Computational Systems Biology Methods and Protocols.7z

maximizing the probability of P(G|D), which means Bayesian

model identifies the optimal network topology that best explains

the expression data. Because the number of possible network topol-

ogies increases with the number of genes in an exponential manner,

it is not feasible to search for all possible networks. Therefore, some

heuristic algorithms, like genetic algorithm and evolutionary algo-

rithm, have been proposed for Bayesian network inference

[18, 19]. One limitation of Bayesian models is that they can’t

present loop motif for networks since they are directed acyclic

graphs. However, feedback loop motifs are prevalent in biological

systems.

In Gaussian graphical model, the gene expression data (D) is

assumed having a Gaussian (normal) distribution, and relationship

between genes is expressed as conditional dependencies through

calculating the partial correlation measurement [20]. Given the

genes X and Y and their k correlated variables Z (Z 1 ,Z 2 ...Zk)

with covariance matrix W, then the relationship between genes X

and Y, termed Pxy.z, can be computed with the following

equations.

rx¼XWZ, ry¼YWZ

Pxy:z¼

covrx;ry



σrxσry

¼

n

Xn

i¼ 1

rx,iry,i

Xn

i¼ 1

rx,i

Xn

i¼ 1

ry,i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

Xn

i¼ 1

r^2 x,i

Xn

i¼ 1

rx,i

vu! 2

u

t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

Xn

i¼ 1

r^2 y,i

Xn

i¼ 1

ry,i

vu! 2

u

t

ð 7 Þ

whererxandryare residual variables X and Y given thek-dimension

Z as controlling variables,nis the number of experiment, andrx,i

andry,iare theith expression value of genes X and Y, respectively.

Please keep in mind that when the number of genes greatly exceeds

the number of experiments, the covariance matrix W can’t be

estimated certainly. Therefore, some regularized regression meth-

ods, such as the LASSO, two-stage adaptive LASSO, and ridge

regression approaches, have been developed to help estimate the

covariance matrix correctly [21, 22], which promotes the applica-

tion of the Gaussian graphical model in network inference problem.

2.1.3 Integrative Inferring
Models

Each inferring method has its strengths and weaknesses because of

various mathematic assumptions, which lead to different bias of

network reconstruction. For example, the information-theoretic

models can detect feedback loop, while the probabilistic graphical

models can’t. Whereas the Bayesian model gives directionality of

each links, information-theoretic models do not. Therefore, com-

bining different inferring models can provide more reliable gene

regulatory networks. For the integrative inferring models, different

inferring methods are applied to reconstruct networks firstly. Then

The Reconstruction and Analysis of Gene Regulatory Networks 141