Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Hong Kong

Abstract

Background

Probabilistic Boolean Network (PBN) is a popular model for studying genetic regulatory networks. An important and practical problem is to find the optimal control policy for a PBN so as to avoid the network from entering into undesirable states. A number of research works have been done by using dynamic programming-based (DP) method. However, due to the high computational complexity of PBNs, DP method is computationally inefficient for a large size network. Therefore it is natural to seek for approximation methods.

Results

Inspired by the state reduction strategies, we consider using dynamic programming in conjunction with state reduction approach to reduce the computational cost of the DP method. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method.

Conclusions

Finding the optimal control policy for PBNs is meaningful. The proposed problem has been shown to be

Background

An important goal for studying genetic regulatory network is to understand the gene behavior and to develop optimal control policy for potential applications to medical therapy. While many models have been proposed for modeling gene regulatory networks, Boolean Networks (BNs)

Many methods in control theory are available for the intervention of PBNs. A gene control model has been proposed in

Datta et al.

Chen et al.

Several reduction methods have been proposed recently. In

Qian et al.

We consider the problem of minimizing the maximum cost in control of PBN and we employ transition probability-based reduction strategy to reduce the network complexity of a PBN. We show that under some condition and in many of our numerical examples, the optimal control sequence obtained from the reduced network is the same as the one in the original network. Then we apply the dynamic programming-based algorithm to the reduced network. The computational complexity of dynamic programming-based algorithm when applied to the original network is ^{n}) (depending on the number of network states) when the number of control nodes

The remainder of the paper is structured ae follows. We first give a brief review on PBNs and the dynamic programming method. We then introduce our state reduction approach together with some theoretical results to support our proposed approach. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method. Finally some discussion will be given to conclude the paper.

A brief review on BNs and PBNs

A BN consists of a set of _{1},_{2},..., _{n}_{i}_{1}, _{2},..., _{n}_{i}_{i}_{i}_{i}_{1}, _{i}_{2},..., _{ik}_{i}_{1},_{i}_{2},...,_{ik}_{i}_{i}_{i}_{1}, _{i}_{2},..., _{ik}_{i}_{i}_{1}, _{2},..., _{n}

Since BN is a deterministic model, a stochastic model is more preferable due to the measurement noise in inferring a gene regulatory network. A stochastic version of BN, PBN _{i}

The state of _{i}

A PBN can be regarded as a finite collection of BNs over a fixed set of nodes, where each BN has a fixed set of Boolean functions **f _{j}**(

Since _{i}^{n}].

The dynamics of a PBN can be studied by using Markov chain theory, see for instance _{ij }

Here

A review on dynamic programming

In this section, we first introduce several definitions to facilitate the discussion. We then introduce the dynamic programming-based algorithm. Suppose a PBN has a set of internal nodes {_{1},_{2},...,_{n}_{n+}_{1},_{n+}_{2},...,_{n+m}_{i}_{ik}

to be the state of network. Then we define control input as

Here we are interested in the following problem: Minimizing the maximum cost in control of PBN.

Given the terminal cost _{M}_{M}^{n}} at terminal time step _{0},_{1},...,_{M}

**Step 0: **Set

_{M}_{M}_{M}_{M }=

**Step 1**:

**Step 2**: For any _{t}^{n}} and _{t}

and

**Step 3**: If **Step 1**; Otherwise, stop.

The state reduction approach

In this section we propose our state reduction method.

Transition probability-based state reduction strategy

Due to the high network complexity of a PBN, one has to deal with matrices of huge size which increases exponentially with the number of internal nodes. Network reduction is therefore an important issue to be addressed in this situation. In

where

Which means that the network will never enter state

The dynamic programming-based algorithm on the reduced network

Since the computational complexity of the dynamic programming is ^{n}) when the number of control nodes

**Proposition 1 **

It is straightforward to see that, starting from the initial state, the network will never enter into transient states to be deleted. Therefore the network will never stop at those states at the terminal time step. This means that the deleted states will not be included in the optimal route, and the cost of deleted states will not be counted. Hence deleting these transient states will not influence the result obtained from the DP method when applied to the reduced network.

Based on transition probability-based strategy, one can iteratively delete those transient states until all the remaining states are critical states. In each step, we need to update the transition matrix for the reduced network by deleting the corresponding row and column from the transition matrix. After making the reduction, one can get a reduced network with a set of states

An analysis of the reduction method when indegree

In a PBN of

All the possible BNs for 2 genes when

**States**

**f _{1}**

**f _{2}**

**
f
_{11}
**

**
f
_{12}
**

**
f
_{13}
**

**
f
_{14}
**

**
f
_{21}
**

**
f
_{22}
**

**
f
_{23}
**

**
f
_{24}
**

00

0

1

0

1

0

1

0

1

01

0

1

1

0

0

1

1

0

10

1

0

0

1

1

0

0

1

11

1

0

1

0

1

0

1

0

In general, we can also compute the number of all the possible BNs for ^{n}. For example, from Table ^{2 }= 16 networks with 2^{2 }^{2 }sizes. When every row contains 1, it means the number of nonzero rows is 2^{2}. To satisfy this condition, we have to choose 2 genes as parent genes and consider every gene has two possible states. Thus, we can deduce that the number of such networks is ^{1}, we just select only one gene as the parent gene and the corresponding selected possibilities are

**Proposition 2 **When the indegree of a BN is one, the distribution of zero row is given in Table

Distribution of number of nonzero rows in BNs when

**Number of nonzero rows in BN**

**Number of BNs in all the (2 n) ^{n}BNs**

2^{n}

^{n}

2^{(n-k)}

In Table ^{n }such kind of BNs. This means that after transition, all the states will still be visited. In calculating the number of BNs satisfying this particular condition, we should ensure that the ^{n }BNs having no zero row. As a matter of fact, if we define a function _{dis}

Therefore to compute the number of BNs when the number of non-zero rows is 2^{n-k}, one should select ^{n-k},

Furthermore, since there are 2^{n }states for ^{n}. One can observe that with the increase of ^{n }BNs is decreasing fast because

Hence this guarantees the efficiency of state reduction.

State reduction for PBN with random perturbation

In this section we discuss the state reduction strategy for PBNs with random perturbation. Let **v**(t), then state at the next time step is determined by the transition matrix without perturbation ^{n}, or by randomly perturbation with probability 1-(1-^{n}. Therefore the transition matrix with perturbation is given by

where

where

To carry out the state reduction strategy, we need to delete all the states which can only be entered by random perturbation. Here we set the threshold for ^{n}. If for some state

is satisfied, then we can delete the state.

Table

Reduction Rates for PBN with Gene Perturbations

12.5%

37.5%

36.0%

35.4%

42.2%

60.5%

42.8%

48.4%

77.5%

67.8%

63.1%

63.8%

84.9%

73.1%

77.0%

72.4%

Results and discussions

In this section, we give some numerical examples to compare the result of dynamic programming-based algorithm on the reduced network with the one on the original network.

A 6-gene example

We first consider a 6-node example. We consider the cases of _{M}_{M}^{5}. When ^{4}. Table

A 6-Node Example for PBN Without Perturbation

**Size**

**Cost**

**CPU Time (sec.)**

**Original**

**Reduced**

**Original**

**Reduced**

**Original**

**Reduced**

32

24

17

17

0.1276

0.0943

32

26

5

5

0.1255

0.0997

32

26

21

21

0.1286

0.1062

32

27

19

19

0.1291

0.1085

32

30

25

25

0.1355

0.1274

32

32

29

29

0.1355

0.1355

16

11

9

9

0.1061

0.0692

16

10

3

3

0.0996

0.0585

16

14

9

9

0.1051

0.0996

16

16

6

6

0.1040

0.0997

16

16

14

14

0.1063

0.1050

16

16

12

12

0.1049

0.1053

A 6-Node Example for PBN With Random Perturbation

**Size**

**Cost**

**CPU Time (sec.)**

**Original**

**Reduced**

**Original**

**Reduced**

**Original**

**Reduced**

32

12

31

31

0.0402

0.0162

32

23

10

10

0.0375

0.0257

32

28

24

24

0.0396

0.0329

32

30

26

26

0.0386

0.0363

32

32

30

30

0.0402

0.0399

32

32

30

30

0.0419

0.0418

16

12

13

13

0.0315

0.0213

16

15

9

9

0.0277

0.0262

16

14

9

9

0.0300

0.0250

16

16

7

7

0.0293

0.0292

16

16

15

15

0.0316

0.0314

16

16

13

13

0.0316

0.0307

A 12-gene example

We then consider a 12-node example. We consider the cases of _{M}_{M}^{11}. When ^{10}. Table

A 12-Node Example for PBN Without Perturbation

**Size**

**Cost**

**CPU Time (sec.)**

**Original**

**Reduced**

**Original**

**Reduced**

**Original**

**Reduced**

2048

426

757

757

63.84

4.07

2048

700

1258

1258

256.68

36.89

2048

502

1830

1830

272.27

23.27

2048

1462

1591

1591

264.39

143.48

2048

1103

2036

2036

279.58

88.80

2048

1801

1987

1987

272.35

243.79

1024

350

607

607

153.85

24.35

1024

444

179

179

148.80

36.10

1024

415

342

342

140.62

29.49

1024

801

328

328

150.21

107.27

1024

759

736

736

146.98

87.76

1024

937

756

756

172.93

146.29

A 12-Node Example for PBN With Random Perturbation

**Size**

**Cost**

**CPU Time (sec.)**

**Original**

**Reduced**

**Original**

**Reduced**

**Original**

**Reduced**

2048

363

1906

1906

18.19

0.89

2048

867

322

322

17.99

3.75

2048

581

2005

2005

18.27

1.92

2048

1340

1931

1931

18.27

8.35

2048

830

1990

1990

18.42

3.59

2048

1882

1847

1847

18.38

15.86

1024

204

286

286

9.67

0.67

1024

508

153

153

9.60

2.81

1024

384

242

242

9.63

1.78

1024

785

486

486

9.69

6.09

1024

747

839

839

9.91

5.68

1024

992

611

611

9.87

9.26

Conclusions

From the experiment results, one can see that applying dynamic programming-based algorithm on the reduced network can reduce the computational complexity. The performance of the algorithm on the reduced network depends on the parameters of

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

XC came up with the idea. XC and WKC designed the research. XC, HJ and YQ performed the research and analyzed the results. XC, HJ and WKC wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the three anonymous referees for their helpful and encouraging comments and suggestions. The preliminary version has been presented in IEEE Conference on Systems Biology (ISB), Zhuhai, China and published in the conference proceedings

This article has been published as part of