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PROCEEDINGSOpen Access

An estimation method for inference of gene

regulatory net-work using Bayesian network with

uniting of partial problems

Yukito Watanabe*, Shigeto Seno, Yoichi Takenaka, Hideo Matsuda

From The Tenth Asia Pacific Bioinformatics Conference (APBC 2012)

Melbourne, Australia. 17-19 January 2012

Abstract

Background: Bayesian networks (BNs) have been widely used to estimate gene regulatory networks. Many BN

methods have been developed to estimate networks from microarray data. However, two serious problems reduce

the effectiveness of current BN methods. The first problem is that BN-based methods require huge computational

time to estimate large-scale networks. The second is that the estimated network cannot have cyclic structures,

even if the actual network has such structures.

Results: In this paper, we present a novel BN-based deterministic method with reduced computational time that

allows cyclic structures. Our approach generates all the combinational triplets of genes, estimates networks of the

triplets by BN, and unites the networks into a single network containing all genes. This method decreases the

search space of predicting gene regulatory networks without degrading the solution accuracy compared with the

greedy hill climbing (GHC) method. The order of computational time is the cube of number of genes. In addition,

the network estimated by our method can include cyclic structures.

Conclusions: We verified the effectiveness of the proposed method for all known gene regulatory networks and

their expression profiles. The results demonstrate that this approach can predict regulatory networks with reduced

computational time without degrading the solution accuracy compared with the GHC method.

Background

Finding gene regulations is an important objective of

systems biology [1,2]. Causal gene regulatory interac-

tions are widely described using gene regulatory net-

works. Estimating gene regulatory networks can help

reveal complicated regulations.

Recently, microarray [3,4] has rapidly produced a

wealth of information about gene expression activities.

The volume of data necessitates computational methods

to identify and analyze the underlying gene regulatory

networks [5]. A number of analytical methods have

been proposed to estimate gene regulatory networks

from gene expression profiles. Boolean networks, graphi-

cal Gaussian models (GGM), differential equation

models, and Bayesian networks (BNs) are widely used

models.

A Boolean network is a discrete dynamical network

[6,7]. In a Boolean network, the state of a gene is repre-

sented by a Boolean variable (ON or OFF) and interac-

tions between the genes are represented by Boolean

functions that determine the state of a gene on the basis

of the states of certain other genes. Hence, continuous

gene expression data must be transformed into binary

data before a Boolean network can be estimated, and

much information is lost in this binary encoding. As

gene expression cannot be described adequately by only

two states, Boolean networks are limited by their

definition.

A GGM is an undirected probabilistic graphical model

[8]. This model allows the identification of conditional

independence relations among the nodes under the

* Correspondence: w-yukito@ist.osaka-u.ac.jp

Department of Bioinformatic Engineering, Graduate School of Information

Science and Technology, Osaka University, Osaka, Japan

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© 2012 Watanabe 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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assumption of a multivariate Gaussian distribution of

the data. In a GGM, regulations between genes are esti-

mated by calculating the correlation between pairs of

variables. Therefore, the GGM does not identify the

direction of regulatory relationships between two genes,

but rather only calculates the correlations between their

gene expression data.

A differential equation model describes gene expres-

sion changes as a function of the expression of other

genes and environmental factors [9-11]. Their flexibility

allows the complex relations among components to be

described. In a differential equation model, a gene regu-

lation is described as the function of several gene

expression levels. When the input data includes experi-

mental noise, this model cannot estimate the gene regu-

latory network accurately. Also, if there is not sufficient

data input, overfitting occurs.

BN is a graphical model for representing probabilistic

relationships among a set of random variables [12-16].

These relationships are encoded in the structure of a

directed acyclic graph whose nodes are the random vari-

ables. The relationships between the variables are

described by a joint probability distribution. In a BN,

causal interactions between more than three genes can

be estimated. BN has advantages over the above models

in applications where BN deals better with the experi-

mental noise.

Using a BN, it is hard to estimate a large-scale net-

work because the search space grows exponentially as

the number of genes increases. Therefore, overcoming

this problem has been the focus of much research. The

proposed solutions to this problem can be divided into

three types. The first type limits the number of esti-

mated genes. Even when estimating a large-scale net-

work, part of the network is often attracted. The second

type parallelizes the estimation by supercomputer or

other high-performance computer. Effective parallelizing

makes it possible to estimate large-scale networks. The

third type improve the algorithm itself. These methods

reduce computational time and estimate the network by

a heuristic.

An example of the first type of solution is proposed by

Peña et al. [17]. This method overcomes the problem of

the user having to decide in advance which genes are

included in or excluded from the learning process. The

method receives a seed gene S and a positive integer R

from the user, and returns a BN. It starts the BN from S

genes, then adds the parents and children of all the

genes in the BN R + 1 times, and prunes some genes. In

this way, the user avoids deciding in advance which

genes to include.

A solution of the second type proposed by Tamada et

al. [18] can estimate gene regulatory networks consist-

ing of more than 20,000 genes from gene expression

data. The method uses a supercomputer, and it is mas-

sively parallelized. It repeatedly estimates subnetworks

by hill climbing in parallel for genes selected by neigh-

bor node sampling. The method high-handedly over-

comesthe problemof

supercomputer. Even if a supercomputer can effectively

provide a large-scale network, an estimation method

designed to run on a workstation is also required.

A solution of the third type for estimating gene regu-

latory networks was implemented by Bøttcher et al.

[19]: the greedy hill climbing (GHC) method. By com-

paring networks that differ only by a single directed

edge, either added, removed, or reversed, a GHC

method can estimate networks of larger scale than a

search of all possible networks and do so on a worksta-

tion rather than a supercomputer, thus overcoming two

problems at once. However, the estimation accuracy of

this method is not high, because the method tends to

produce only local optimal solutions.

In this paper, we present a novel BN-based determi-

nistic method with reduced computational time to over-

come the above-mentioned problems. The proposed

method can estimate a network as large-scale as those

estimated by the GHC method, run on a workstation,

and estimate more accurately than the GHC method.

We take another approach to estimate more accurately

than the GHC method. First, our method generates all

the combinational subsets with three genes. Then, we

estimate all possible networks for each subset using the

BN method and unite the networks into a single net-

work including all genes. This approach enables us to

estimate more accurately for the same computational

time than the GHC method.

In order to verify the effectiveness of the proposed

method, we perform two experiments, to evaluate scal-

ability and accuracy: i.e., one to verify the proposed

method can estimate networks as large-scale as those

estimated by the GHC method, and one to verify it can

estimate more accurately than the GHC method. These

experiments are performed using randomly sampled

genes. In addition, we conduct a third experiment to

confirm that our method outperforms the GHC method

using real data.

theBNbyusing the

Results

Bayesian networks

Let D = (V, E) be a directed acyclic graph (DAG), where

V is a finite set of nodes and E is a finite set of directed

edges between the nodes [19]. The DAG defines the

structure of the BN.

Each node v Î V in the graph corresponds to a ran-

dom variable xv. The set of variables associated with the

graph D is then X = {xv}. Often we do not distinguish

between a variable xvand the corresponding node v. To

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each node v with parents pa(v), a local probability distri-

bution, p(xv|xpa(v)), is attached. The set of local probabil-

ity distributions for all variables in the network is P. A

BN for a set of random variables X is the pair (D,P).

Directed edges in D encode conditional dependencies

between the random variables X through the factoriza-

tion of the joint probability distribution.

?

As a measure of how well a DAG D represents the

conditional dependencies between the random variables,

we use the relative probability

p(x) =

v∈V

p?xv|xpa(v)

?.

(1)

p(D,d) = p(d|D)p(D),

and refer to it as a network score, where d is data and

p(d|D) is called the likelihood of D.

The log network score contribution of a node is evalu-

ated whenever the node is learned. The log network

score N(D) is given by

(2)

N(D) = logp(D,d).

(3)

The number of possible DAGs grows exponentially

with the number of nodes, and the problem of identify-

ing the network with the highest score is NP-hard. If

the number of random variables in a network is large, it

is not computationally possible to calculate the network

score for all possible DAGs. For these situations, the

search strategy GHC method is implemented.

The GHC method is as follows.

1. Select an initial DAG D0randomly from which to

start the search.

2. Calculate the Bayes scores of D0and all possible

networks that differ by only one directed edge, that

is, an edge is added to D0, an edge in D0is deleted,

or the direction of an edge in D0is reversed.

3. Among all these networks, select the one that

increases the Bayes score the most.

4. If the Bayes score was not improved, stop the

search. Otherwise, make the select network D0and

repeat from step 2.

In the GHC method, we can limit the maximum num-

ber of these steps in the search algorithm. Also, the

search algorithm can restart an arbitrary number of

times. More details on the parameter setting will be

described later in this paper.

Methods

We propose a new method to estimate a gene regulatory

network with reduced computational time. The pro-

posed method is composed of three steps: dividing the

whole problem into partial problems, estimating gene

regulatory networks of partial problems, and uniting the

estimated networks. In this section, we describe our

BN-based method using the analysis of a set of expres-

sion data as an example. This example includes five

genes V = {vi|1 ≤ i ≤ 5}. A conceptual representation of

our approach is presented in Figure 1. We call a search

of all possible networks an exhaustive search to distin-

guish it from the GHC method.

Step 1: Dividing the whole problem into partial problems

Our approach first divides the set of all genes V into all

the combinational subset with three genes (triplets) t =

{vi, vj, vk Î V|1 ≤ i <j <k ≤ 5}. For example, our

approach obtains5C3= 10 partial problems {v1, v2, v3}.

{v1, v2, v4}, ..., {v3, v4, v5}.

Step 2: Estimating gene regulatory networks

After making partial problems, we next calculate inde-

pendently the scores of all the possible networks of each

partial problem by exhaustive search and obtain esti-

mated DAGs G. The number of possible alternative net-

works for a triplet {v1, v2, v3} is 33= 27 because there

are three cases for each potential edge (vi, vj) (1 ≤ i <j ≤

3): a directed edge from vito vj, a directed edge from vj

to vi, and no edge.

Let c = (D, SD, RD) be a tuple, where D Î G is a DAG,

SD= p(D, d) is a score of D, where p(D, d) is given by

Equation 2, and RDis a rank of D.

We add tuples of all the partial problems to Z, where

Z is a set of c. For example, when we have 10 partial

problems {v1, v2, v3}.{v1, v2, v4}, ... , {v3, v4, v5}, we add

270 tuples of networks to Z.

Step 3: Uniting estimated partial problems

To solve the original problem, this step unites three-

gene networks into a single gene regulatory network.

The policy of the step is to classify relationships

between genes, i.e., determine (vi, vj) (1 ≤ i <j ≤ 3) into

one of the three edge types (a directed edge from vito

vj, a directed edge from vjto vi, or no edge between vi

and vj) according to the score calculated in Step 2.

To select an edge type between genes viand vj, we

calculate an edge (vi, vj) value for each of the three

types t using the following:

?

where D has edge (vi, vj). Then we select one edge

type that has the highest total value.

When two or more edge types have the highest total

value, we use edge scores of the partial problems whose

ranks are 2 or more.

Algorithm

Input: V = V1, ..., Vn: a set of genes, GEP: gene expres-

sion profiles of V

(D,SD,1)∈Z

SD,

(4)

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Output: GV: DAG including genes V

Variable:Z: a set of tuples (graph, score, rank)

1: Make a collection of set V that includes all the sub-

sets of V with three elements

2-1: for each U in V do

2-2: Make a collection of set Duthat includes all the

DAGs of U

2-3:for each D in Dudo

2-4: calculate rank RDand score SDwith GEP

2-5:add (D, SD, RD) to Z

2-6: end for

2-7: end for

3-1: i ¬ 1

3-2: repeat

3-3:for each edge between genes (x, y) in D of (D,

SD, i) do

3-4: add all SDof (D, SD, i) for each of the three

edge types

3-5: if one edge type has the highest total SDthen

3-6: add an edge between genes (x, y) to GV

3-7: end if

3-8:

total SDthen

3-9:

GV, where w is a gene ≠ x, y do

3-10:

of (D, SD, i), where D includes genes x, y, and w.

3-11:end for

3-12: add edge (x, y) selected in (3-10) with the

highest SDto GV

3-13:end if

3-14:end for

3-15:i¬i+1

3-16: until directions of all edges in GVare assigned

3-17: return GV

A flowchart of the algorithm can be found in Figure 2.

if two or more edge types have the highest

for each edge between genes (x or y, w) in

select edge between genes (x, y) from D

Computational experiments

To verify the effectiveness of the proposed method, we

performed three experiments. The first experiment

determines computational time for different numbers of

genes. The purpose of this experiment is to verify that

Figure 1 Conceptual representation of our approach. Yellow circles represent genes. Blue circles represent partial problems. Small directed

edges represent regulatory relationships between genes. Large directed edges represent the flow of the method.

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the proposed method is able to estimate gene regulatory

networks that are as large-scale as those estimated by

the GHC method. The second experiment demonstrates

that the proposed method is more accurate than the

GHC method. The third experiment shows, through an

example, that our algorithm works well for inferring real

gene regulatory networks. We estimate the networks,

including the known gene regulatory network, and com-

pare the network estimated by the proposed method

and that by the GHC method.

Implementation, system, and materials

Steps 1 and 2 are implemented using the deal package

version 1.2-33 written in R. We use R 2.10.1. Step 3 is

implemented using Perl 5.10.1.

The GHC method is implemented in the deal package

version 1.2-33. In these experiments, the maximum

number of actions, i.e., adding, deleting, or reversing a

directed edge, is set at 50 and the number of restarts is

set at 0. We call these parameters the default parameter

set.

We performed all the experiments on a computer with

Intel Core2 Duo 6600 CPU 2.40 GHz processors with

3.0 GB memory. The operation system is Ubuntu 10.04.

We used a dataset of two time-series gene expression

profiles including 45102 genes from a mouse adipocyte

and osteoblast. The number of time points is 62.

Experiment 1 We verified that the proposed method

can estimate gene regulatory networks as large-scale as

Figure 2 Flowchart of the algorithm. Circles represent start and end points. Rectangles represent generic processing steps. Diamonds

represent decision steps.

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those estimated by the GHC method. We used the pro-

posed method, an exhaustive search, and the GHC

method, and compared the estimation time for from 3

to 70 genes. In this experiment, we selected genes from

the gene expression profile from a mouse adipocyte by

random sampling. We ran this process 50 times and cal-

culated the mean estimation time. The results are sum-

marized in Figure 3.

In Figure 3, the horizontal axis corresponds to the

number of genes and the vertical axis corresponds to

the logarithm of the estimation time. The proposed

method was able to estimate the network including 70

genes, and the estimation times were almost the same

as those of the GHC method. The estimation time of

the proposed method was shorter than that of the GHC

method for 40 or more genes. The estimation time of

the proposed method was longer than that of the GHC

method for 15 or fewer genes. The estimation time of

the exhaustive search was very large by 5 genes.

Experiment 2 We verified that the estimation accuracy

of the proposed method is higher than that of the GHC

method for nearly identical estimation times. We com-

pared the estimation results of the exhaustive search

with the results of the proposed method and the GHC

method. In this experiment, we selected five genes ran-

domly from the gene expression profile 100 times from

a mouse adipocyte and osteoblast. We estimated the

network of these five genes by the proposed method

and the GHC method. There are 59049 DAGs for five

genes, and all the DAGs are ranked by the scores of the

exhaustive search. The ranking was used to evaluate the

networks estimated by the proposed method and the

GHC method. The results are listed in Figure 4.

The two bar charts in Figure 4 show the ranks of 100

networks estimated by the proposed method and the

GHC method. The left bar chart is the results for adipo-

cyte, and the right are those for osteoblast. The

correspondence count is the number of times that the

network estimated by the proposed method or the GHC

method corresponded with the network of the exhaus-

tive search. The ranking in the exhaustive search is the

ranking of the networks estimated by the exhaustive

search. The networks are ranked by the scores of the

exhaustive search. As there are 59049 DAGs for five

nodes, the ranks are from 1st to 59049th.

The correspondence count of the proposed method

from the 1st to 10th networks of the exhaustive search

exceeded 50. For the correspondence count from the

30001th to the 59049th network of the exhaustive

search, the GHC method exceeded 50 and the proposed

method was less than 10.

Experiment 3 We used a known gene regulatory net-

work and verified that the proposed method can esti-

mate more accurately than the GHC method with the

same or less computational time. We compared the reg-

ulations estimated by the proposed method with those

of the GHC method. In this experiment, we used 40

genes from the gene expression profile from a mouse

adipocyte. Of these, 7 genes are Pparg and the genes

that regulate or are regulated by Pparg in adipocyte.

These are shown in Figure 5(a). The remaining 33 genes

were selected by random sampling. The results and

known networks are shown in Figure 5. In this experi-

ment, we used two parameter sets for the GHC method.

One is the default parameter set. In the other parameter

set, the maximum number of actions is 100 and the

number of restarts is 10, which will return a better net-

work but requires about 20-fold longer computational

time than the default.

In Figure 5, results of the default and other parameter

set are shown as networks (b) and (c), respectively. We

call (c) the network estimated by the highly accurate

GHC method in this experiment. Network (d) is esti-

mated by the proposed method. The edges in networks

Figure 3 Comparison of the estimation time. The estimation time of the exhaustive search, the GHC method, and the proposed method.

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(b), (c), and (d) are categorized according to the edges of

network (a). The red edges are also in network (a), the

blue edges have a different direction from those in net-

work (a), and the black edges have no relationship in

network (a).

Figure 5 shows that the proposed method was able to

estimate more correctly than the GHC method. The

sensitivity and selectivity of the proposed method were

33% and 30%, those of the GHC method were 0% and

0%, and those of the high accurate GHC method were

11% and 14%. Networks (b), (c), and (d) have many

edges that the known gene regulatory network does not

have, but these edges describe indirect regulations. For

example, in Figure 5(d), there is a black edge from C/

EBPa to Stat1. The edge describes the indirect regula-

tion from C/EBPa to Stat1 via Pparg because there are

edges from C/EBPa to Pparg and from Pparg to Stat1

in Figure 5(a).

Discussion

The GHC method tends to produce local optimal solu-

tions. For example, in Figure 4, the results of the GHC

method have two peaks, corresponding to the classes of

1-10 and 30001-59049. We cannot completely avoid

selecting a local optimal solution when using the GHC

method, because the solution accuracy depends on the

initial DAG from which the search is started. To obtain

the best network when using the GHC method, the esti-

mation must be repeated using different initial DAGs.

In contrast, the proposed method can produce one

result as the best network.

The results of our experiments indicate that dividing

the set of all genes and uniting the network results can

estimate more accurately than the GHC method. With

the GHC method, the maximum number of actions, i.e.,

adding, deleting, or reversing a directed edge, and the

number of restarts can be adjusted. If these parameters

are increased as much as possible, the estimation accu-

racy can be made comparable to that of the exhaustive

search. However, this would spoil the advantage of the

GHC method that it can estimate with high speed. The

GHC method selects the action that increases the net-

work score the most; therefore, a regulation that

increases the network score only slightly is rarely

selected. In this sense, the search of the GHC method is

considerably biased. This aspect becomes pronounced

when the limiting parameters are set strictly. With the

proposed method, regulations that have a positive effect

will be selected independently of whether that effect is

slight or strong. For example, in Figure 5, the regulatory

relationship between Pparg and C/EBPb could not be

estimated by the GHC method, even if the parameters

of the restart and the actions were significantly

increased.

We verified that the proposed method can estimate

networks as large-scale as those estimated using the

GHC method. We spend at most 0.1 second to estimate

the network of one partial problem with three genes

Figure 4 Comparison of the estimated network. Frequency that the networks estimated by the GHC method and the proposed method

correspond to those of the exhaustive search (from 1 to 59049).

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and repeat the estimationnC3times in the proposed

method. Therefore, the proposed method can estimate

the network with a low amount of memory compared

with the GHC method, which, like the exhaustive

search, requires much memory. When we estimate a

network for a data set from a large number of genes

using the GHC method, it is easy to run out of memory,

making the actual computational time longer than the

theoretical time.

Conclusions

In this study, we present a novel BN-based deterministic

method with reduced computational time. We con-

firmed experimentally that the proposed method can

reduce the computational time drastically without

degrading the solution accuracy. The proposed method

can estimate networks as large-scale as those estimated

by the GHC method. Furthermore, the proposed

method can estimate more accurately than the GHC

method, even if the computational time of the GHC

method is increased to more than 20 times that of the

proposed method.

Acknowledgements

This work was partially supported by Grant-in-Aid for Scientific Research

(22680023 and 22310125) from the Japan Society for the Promotion of

Science (JSPS), and by the HPCI STRATEGIC PROGRAM Computational Life

Science and Application in Drug Discovery and Medical Development from

Figure 5 Comparison of the network including Pparg and genes that regulate or are regulated by Pparg. (a) is the known gene

regulatory network. (b) is the network estimated by the GHC method with the maximum number of actions set at 50 and the number of

restarts set at 0. (c) is the network estimated by the GHC method with the maximum number of actions set at 100 and the number of restarts

set at 10. (d) is the network estimated by the proposed method. Blue circles represent genes. Red edges indicate edges also in network (a), blue

edges indicate edges with a different direction from those in network (a), and black edges indicate that there are no such relationships in

network (a).

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the Ministry of Education, Culture, Sports, Science and Technology of Japan

(MEXT).

This article has been published as part of BMC Genomics Volume 13

Supplement 1, 2012: Selected articles from the Tenth Asia Pacific

Bioinformatics Conference (APBC 2012). The full contents of the supplement

are available online at http://www.biomedcentral.com/1471-2164/13?

issue=S1.

Authors’ contributions

YW implemented the algorithm and performed the analyses. YW, SS, YT, and

HM conceived and designed the experiments and wrote the paper.

Competing interests

The authors declare that they have no competing interests.

Published: 17 January 2012

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doi:10.1186/1471-2164-13-S1-S12

Cite this article as: Watanabe et al.: An estimation method for inference

of gene regulatory net-work using Bayesian network with uniting of

partial problems. BMC Genomics 2012, 13(Suppl 1):S12.

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