Biomedical Engineering Reference
In-Depth Information
carrier-substrate complex from outside to inside has the same reaction rate,
. Using the
law of mass action, we get
q
S
o
¼
K
q
S
o
q
C
o
þ
K
1
q
P
o
q
C
o
¼
K
1
q
S
o
q
C
o
þ
K
1
q
P
o
þ
K
2
q
C
i
K
2
q
C
o
q
P
o
¼
K
1
q
S
o
q
C
o
K
1
ð
þ
K
Þ
q
P
o
þ
K
q
P
i
1
2
2
ð
8
:
81
Þ
q
P
i
¼
K
1
þ
K
2
ð
Þ
q
P
i
þ
K
2
q
P
o
þ
K
1
q
S
i
q
C
i
q
S
i
¼
K
1
q
S
i
q
C
i
þ
K
q
P
i
1
q
C
i
¼
K
1
q
S
i
q
C
i
þ
K
q
P
i
þ
K
q
C
o
K
q
C
i
1
2
2
Since the carrier is not consumed in the reaction, then the total carrier is a constant, given as
q
C
o
þ
q
C
i
þ
q
P
o
þ
q
P
i
¼
:
Naturally, we can add an input to the system in Eq. (8.81) or simplify using the quasi-
steady-state approximation as before. In addition, the substrate can be involved in other
reactions inside the cell, such as moving into an organelle (mitochondria) via diffusion
and then experiencing an enzyme reaction.
z
Glucose Transport
Consider the transport of glucose across the cell membrane. We know that glucose does
not diffuse across the cell membrane but is transported across the cell membrane by a car-
rier-mediated transport process. Glucose binds to a protein that transports it across the
membrane, allowing it to pass into the cytosol. This process does not use any energy. Using
the model illustrated in Figure 8.21, we have
ð
8
:
82
Þ
where
G
o
and
G
i
is glucose outside and inside the cell, respectively;
C
o
is the carrier on the
is the carrier on the inside of the membrane;
is the bound
outside of the membrane; C
i
P
o
substrate and carrier complex on the outside of the membrane; and
P
i
is the bound sub-
strate and carrier complex on the inside of the membrane. Note that there is no reverse reac-
tion for glucose in Eq. (8.82), since glucose does not leave the cell. We assume that glucose is
consumed at a constant rate
inside the cell during cell respiration (described in the next
section) and that glucose is available in the interstitial fluid at a rate of
J
i
J
o
.
The equations that
describe this system are given by
q
G
o
¼
K
1
q
G
o
q
C
o
þ
J
o
q
C
o
¼
K
q
C
o
q
P
o
¼
K
1
q
G
o
q
C
o
K
2
q
P
o
þ
K
2
q
P
i
q
G
i
¼
K
q
G
o
q
C
o
þ
K
q
C
i
K
1
2
2
ð
8
:
83
Þ
q
P
i
J
i
1
q
C
i
¼
K
q
P
i
þ
K
q
C
o
K
q
C
i
1
2
2
q
P
i
¼
K
2
q
P
o
K
1
q
P
i
