Biomedical Engineering Reference
In-Depth Information
where
f
D
γx
is the vector field of the system on a regular domain
D
.In
Fig.
2.3
, there are only two regular domains (
A
and
B
) whose boundary contains
the switching domain defined by
x
2
=
θ
2
, and hence the convex hull
H
(
x
) is the
cone formed by the two vectors
f
A
(
x
)
−
−
γx
.
A solution of Eq. (
2.5
)on[0
,T
]
in the sense of Filippov
is an absolutely
continuous (w.r.t.
t
) function
ξ
(
t
;
x
0
) such that
ξ
(0;
x
0
)=
x
0
γx
and
f
B
(
x
)
−
and
dξ/dt
∈
H
(
ξ
)
for almost all
t
∈
[0
,T
]. It may give rise to a sliding mode along the plane defined
by the threshold. See for instance [
9
] for a deeper analysis of solutions of PWA
systems.
2.2.3
Methodological Developments
In general, there are
n
molecular species in a system,
x
=(
x
1
,...,x
n
)
t
n
+
,
and the changes in the concentration of species
i
result from a balance between
production or synthesis processes (
g
i
(
x
)
≥
0) and degradation or transformation
(
d
i
(
x
)
≥
0) into other species. Each species is thus characterized by an equation of
the form:
∈
R
dx
i
dt
=
g
i
(
x
)
−
d
i
(
x
)
.
(2.6)
The main problem is the choice of appropriate mathematical expressions of
g
i
(
x
)
and
d
i
(
x
). These will depend on the modeling formalism and should reflect the
observed dynamical behavior. For instance, for PWA systems,
g
i
(
x
)=
f
i
(
x
) and
d
i
(
x
)=
γ
i
x
i
, as seen in Eq. (
2.4
).
2.2.3.1
Modeling Transcription and Translation
In the classical sequence of events, transcription is initiated upon binding of a
transcription factor to the gene. Besides transcription factors, other proteins may
bind to specific sites of the gene and contribute to enhance (respectively, reduce) the
transcription rate. Such proteins are called
activators
(respectively,
repressors
). In
general, the binding of
m
molecules of protein
A
to the promoter site of a gene (
D
)
to form a new complex (
C
), is represented as
k
1
k
2
D
+
mA
C,
(2.7)
and can be modeled through the
law of mass-action
, where each reaction rate is
proportional to a product of concentrations according to the stoichiometry:
C
=
k
1
DA
m
− k
2
C,
(2.8)
D
=
−
C.
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