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BioMed Central

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BMC Bioinformatics

Open Access

Proceedings

Comparison of probabilistic Boolean network and dynamic

Bayesian network approaches for inferring gene regulatory

networks

Peng Li1, Chaoyang Zhang*1, Edward J Perkins2, Ping Gong3 and

Youping Deng*4

Address: 1School of Computing, University of Southern Mississippi, Hattiesburg, MS 39406, USA, 2Environmental Laboratory, U.S. Army Engineer

Research and Development Center, 3909 Halls Ferry Rd. Vicksburg, MS, 39180, USA, 3SpecPro Inc., 3909 Halls Ferry Rd, Vicksburg, MS, 39180,

USA and 4Department of Biological Sciences, University of Southern Mississippi, Hattiesburg, MS 39406, USA

Email: Peng Li - peng.li@usm.edu; Chaoyang Zhang* - chaoyang.zhang@usm.edu; Edward J Perkins - Edward.J.Perkins@erdc.usace.army.mil;

Ping Gong - Ping.Gong@erdc.usace.army.mil; Youping Deng* - youping.deng@usm.edu

* Corresponding authors

Abstract

Background: The regulation of gene expression is achieved through gene regulatory networks

(GRNs) in which collections of genes interact with one another and other substances in a cell. In

order to understand the underlying function of organisms, it is necessary to study the behavior of

genes in a gene regulatory network context. Several computational approaches are available for

modeling gene regulatory networks with different datasets. In order to optimize modeling of GRN,

these approaches must be compared and evaluated in terms of accuracy and efficiency.

Results: In this paper, two important computational approaches for modeling gene regulatory

networks, probabilistic Boolean network methods and dynamic Bayesian network methods, are

compared using a biological time-series dataset from the Drosophila Interaction Database to

construct a Drosophila gene network. A subset of time points and gene samples from the whole

dataset is used to evaluate the performance of these two approaches.

Conclusion: The comparison indicates that both approaches had good performance in modeling

the gene regulatory networks. The accuracy in terms of recall and precision can be improved if a

smaller subset of genes is selected for inferring GRNs. The accuracy of both approaches is

dependent upon the number of selected genes and time points of gene samples. In all tested cases,

DBN identified more gene interactions and gave better recall than PBN.

Background

The development of high-throughput genomic technolo-

gies (i.e., DNA microarrays), makes it possible to study

dependencies and regulation among genes on a genome-

from Fourth Annual MCBIOS Conference. Computational Frontiers in Biomedicine

New Orleans, LA, USA. 1–3 February 2007

Published: 1 November 2007

BMC Bioinformatics 2007, 8(Suppl 7):S13 doi:10.1186/1471-2105-8-S7-S13

<supplement> <title> <p>Proceedings of the Fourth Annual MCBIOS Conference. Computational Frontiers in Biomedicine</p> </title> <editor>Dawn Wilkins, Yuriy Gusev, Raja Loganantharaj, Susan Bridges, Stephen Winters-Hilt, Jonathan D Wren (Senior Editor)</editor> <note>Proceedings</note> </supplement>

This article is available from: http://www.biomedcentral.com/1471-2105/8/S7/S13

© 2007 Li et al; licensee BioMed Central Ltd.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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wide scale. In last decade, the amount of gene expression

data has increased rapidly necessitating development of

computational methods and mathematical techniques to

analyze the resulting massive data sets. In order to under-

stand the functioning of cellular organisms, why compli-

cated response patterns to stressors are observed, and

provide a hypothesis for experimental verification, it is

necessary to model gene regulatory networks (GRNs).

Currently, clustering, classification and visualization

methods are used for reconstruction or inference of gene

regulatory networks from gene expression data sets. These

methods generally group genes based on the similarity of

expression patterns. Based on large-scale microarray data

retrieved from biological experiments, many computa-

tional approaches have been proposed to reconstruct

genetic regulatory networks, such as Boolean networks

[1,2], differential equations [1,3], Bayesian networks [4-6]

and neural networks [7]. Among these approaches,

Boolean network methods and Bayesian network meth-

ods have drawn the most interest in the field of systems

biology.

Much recent work has been done to reconstruct gene reg-

ulatory networks from expression data using Bayesian net-

works and dynamic Bayesian network (DBN). Bayesian

network approaches have been used in modeling genetic

regulatory networks because of its probabilistic nature.

However, drawbacks of Bayesian network approaches

include failure to capture temporal information and mod-

eling of cyclic networks. DBN is better suited for character-

izing time series gene expression data than the static

version. Perrin et al. [8] used a stochastic machine learn-

ing method to model gene interactions and it was capable

of handling missing variables. Zou et al. [9] presented a

DBN-based approach, in which the number of potential

regulators is limited to reduce search space. Yu et al. [10]

developed a simulation approach to improve DBN infer-

ence algorithms, especially in the context of limited quan-

tities of biological data. In [11], Xing and Wu proposed a

higher order Markov DBN to model multiple time units in

a delayed gene regulatory network. Recently, likelihood

maximization algorithms such as the Expectation-Maxi-

mization (EM) algorithm have been used to infer hidden

parameters and deal with missing data [12].

The Boolean Network model, originally introduced by

Kauffman [1,13,14] is also very useful to infer gene regu-

latory networks because it can monitor the dynamic

behaviour in complicated systems based on large

amounts of gene expression data [15-17]. One of the

main objectives of Boolean network models is to study the

logical interactions of genes without knowing specific

details [17,18]. In a Boolean network (BN), the target

gene is predicted by other genes through a Boolean func-

tion. A probabilistic Boolean network (PBN), first intro-

duced by Shmulevich et al. in [16,19] is the stochastic

extension of Boolean network. It consists of a family of

Boolean networks, each of which corresponds to a contex-

tual condition determined by variables outside the model.

As models of genetic regulatory networks, the PBN

method has been further developed by several authors. In

[20], a model for random gene perturbations was devel-

oped to derive an explicit formula for the transition prob-

abilities in the new PBN model. In [21], intervention is

treated via external control variables in a context-sensitive

PBN by extending the results for instantaneously random

PBN in several directions. Some learning approaches for

PBN have also been explored [22-24]. Considering the

same joint probability distribution over common varia-

bles, several fundamental relationships of two model

classes (PBN and DBN) have been discussed in [25].

In this paper, two important computational approaches

for modeling gene regulatory networks, PBN and DBN,

are compared using a biological time-series dataset from

the Drosophila Interaction Database [26] to construct a

Drosophila gene network. We present the PBN and DBN

approaches and GRN construction methods used and dis-

cuss the performance of the two approaches in construct-

ing GRNs.

Results

A real biological time series data set (Drosophila genes

network from Drosophila Interaction Database) was used

to compare PBN and DBN approaches for modeling gene

regulatory networks [27,28]. The raw data was preproc-

essed in the same way as given in [29]. There were 4028

gene samples with 74 time points available in Drosophila

melanogaster genes network through the four stages of the

life cycle: embryonic, larval, pupal and adulthood [27].

An example network of drosophila muscle development

is given in [29], in which muscle-specific protein 300

(Msp-300) is treated as hub gene in their inferred network.

We used a different subset of the genes which participate

in the development of muscle. Particularly, Mlp84B and

other genes which contribute to larval somatic muscle

development were used to infer gene regulatory networks.

The D. melanogaster gene Muscle LIM protein at 84B

(abbreviated as Mlp84B) has also been known in FlyBase

as Lim3. It encodes a product with putative protein bind-

ing involved in myogenesis which is a component of the

cytoplasm. It is expressed in the embryo (larval somatic

muscle, larval visceral muscle, muscle attachment site,

pharyngeal muscle and two other listed tissues). Table 1

shows the scores of Mlp84B interacting with other related

genes [26].

Here, we first selected 12 genes to infer GRNs using PBN

and DBN. The constructed GRNs are shown in Figure 1.

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There exists 18 interactions totally within this small larval

somatic muscle network [26]. 10 and 12 interactions in

the network have been successful identified. Most interac-

tions between Mlp84B and genes with high confidence

have been inferred.

More comparison results of PBN(n, e) and DBN(n, e) are

given in Table 2, where n is the number of nodes (genes)

in network and e the number of edges (interactions)

among the nodes. PBN(30, 60) means that there are 30

nodes and 60 edges in that PBN simulation. To analyze

the effect of network size on the inference accuracy, four

(n, e) combinations, (12, 18), (20, 35), (30, 60) and (40,

80), were considered for inferring gene network. For each

combination, we randomly selected five subsets of genes

of the same numbers of genes and edges from the Dro-

sophila gene network. For each subset genes, we inferred

a gene network and retrieved the number of correct edges

Ce, miss errors Me, and false alarm errors Fe. For each

combination (n, e), the average and range of Ce, Me and

Fe were calculated, as given in Table 2. A correct edge is the

one that exists in a real network (i.e. the Drosophila gene

network) and is successfully identified by the inference

methods. Miss error is defined as the edge between two

genes that exists in a real network, but the inference algo-

rithms miss or make wrong orientations. False alarm error

is the edge that the inference algorithms create but does

not exist in the real network.

We used the benchmark measures recall R and precision P

to evaluate performances of inference algorithms for PBN

and DBN. While different definitions for recall and preci-

sion exist [30], in this paper, R is defined as Ce/(Ce + Me)

and P is represented as Ce/(Ce + Fe). The selection of sub-

set genes in network was based on the current existing

gene interactions and network diagram in the Drosophila

genes network [26].

The results in Table 2 show that for the same (n, e) case,

DBN reduces miss errors but increases false alarms errors

slightly. For all cases, DBN can identify more corrected

edges than PBN and hence improve recall. The precision

Table 1: The interactions and scores of Mlp84B with other genes

High ConfidenceScores Other interactionsScores

CG10722

CG13501

CG17440

CG7046

CG7447

CG11115 (Ssl1)

0.5642

0.9005

0.5811

0.6626

0.5411

0.7917

Cdk7

Impe1

Pfk

TfIIB

Stck

tup

0.3569

0.1108

0.3155

0.2436

0.2523

0.1094

Drosophila larval somatic muscle development network

Figure 1

Drosophila larval somatic muscle development network. The genetic network inferred by PBN. (b)The genetic net-

work inferred by DBN

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of DBN is better in three cases but worse in one case than

PBN. For both PBN and DBN methods, recall and preci-

sion decrease if the number of genes increases. One can

see that if more genes are selected for inferring GRNs, the

network contains more edges and it is more difficult to

successfully identify the interactions among genes. While

the DBN method can give better recall of identifying

genetic network interactions, it is more time-consuming

than PBN.

Discussion

It is challenging to infer GRNs from time series gene

expression data. Among thousands of genes, each gene

interacts with one or more other genes directly or indi-

rectly through complex dynamic and nonlinear relation-

ships, time series data used to infer genetic networks have

low-sample size compared to the number of genes, and

gene expression data may contain a substantial amount of

noise. Different approaches may have different perform-

ances for different datasets. Moreover, inference accuracy

depends not only upon models but also on inference

schemes. In this paper, we only select two representative

inference algorithms for PBN and DBN to model the

GRNs, respectively. It is desirable to perform a more com-

prehensive evaluation of the two approaches with differ-

ent inference methods and to develop the more robust

algorithm and techniques to improve the accuracy of

inferring GRNs.

Conclusion

PBN-based and DBN-based methods were used for infer-

ring GRNs from Drosophila time series dataset with 74

time points obtained from the Drosophila Interaction

Database. The results showed that accuracy in terms of

recall and precision can be improved if a smaller subset of

genes is selected for inferring GRNs. Both PBN and DBN

approaches had good performance in modeling the gene

regulatory networks. In all tested cases, DBN identified

more gene interactions and gave better recall than PBN.

The accuracy of inferring GRNs was not only dependent

upon the model selection but also relied on the particular

inference algorithms that were selected for implementa-

tion. Different inference schemes may be applied to

improve accuracy and performance.

Methods

Boolean network and probabilistic Boolean network

In a BN, the expression level of a target gene is function-

ally related to the expression states of other genes using

logical rules, and the target gene is updated by other genes

through a Boolean function. There are only two gene

expression levels (states) in a Boolean network (BN): on

and off, which are represented as "activated" and "inhib-

ited". A probabilistic Boolean network (PBN) consists of

a family of Boolean networks and incorporates rule-based

dependencies between variables. In a PBN model, BNs are

allowed to switch from one to another with certain prob-

abilities during state transitions. Since PBN is more suita-

ble for GRN reconstruction from time series data and a

Boolean network is just a special case of PBN and we only

consider PBN for comparison.

Boolean network

We use the same definition as in [2,18] for a Boolean net-

work. A Boolean network G(V, F) is defined by a set of

nodes (variables) representing genes V = {x1, x2,..., xn}

(where xi ∈ {0, 1} is a binary variable) and a set of

Boolean functions F = {f1, f2,..., fn}, which represents the

transitional relationships between different time points. A

Boolean function

f xx

j i j i

(,

( )

12

with k(i) spec-

ified input nodes is assigned to node xi. The gene status

(state) at time point t + 1 is determined by the values of

some other genes at previous time point t using one

Boolean function fi taken from a set of Boolean functions

F. So we can define the transitions as

x

ji

k i

( )

,...,)

( )( )

x t

i

f x

(

t x

( ),

txt

j i

( )

1

j i

2

ji

k i

( )

()( ),...,( ))

( ) ( )

+=

1

Table 2: Comparison of PBN and DBN methods using different sample networks

Miss errors Me

False alarm errors Fe

Correct edges Ce

Accuracy (%) (R, P) Time(s) T

min maxavg minmax avgminmax avgrecall precisionavg

PBN(12,18)

PBN(20,35)

PBN(30,60)

PBN(40,80)

DBN(12,18)

DBN(20,35)

DBN(30,60)

DBN(40,80)

2

12

33

48

3

13

30

46

9

22

41

63

8

17

39

57

6.4

16.8

36.0

55.4

5.8

15.2

33.6

51.2

0

3

7

4

1

4

11

5

4

6

10

6

3

7

15

9

2.4

4.8

8.0

5.6

2.2

5.4

12.6

7.4

6

11

17

18

9

14

24

28

9

15

20

22

11

18

30

34

7.8

13.6

18.4

19.6

10.4

16.8

20.2

22.8

54.9

44.7

33.8

26.1

64.2

52.5

37.5

30.8

76.5

73.9

69.6

77.8

82.5

75.7

61.6

75.5

13.2

19.7

27.9

39.2

20.1

36.0

50.6

87.6

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where each xi represents the expression value of gene i, if

xi = 0, gene i is inhibited; if xi = 1, it is activated. The vari-

able jk(i) represents the mapping between gene networks at

different time points. Boolean function F represents the

rules of regulatory interactions between genes.

Probabilistic Boolean network

Probabilistic Boolean network inference is the extension

of Boolean network methods to combine more than one

possible transition Boolean functions, so that each one

can be randomly selected to update the target gene based

on the selection probability, which is proportional to the

coefficient of determination (COD) of each Boolean func-

tion. Here we briefly give the same notation of PBN as in

[19]. The same set of nodes V = {x1, x2,..., xn} as in a

Boolean network is used in a PBN G(V, F), but the list of

function sets F = {f1, f2,..., fn} is replaced by F = {F1, F2,...

Fn}, where each function set com-

posed of l(i) possible Boolean functions corresponds to

each node xi. A realization of the PBN at a given time point

is determined by a vector of Boolean functions. Each real-

ization of the PBN maps one of the vector functions

n

= (,,...)

( ) ( )( )12

, 1 ≤ k ≤ N, 1 ≤ k(i) ≤ l(i), where

and N is the number of possible realizations.

Given the values of all genes in network at time point t

and a realization fk, the state of the genes after one updat-

ing step is expressed as

(x1(t +1), x2(t +1),... xn(t +1)) = fk(x1(t), x2(t),... xn(t))

Let f = (f(1), f(2),... f(n)) denote a random vector taking val-

ues in F1 × F2 ? × Fn. The probability that a specific transi-

i ( )

tion function , (1 ≤ j ≤ l(i)) is used to update gene i is

equal to

Given genes V = {x1, x2,..., xn}, each xi is assigned to a set

=

{

of Boolean functions to update tar-

get gene. The PBN will reduce to a standard Boolean net-

work if l(i) = 1 for all genes. A basic building block of a

PBN describing the updating mechanism is shown in Fig-

ure 2.

Construction of GRNs from PBN

The Coefficient of Determination (COD) is used to select

a list of predictors for a given gene [19,23]. So far, most

learning methods for reconstructing gene regulatory net-

work use COD to select predictors for each target gene at

any time point t. COD has also been used previously for

the steady state data sets. Here we use upper case letters to

represent random variables: Let Xi be the target gene,

ii

l i

12

( )

,,,

…

be sets of genes and be

available Boolean functions. Thus, the optimal predictors

i

1

(

of Xi can be defined by

and the probabilistic error measure can be represented as

( ) ( )

X fX

i

k

k

i ( )

. For each k, the COD for Xi relative to the

conditioning set is defined by

where εi is the error of the best estimate of Xi [23].

Now, if a class of gene sets which have

high CODs has been selected, we can use the optimal

ii

12

,,

…

Boolean functions as the rule set for gene

Xi, with the probability of being chosen (see(3)).

Then the approximations are given by

According to the above expressions [19,23], those

Boolean functions corresponding to the highest CODs

will be selected in the probabilistic network. The selected

Boolean functions are used to predict the gene expression

status at the next time point, and they also will be used to

reconstruct gene regulatory networks.

Bayesian networks and dynamic Bayesian networks

Among the many computational approaches that infer

gene regulatory networks from time series data, Bayesian

network analysis draws significant attention because of its

probabilistic nature. DBN is the temporal extension of

Bayesian network analysis. It is a general model class that

is capable of representing complex temporal stochastic

processes. It captures several other often used modeling

frameworks as its special cases, such as hidden Markov

models (and its variants) and Kalman filter models.

Ff

i

j

i

jl i

=

=

{}

( )

, ,... ( )1 2

ffff

k

kkk n

( )1 ( )2 ( )

fF

k i

( )

i

i

( )∈

fj

cfff

j

i

i

j

i

k f

:

f

k

k i

( )

i

j

i

( )

( )

( )

Pr{} Pr{}

( )( )

====

=

∑

f

Ff

i

j

i

jl i

=

}

( )

, ,... ( )1 2

XXX

i

( )( )( )

fff

ii

l i

( )

i

12

( )( )( )

,,,

…

fXfXfX

iii

l i

( )

i

l i

( )

i

122

( )( )( )( )( )( )

),(),,()

…

ε(,())

ii

Xk

ω

εε

ε

k

i

ii

k

i

k

i

i

X fX

=

− (,( ))

( )( )

XXX

ii

l i

( )

i

12

( )( ) ( )

,,,

…

fff

l i

( )

i

( )( )( )

,

fj

i ( )

ck

i

k

i

j

i

j

l i

( )

=

( )

=

∑

ω

ω

1

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Bayesian network

Given a set of variables U = {x1, x2,... xn} in gene network,

a Bayesian network, for U is a pair B = (G, Θ) which

encodes a joint probability distribution over all states of

U. It is composed of a directed acyclic graph (DAG) G

whose nodes correspond to the variables in U and Θ

which defines a set of local conditional probability distri-

butions (CPD) to qualify the network. Let Pa(xi) denote

the parents of the variables xi in the acyclic graph G and

pa(xi) denote the values of the corresponding variables.

Given G and Θ, a Bayesian network defines a unique joint

probability distribution over U given by

For more detail on Bayesian networks, see [24].

Dynamic Bayesian network

A DBN is defined by a pair (B0, B1) represents the joint

probability distribution over all possible time series of

variables X = {X1, X2,... Xn}, where Xi(1 ≤ i ≤ n) represents

the binary-valued random variables in the network,

besides, we use lower case xi (1 ≤ i ≤ n) denotes the values

of variable Xi. It is composed of an initial state of Bayesian

network B0 = (G0, Θ0) and a transition Bayesian network

B1 = (G1, Θ1), where B0specifies the joint distribution of

the variables in X(0) and B1 represents the transition prob-

abilities Pr{X(t + 1) | X(t)} for all t. In slice 0, the parents

of Xi(0) are assumed to be those specified in the prior net-

work B0, which means Pa(Xi(0)) ⊆ X(0) for all 1 ≤ i ≤ n; in

slice t + 1, the parents of Xi(t + 1) are nodes in slices t,

Pa(Xi(t + 1)) ⊆ X(t) for all 1 ≤ i ≤ n and t ≥ 0, as stated in

[25], the connections only exist between consecutive

slices. The joint distribution over a finite list of random

variables X(0) ∪ X(1) ∪ ? ∪ X(T) can be expressed as

[24,25]

An example of a DBN is shown in Figure 3.

Construction of GRNs from DBN

Given a set of training gene data, how the network struc-

ture is found that best fits the data is called learning the

structure of a dynamic Bayesian network. The goal of con-

structing a network is to find the model with maximum

likelihood (i.e., REVEAL algorithm in [3] and its improve-

ment in [9]). The network we want to learn is the transi-

tion network, i.e., the network defining dependencies

between the adjacent time slices X(t) and X(t + 1). The

training set of data is composed of all adjacent time-slices

X(t) and X(t + 1).

Pr{ ,,...}Pr{| ( )}

x

i

pa

x x

1

xx

ni

i

2

1

=

=∏

n

Pr{ ( ), ( ),..., ( )}xx01

Pr{ ( )}x Pr{ ()| ( )}x1

Pr{

x T

x

t

T

∏

0

0

1

=+

=

=

−

tt

xi iijj

j

n

ti

n

X x t X t

( )|0( ( ))}0 Pr{ ( )|(())}11

10

1

1

pa pa

T

∏

×++

==

−

=

∏∏

A basic building block of a PBN

Figure 2

A basic building block of a PBN.

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Algorithms for learning gene network structure have

focused on networks with complete data. Structural

Expectation Maximization (SEM) is developed to handle

data with hidden variables and missing values. One of the

algorithms to infer network structure from training data is

based on the mutual information analysis of the data. For

each node, this algorithm learns the optimal parent set

independently by choosing the parent set that maximizes

a scoring function. The scoring function is defined by

I(X, Pa(X))/max{H(X), H(Pa(X))}

where I(X, Y) is the mutual information between X and Y,

and H(X) is the entropy of X. With parent set of genes in

DBN, GRNs can be constructed [31].

For each inferred network, scoring metrics are used to

evaluate the probabilistic scores which explain relation-

ships in the given data sets. There are two popular Baye-

sian scoring metrics: the BDe (Bayesian Dirichlet

equivalence) score [32] and the BIC (Bayesian informa-

tion criterion) score [33]. Then, the network with highest

score will be identified using search heuristics, which have

three widely used methods: greedy search, simulated

appealing and a genetic algorithm [10].

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

PL implemented the algorithms and inferred gene net-

works. PL and CZ performed the statistical analysis and

drafted the manuscript. CZ and YD coordinated the study.

EP, PG and YD gave suggestions to improve the methods

and revised the manuscript. All authors read and

approved the final manuscript.

Acknowledgements

This work was supported by the Army Environmental Quality Program of

the US Army Corps of Engineers under contract #W912HZ-05-P-0145.

Permission was granted by the Chief of Engineers to publish this informa-

tion. The project was also supported by the Mississippi Functional Genom-

ics Network (DHHS/NIH/NCRR Grant# 2P20RR016476-04).

This article has been published as part of BMC Bioinformatics Volume 8 Sup-

plement 7, 2007: Proceedings of the Fourth Annual MCBIOS Conference.

Computational Frontiers in Biomedicine. The full contents of the supple-

ment are available online at http://www.biomedcentral.com/1471-2105/

8?issue=S7.

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