ 37.5% ⎥ V ⎢ ⎥. The probability of each state equals to S p = n+m+l−1 = ⎢ ⎥ 2 ⎢ 6.25% ⎥ ⎢ 6.25% ⎥ ⎣ ⎦ 0% 12.5% Suppose that the input u(k) is a vector, whose value equals to 1 with probability 70%, and value equals to 0 with probability 30%, so the input u(k) can be presented as u(k ) = [00..73].
⎛⎡0.7
Vp =
L u ( k ) 1 2n 2n
⎜⎢ 0 ⎜⎢ 0 ⎜⎢ ⎜⎢0.3 = ⎜⎢ ⎜⎢ 0 ⎜⎢ 0 ⎝⎣ 0 0
⎡2.1⎤
⎡26.25%⎤
0 0.6
0% 7.5%
0 0.7 0 0.3 0 0 0 0
0.7 0 0 0.3 0 0 0 0
0 0.7 0 0.3 0 0 0 0
0.7 0 0 0.3 0 0 0 0
0 0.7 0 0.3 0 0 0 0
0 0 0 0 0.7 0 0 0.3
⎤
⎞
0 0 ⎥ ⎟ ⎟ 0 ⎥ ⎥ ⎟ 0 ⎥ ⎟ 18 ⎟/23 ⎥ 0 ⎥ ⎟ ⎟ 0.7⎥ ⎦ ⎠ 0 0.3
⎢2.1⎥ ⎢26.25%⎥ ⎢0⎥ ⎢ 0% ⎥ ⎢ ⎥ ⎢ ⎥ 1 . 8 ⎢ ⎥ ⎢ 22.5% ⎥ = ⎢ ⎥/ 8 = ⎢ ⎥ ⎢0.7⎥ ⎢ 8.75% ⎥ ⎢0.7⎥ ⎢ 8.75% ⎥ ⎣ ⎦ ⎣ ⎦ So, it can be concluded that the simulation results verify the efficiency of our methods. 5. Conclusion There is a lack of studies for the state analysis of BCN with impulsive and uncertain disturbances. However, for many evolution processes in the real world may experience abrupt changes of states at certain time instants. This is especially common in biological networks. These disturbances always cause the inaccuracy between the theoretical calculation and
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practical measurement. In this note, we introduce the probability in the analysis of the state of BCN, mathematically, the STP is a useful tool. The simulation results validation the efficiency for this model. So, it is can be concluded that the method in this note can resolve the concretizing problem of BCN, which can bridge the gap for the modeling of GRN based on BCN. Acknowledgments This paper is supported by the National Natural Science Foundation of China (Grant nos. 31560622, 31260538, 30960246, 61563038, and 71261015), Inner Mongolia Colleges and Universities Scientific and Technological Research Projects (Grant no. NJZY132), Inner Mongolia Talent Development Funded Project, Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (No. NMGIRT-A1609), Program for Science and Technology Innovation Guide Awards fund of Inner Mongolia Autonomous Region, Inner Mongolia University of Finance and Economics Institute of Big Data Research, the Shandong Province Outstanding Young Scientists Research Award Fund Project (Grant no. BS2015DX006), the Shandong Academy of Sciences Youth Fund Project (Grant no. 2016QN003) and Natural Science Foundation of Shandong Province (Grant no. ZR2016YL011). References [1] J M Bower, H Bolouri, Computational modeling of genetic and biochemical networks, Am. Math. Mon. (1) (2002) 58. [2] H Zang, T Zhang, Y Zhang, Bifurcation analysis of a mathematical model for genetic regulatory network with time delays, Appl. Math. Comput. 260 (C) (2015) 204–226. [3] H Huang, T Huang, X Chen, Global exponential estimates of delayed stochastic neural networks with Markovian switching, Neural Netw. Off. J. Int. Neural Netw. Soc. 36 (8) (2012) 136–145. [4] S Huang, D E Ingber, Shape-dependent control of cell growth, differentiation and apoptosis: switching between attractors in cell regulatory networks, Exp. Cell Res. 261 (1) (20 0 0) 91–103. [5] I Shmulevich, H LaHdesmaKi, E R Dougherty, et al., The role of certain Post classes in Boolean network models of genetic networks, Proc. Natl. Acad. Sci. U. S. A. 100 (19) (2003) 10734–10739. [6] Q Meng, H Jiang, Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays, Neurocomputing 74 (1-3) (2010) 362–368. [7] I Shmulevich, E R Dougherty, Z Wei, Gene perturbation and intervention in probabilistic Boolean networks, Bioinformatics 18 (10) (2002) 1319–1331. [8] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol. 22 (3) (1969) 437–467. [9] B Gao, L Li, H Peng, et al., Principle for performing attractor transits with single control in Boolean networks, Phys. Rev. E 88 (6) (2013). [10] B Drossel, T Mihaljev, F Greil, Number and length of attractors in a critical Kauffman model with connectivity one, Phys. Rev. Lett. 94 (8) (2004). [11] B Gao, Y Shi, C Yang, et al., STP-LWE: a variant of learning with error for a flexible encryption, Math. Probl. Eng. 2014 (2) (2014) 55–57. [12] K Mathiyalagan, H P Ju, R Sakthivel, Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities, Appl. Math. Comput. 259 (C) (2015) 967–979. [13] F Li, J Sun, Stability and stabilization of Boolean networks with impulsive effects, Syst. Control Lett. 61 (1) (2012) 1–5. [14] D Cheng, Disturbance decoupling of Boolean control networks, IEEE Trans. Autom. Control 56 (1) (2011) 2–10. [15] H Chen, J Sun, Global stability and stabilization of switched Boolean network with impulsive effects, Appl. Math. Comput. 224 (4) (2013) 625–634. [16] H Chen, B Wu, J Lu, A minimum-time control for Boolean control networks with impulsive disturbances, Appl. Math. Comput. 273 (C) (2016) 477–483. [17] C Hao, X Li, J Sun, Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, IEEE Trans. Autom. Control 60 (3) (2015) 806–811. [18] X Wang, X Li, J Lu, Control and flocking of networked systems via pinning, IEEE Circuits Syst. Mag. 10 (3) (2010) 83–91. [19] H. Li, L. Xie, Y. Wang, On robust control invariance of Boolean control networks, Automatica 68 (2016) 392–396. [20] B. Gao, Z. Deng, D. Zhao, Competing spreading processes and immunization in multiplex networks, Chaos Solitons Fractals 93 (2016) 175–181. [21] D Cheng, H Qi, A linear representation of dynamics of Boolean networks, IEEE Trans. Autom. Control 55 (10) (2010) 2251–2258. [22] D Cheng, H Qi, Z Li, Controllability and observability of Boolean control networks, Automatica 45 (7) (2011) 1659–1667. [23] L Rui, Y Meng, T Chu, State feedback stabilization for Boolean control networks, IEEE Trans. Autom. Control 58 (58) (2013) 1853–1857.