Biomedical Engineering Reference
In-Depth Information
Figure 3.2
Reaction of calcium binding to a buffer.
equation as in Figure 3.2: in which calcium binds to buffer with a forward rate con-
stant
k
f
and unbinds with a backward rate constant
k
b
. This buffering flux describing
the rate of change for calcium can be represented by the equation
J
buffer
=
−
k
f
[
Ca
][
B
]+
k
b
[
CaB
]
(3.2)
In addition to the [Ca] balance equation, it is necessary to describe [B] and [CaB] by
differential equations for their rate of change. If it is assumed that the total amount of
buffer ([
B
total
]) remains constant, then we can assume that ([
B
total
] = [B] + [CaB]),
which allows the reduction of the total number of differential equations by one for
each buffer. Typically the rate constants
k
f
and
k
b
are very fast compared to other
cellular processes. This requires for a very small time step to be used when solv-
ing the differential equations associated with buffering. Because of the difference in
time scale between these reactions and other cellular processes, this will increase the
computational time necessary to solve the problem. A solution to this problem is the
rapid buffering approximation, suggested by Wagner and Keizer [53], which assumes
that the buffering reaction (Eq. 3.2) is in equilibrium. Using a steady state approx-
imation with the differential equations describing buffering, and the chain rule, the
equation describing the rate of change for total cytosolic calcium with respect to
time can be broken down into a term for the rate of change for the free calcium with
respect to time multiplied by the rate of change of the total calcium with respect to
the free calcium. This yields a buffering factor c that scales the other fluxes in the
calcium balance equation to account for the fraction of other fluxes that is not bound
by the buffer.
1
−
1
[
B
S
,
total
]
K
S
,
eq
[
B
M
,
total
]
K
M
,
eq
[
B
E
,
total
]
K
E
,
eq
=
+
2
+
2
+
c
(3.3)
(
K
S
,
eq
+[
Ca
])
(
K
M
,
eq
+[
Ca
])
(
K
E
,
eq
+[
Ca
])
2
where
K
S
,
eq
,
and
K
E
,
eq
are the disassociation constants for calcium and the sta-
tionary, mobile, and exogenous buffers, respectively. The total concentrations for the
stationary, mobile, and exogenous buffers are [
B
S
,
total
], [
B
M
,
total
], and [
B
E
,
total
],
respectively. Additional terms can be added to account for different buffers and cal-
cium binding dyes on calcium dynamics. The rapid buffering approximation also
accounts for the effect of buffering on diffusion by altering the diffusive flux to
K
M
,
eq
,
∂
2
[
Ca
]
J
diffusion
=
D
Ca
(3.4)
∂
x
2
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