Biology Reference
In-Depth Information
and it is only appropriate under certain conditions, such as constant temperature, well-
mixed solution, large number of molecules, etc. In this section, we briefly describe
how to write differential equation models from a reaction mechanism.
Reactions can be classified based on their reactants. The reaction
A
→
P
is called
uni-molecular, since one reactant
A
becomes a product
P
. The reaction
A
P
is
called bi-molecular, since there are two reactants
A
and
B
becoming a product
P
. A tri-
molecular reaction looks like
A
+
B
→
P
, and has three reactants
A, B,
and
C
.
According to the mass-action kinetics, a reaction rate is proportional to the prob-
ability of collision of the reactants involved. At higher concentration, the collisions
occur more often. This probability can be taken to be proportional to each reactant
concentration.
Now consider a uni-molecular reaction in which
A
becomes
P
,
+
B
+
C
→
k
−→
A
P
.
(2.1)
Here
k
is called a kinetic rate constant describing how likely it is for this reaction to
occur and produce the product P. According to mass action kinetics, the rate of this
reaction can be written as
d
[
P
]
v
=
=
k
[
A
]
.
dt
Here
represent concentrations of
A
and
P
, respectively;
k
is a first order
rate constant. Units for
[
A
]
and
[
P
]
are concentrations and unit of
d
[
P
]
dt
[
A
]
and
[
P
]
is concentration
per time. Therefore, unit of the rate constant
k
has to be time
−
1
.
A bi-molecular reaction looks like,
k
−→
A
+
B
P
.
(2.2)
In this reaction the reactants
A
and
B
react and become a product
P
with a rate constant
k
. The rate constant quantifies how likely it is that the collision of
A
and
B
ends up
with the product
P
. The rate of this reaction is
d
[
P
]
v
=
=
k
[
A
][
B
]
.
(2.3)
dt
)
−
1
.
The reactions given in both Eq. (
2.1
) and Eq. (
2.2
) are uni-directional reactions.
In theory, all chemical reactions are reversible. Now let's consider a two-directional
reaction like
Here
k
is a second order rate constant and its unit becomes
(
concentration
×
time
B
k
1
A
+
k
2
P
.
(2.4)
In this reaction,
A
and
B
react and become
P
with an associated rate constant
k
1
.On
the other hand,
P
can break down and produce
A
and
B
. The rate constant for the
backward reaction is
k
2
.Here
P
is produced at a rate
v
f
=
k
1
[
A
][
B
]
and consumed
at a rate
v
b
=
k
2
[
P
]
. Hence, how the concentration of
P
changes over time is given
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