- Dynamic Modeling 107
In the discrete dynamic model of gene regulation (9.4), the regulatory
genes that potentially regulates target geneicould be all genes or some
subset of genes among the genes of interest. Some other kinds of data
such as ChIP-on-chip (ChIP-chip) or ChIP-sequencing (ChIP-seq) data
demonstrating the protein-DNA interaction information may be integrated
to screen the potential genes that regulates target genei.Inthatcase,the
number of genes potentially regulating genei,Ni, would be different for each
gene. Furthermore, in addition to the state variables (xi(t),xj(t)’s) in (9.4),
there are many model parameters, i.e.,aij’s,λi,ki, which characterize the
interactions among the system components and environmental effects. Since
the main objective is to investigate gene regulations between the genes of
interest, the regulatory abilitiesaij’s, whose values quantify the regulatory
strength between geneiand genej, are of special importance. These model
parameters would be identified with the aid of experimental data in the next
section.
3.3. Model parameter identification
Following formation of the discrete dynamic models describing temporal
expression of the selected genes of interest, the parameters in the models
require identification using the transcriptomics data. The model parameter
identification means adjusting the parameters of the model until the behavior
of the model matches the generated experimental data. Maximum likelihood
estimation method is one of the frequently used methods for parameter
identification,^6 which is introduced here. The discrete dynamic model (9.4)
can be written in the following regression form:
xi(t+1)=[x 1 (t) ··· xNi(t) xi(t)1]·⎡
⎢⎢
⎢⎢
⎢
⎢⎢
⎣ai 1
..
.
aiNi
(1−λi)
ki⎤
⎥⎥
⎥⎥
⎥
⎥⎥
⎦+εi(t)≡φi(t)·θi+εi(t) (9.5)whereφi(t) denotes the regression vector which can be obtained from the
transcriptomics data, andθi is the parameter vector to be estimated for
target gene i. In order to avoid the danger of overfitting the estimated
parameters, the original data points are interpolated toLdata points by the
cubic spline method^7 (Lshould be larger than the number of parameters to
be estimated). In other words, there are{xi(l+1),φi(l)}data point pairs for