Biomedical Engineering Reference
In-Depth Information
As the population of the chemical species increases, the second term in the above
equation has a negligible effect compared with the first one: indeed, this one scales
linearly with the size of the system while the second term scales only sublinearly.
Leaving out the stochastic term of the above equation, the time evolution of the
system can be represented by the
reaction rate equations
, a continuous and deter-
ministic approach:
X
d
X
d
D
v
j
a
j
.
X
.t //
t
j D1
Note that considering the notation of Sect.
6.4
,
P
j D1
v
j
a
j
.
X
.t // D
Nf
. According
to this formalism, the functions
are called
kinetic laws
and define the rate
of the biochemical reactions; moreover, the system state is usually expressed in
terms of molecules concentration. A general form of kinetic laws is
a
j
.
X
.t //
Y
X
ˇ
i
i
a
j
.
.t // D k
j
.t /;
X
i D1
where the real valued elements
ˇ
i
are, respectively, the kinetic constants and
the kinetic orders. Following this definition, the rate of the process
k
j
and
r
j
is determined
by the kinetic constant and the product among the molecular species concentrations.
For example, considering the reaction
r
9
W s
4
C s
5
! s
6
C s
7
, a possible kinetic
law is:
a
9
D k
9
X
2
X
4
X
5
s
7
is determined by a specific kinetic constant,
k
9
, and by the concentration of reactants
s
6
and
i.e., the rate of production of
s
2
.
A number of different modeling formalisms arise from this general form [
26
].
On the one hand, if kinetic orders assume only integer values, we obtain the
con-
ventional kinetic models
. In turn, by introducing particular approximations in this
formalism, it is possible to derive a number of specific equations such as the
Michaelis-Menten equation. On the other hand, if kinetic orders can be real valued,
we have the class of
power law models
. If the values are only positive we have
de-
tailed power law models
, while if the values can be both negative and positive we
have
simplified power law models
.
S-Systems
(“synergism and saturation”) models
can be derived from simplified power law models, by the aggregation of all the pos-
itive kinetic laws (
v
i;j
>0
s
4
,
s
5
and enzyme
) in a unique input flux and all the negative kinetic laws
(
v
i;j
<0
) in a unique output flux. According to this formalism each ODE is defined
as follows:
Y
Y
d
X
d
ˇ
i;j
j
ˇ
i;j
j
k
i
D k
i
X
X
t
j D1
j D1
The map depicted in Fig.
6.2
summarizes the relationships among the mathemat-
ical descriptions presented above. By introducing successive approximations, it is
possible, first, to switch from a discrete and stochastic description to a continuous
and stochastic one and, second, to derive a continuous and deterministic approach.
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