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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):S13doi: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>
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© 2007 Li et al; licensee BioMed Central Ltd.
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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 Confidence ScoresOther 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 ij 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 min maxavg minmaxavgrecallprecisionavg
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
j l i
=
=
{}
( )
, ,... ( )1 2
ffff
k
kk k 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
j l 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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