Computational Systems Biology Methods and Protocols.7z

x∗iðÞ¼t xiðÞþt uiðÞt, i¼1,2,...m, t¼1, 2,...k

xiðÞ¼tþ 1

Xm

j¼ 1

bi,jfxjðÞt



þviðÞt,fxjðÞt



¼

1

1 þexj

ð 9 Þ

wherebi,jpresents nonlinear relationship between genesiandjand

the other parameters are the same with those of the linear state-

space models. One popular way of solving these nonlinear state-

space models is the extended Kalman filter (EKF) method, which

gives an approximating solution to the nonlinear system through

first-order linear approximation. In addition, the variants of Kalman

filter algorithm such as the cubature Kalman filter (CKF),

unscented Kalman filter (UKF), and particle filter algorithm are

proposed for solving the problem of the nonlinear state-space

models [25].

2.2.3 Time-Delayed
Dependency Models

In biological system, there exist time-delayed phenomena of regu-

lation from regulators to downstream targets due to time require-

ment of the process of transcription and translation. Therefore, the

time-delayed dependency models are developed to update regula-

tion functions of the linear and nonlinear state-space models [26].

xiðÞ¼tþ 1

Xm

j¼ 1

ai,jxjðÞþt

Xq

p¼ 1

ci,pxpðÞþtτ viðÞt

xiðÞ¼tþ 1

Xm

j¼ 1

bi,jfxjðÞt



þ

Xq

p¼ 1

ci,pxpðÞþtτ viðÞt,fxjðÞt



¼

1

1 þexj

ð 10 Þ

whereqis the total numbers of regulators in the system andτis the

time-delayed value. In practice, the value ofτcan be determined by

the Akaike’s information criterion (AIC) so as to avoid over-fitting

the models.

3 Analysis Approaches of Network Topology Structure

After building up a gene regulatory network, the topology analysis

is conducted to reveal structure of the network and thus give more

insights into property of a biological system, which can help biolo-

gist explore function of interested genes and investigate mechanism

of the system [27, 28]. Here, we will give some basic concept of

network topology, and some structure analysis based on these con-

cepts will be introduced in the following sections.

In terms of mathematics, the network is a graph consisting of

nodes and edges. The former are molecules of a biological system,

while the later are links between these molecules. The most ele-

mentary concept of nodes in network is degree (n), which describes

the number of links a node connected to others. Specially, there are

incoming (nin) and outgoing (nout) degrees for a node in a directed

The Reconstruction and Analysis of Gene Regulatory Networks 143