Biomedical Engineering Reference
In-Depth Information
The Q-matrix now becomes
−
(
α
+
k
+
B
x
B
)
0
k
+
B
x
B
$
%
−
(
β
+
k
−
1
)
k
−
1
0
Q
=
(5.2)
0
k
+
1
x
A
−
k
+
1
x
A
0
k
−
B
0
0
−
k
−
B
Note that the transition rate from open to blocked, like the binding transition, is
given by the product of the rate constant and the drug concentration, this time the
blocker concentration
x
B
. Now when the channel is observed to be shut we cannot
tell whether it is blocked or closed, although we do know that the proportion of shut
times that are blocked is
k
+
B
x
B
/
(
α
+
, the probability of leaving state 1 for
state 4 rather than state 2. Moreover, depending on the relative values of some of
the transition rates (for example if
k
−
B
is somewhat greater than
k
+
B
x
B
)
), it may be that
short shut times are more likely to be the result of blocking and long shut times more
likely to be the result of shutting, i.e., a sojourn among the pair of shut states
(
2
,
3
)
.
5.3.3
A five-state model
We now consider a model, introduced by [29], that has been used by several authors
to describe the nicotinic acetylcholine receptor. In this mechanism there may be one
agonist molecule (
A
) or two molecules (
A
2
) bound to the shut receptor (
R
)orthe
open receptor (
R
∗
). In the following diagram that represents this model three shut
states
on the top; on
the right two agonist molecules are bound, one in the middle and none on the left. If
it were possible for the channel to open in the absence of bound agonist then there
would be another open state at the top left of the diagram.
Note that the rate constant for binding one molecule when the channel is free is
written as 2
k
+
1
because there are two free receptor sites; similarly, the dissociation
rate constants for the unbinding of one of two occupied receptor sites are written as
The transition rate matrix is
(
3
,
4
,
5
)
are shown on the bottom row and two open states
(
1
,
2
)
Q
=
$
%
k
+
2
x
A
)
k
+
2
x
A
(
α
+
0
0
1
1
2
k
−
2
2
k
−
2
)
−
(
α
+
0
0
2
2
0
−
(
β
+
2
k
−
2
)
2
k
−
2
0
2
2
0
k
+
2
x
A
−
(
β
+
k
+
2
x
A
+
k
−
1
)
k
−
1
1
1
0
0
0
2
k
+
1
x
A
−
2
k
+
1
x
A
(5.3)
10
7
M
−
1
s
−
1
;
k
+
2
=
k
+
2
=
10
k
+
1
, so that when
one agonist molecule is bound the second receptor site is more likely to bind another
agonist molecule; dissociation rates for the shut conformation
k
−
1
=
In particular, suppose that
k
+
1
=
5
×
2000
s
−
1
.
k
−
2
=
15
s
−
1
Suppose the opening and shutting rates of the singly occupied state are
=
,
1
3000
s
−
1
15000
s
−
1
=
while those for the doubly occupied state are
=
,
α
=
1
2
2
500
s
−
1
:
thus the singly occupied state is slow to open and quick to shut, while
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