Biomedical Engineering Reference
In-Depth Information
as Michaelis-Menten kinetics. The models presented here assume that temperature and
other conditions remain constant unless otherwise indicated.
This section introduces a standard approximation that greatly simplifies the analysis of
biochemical reactions and predictions of how fast a reaction will occur. As shown, the
quasi-steady-state approximation provides an excellent representation of the system's dif-
ferential equations, which are problematic, since they involve stiff differential equations.
By assuming a quasi-steady-state approximation, the set of differential equations is reduced
to a set of algebraic equations. When the early pioneers developed the theory of enzyme
reactions, the solution of the differential equations was not possible by either simulation
or direct solution. Thus, the quasi-steady-state approximation allowed a rather complete
description of the enzyme reaction except for the initial stage involving the formation of the
complex. The set of algebraic equations also provides a mechanism to measure the parameters
of the reaction.
Stiff differential equations require simulation solutions with an extremely small step size
using the standard integrators. Consider Figure 8.4 and the graph for
q
B
changes quickly, and after that, it changes slowly. The step size needs to be very small,
after which a small step size is not needed, since
q
B
:
From 0 to .02 s,
is slowly changing. Thus, simulations
take a very long time to run using the standard integrator. If the step size is too large, the
simulation solution is incorrect because small errors amplify as it reaches steady state.
The default integrator
ode45
in SIMULINK is not a good choice for stiff problems because
it is inefficient. The integrator
ode23tb
is a better choice for stiff problems, which manip-
ulates the step size using an efficient algorithm. We will explore this issue later in this
section.
The capacity-limited reaction model uses a two-step process given by Eq. (8.33):
q
B
ð
8
:
33
Þ
The enzyme mediated reaction first has substrate
S
combining with enzyme
E
to form the
unstable complex
ES
. Then the complex
ES
breaks down into the product
P
and
E
.
The law of mass action equation for Eq. (8.33) is
q
S
¼
K
1
q
S
q
E
þ
K
1
q
ES
q
ES
¼
K
1
q
S
q
E
K
1
þ
K
2
ð
Þ
q
ES
ð
8
:
34
Þ
q
P
¼
K
2
q
ES
Since
q
E
ð
0
Þ¼
q
E
þ
q
ES
or
q
E
¼
q
E
ð
0
Þ
q
ES
, we have
q
E
¼
q
ES
:
Thus,
q
E
¼
K
q
S
q
E
þ
K
1
ð
þ
K
Þ
q
ES
ð
8
:
35
Þ
1
2
To begin with the classical description, we assume that the system is closed—that is,
q
S
ð
0
Þ¼
q
S
þ
q
ES
þ
q
P
and
q
E
ð
0
Þ¼
q
E
þ
q
ES
. Note that
E
is not consumed in the reaction.
Moreover, it is assumed that the complex
ES
increases quickly to a maximum,
q
ES
max
and
then changes very slowly after that. Therefore, we assume complex
ES
is in a quasi-
steady-state mode shortly after the reaction starts—that is,
q
ES
¼
0. We also assume that
