Biomedical Engineering Reference
In-Depth Information
Boolean functions at the current time point. The deterministic nature of the Boolean
network allows for fast analysis, and the application of logic synthesis tools. While
Boolean networks exhibit many observed characteristics of gene regulatory networks,
Boolean networks cannot model the continuous levels of gene expression values.
In the context of RNA and protein production, actual gene expression is more
complex and is measured as a continuous value from measurement techniques such
as microarrays. Other models have been proposed to model the GRN with continu-
ous value gene expression such as Differential Equations [
7
], Linear Equations [
8
],
Continuous Networks [
9
], and Stochastic Gene Networks [
10
]. While such models
can determine continuous functions to model the gene expresion data, continuous
functions in general cannot capture the Boolean-like behavior of genes.
In [
11
], a method was proposed to model continuous gene expression. This model
is based on Zhegalkin polynomial functions [
12
]. Zhegalkin functions are an alterna-
tive representation of Boolean functions, using continuous values. These Zhegalkin
functions can represent any Boolean function having an output value within the unit
interval [0,1] if input variables are also within the unit interval. In [
11
], it was demon-
strated how Zhegalkin functions can be used to model the next state equation for a
given a predictor (target gene and input genes), and time-series expression data for
yeast model dataset.
Our approach also uses Zhegalkin functions to infer gene predictors and func-
tions from normalized continuous gene expression data. As opposed to [
11
] which
uses a linear expression function, our method uses a sigmoid expression function to
more accurately represent the gene expression As a result, our method captures the
Boolean-like behavior of the GRN, unlike [
11
]. Another key difference is that [
11
]
only finds a single regulating function for a given predictor and gene expression data,
while our methods finds the best predictor and regulating function for a target gene
given just the gene expression data, by searching across all possible predictors and
functions in the GRN through a ranking of best fitting predictors using mean-squared
error as a metric.
4.2
Approach
4.2.1
Network Model
As described earlier, the Boolean network model cannot be used with continuous gene
expression values. Instead we use a model similar to BN but which uses continuous
expression values and modified Zhegalkin functions (Sect.
4.2.2
) in place of Boolean
values and Boolean functions. In the model, we define a set of nodes
{
x
1
,
x
2
,
...
,
x
n
}
and modified Zhegalkin functions
. Each node
x
i
is a gene, and each
gene is associated with a Zhegalkin function
z
i
. The value of a gene is a continuous
variable,
x
i
{
z
1
,
z
2
,
...
,
z
n
}
∈
[0, 1], and is updated according to the associated Zhegalkin function
z
i
. Thus, the gene state
x
i
can represent varying levels of expressions from fully
expressed (
x
i
=
1), not expressed (
x
i
=
0), and any expression level in between.
