Biology Reference
In-Depth Information
GRNs [
10
,
11
]. As it has also been observed that GRNs tend to be ordered systems
with a limited number of regimes, there are methods which filter their results to only
those state spaces with few steady states and short limit cycles (see Section
3.2
for
definitions). Other aspects of GRNs, such as the negative or positive feedback loops
and oscillatory behavior, are also considered in model selection.
3.2
POLYNOMIAL DYNAMICAL SYSTEMS (PDSs)
Different mathematical frameworks have been proposed for the modeling of GRNs,
with continuous models using differential equations being the most common
approach. While the behavior of a biological system may be seen as continuous in
that change of concentration of biochemicals can be modeled with continuous func-
tions, the technology to record observations of the system is not continuous and the
available experimental data consist of collections of discrete instances of continuous
processes. Furthermore, discrete models where a variable (e.g., a gene) could be in
one of a finite number of states are more intuitive, phenomenological descriptions
of GRNs and, at the same time, require less data to build. The framework of finite
dynamical systems (FDS) provides the opportunity to model a GRN as a collection
of variables that transition discretely from one state to the next.
Definition 3.1.
A
finite dimensional system
of dimension
n
is a function
F
=
S
n
S
n
, where each
f
i
S
n
(
f
1
,...,
f
n
)
:
→
:
→
S
is called a
local
(or
transition
)
function, and
S
is a finite set.
For each variable
x
i
, its local function
f
i
determines the state of
x
i
in the next
iteration based on the current state of all variables, including possibly
x
i
itself. If we
further require that the state set
S
be a finite field, then a result in [
12
] guarantees that
the local functions of an FDS can be expressed as polynomial functions. Working
exclusively with polynomials facilitates the modeling process significantly, as we
shall see in later sections. Fortunately, from basic abstract algebra we know that a
restriction on the size of
S
can turn it into a
finite field
by requiring that the cardinality
of
S
be a power of a prime integer. An example of a finite field is the set of integers
modulo
p
, denoted
Z
p
, where
p
is prime. The elements of
Z
p
are the equivalence
classes of remainders upon division by
p
, namely
, and addition
and multiplication of these classes corresponds to addition and multiplication of
representatives from the classes modulo
p
.
Example 3.1.
[
0
]
,
[
1
]
,...,
[
p
−
1
]
The finite field
Z
5
has elements
[
0
]
,
[
1
]
,
[
2
]
,
[
3
]
, and
[
4
]
.Wesee
that
[
2
]+[
4
]=[
1
]
since 2
+
4
=
6
≡
1 mod 5. Similarly,
[
2
]·[
4
]=[
3
]
since
2
·
4
3 mod 5.
For simplicity, we will write
m
instead of
=
8
≡
[
m
]
to represent the elements of a finite
field.
If the state set for an FDS
F
is a finite field
F
, we call
F
a
polynomial dynamical
system
(PDS).
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