Biomedical Engineering Reference
In-Depth Information
Figure 5.3
CK mechanism.
experiments. Similarly, the channel-shutting rate constant is known to be dependent
on membrane potential (for muscle-type nicotinic receptors), so it will stay constant
only if the membrane potential stays constant (i.e., only as long as we have an effec-
tive voltage clamp).
5.3.1
A three-state mechanism
If a ligand must be bound before the ion channel can open, at least three discrete
states are needed to describe the channel mechanism. The mechanism of Castillo
and Katz [22], the CK mechanism, has two shut states and one open state; this is
usually represented as in Figure 5.3 where
R
represents a shut channel,
R
∗
an open
channel, and
A
represents the agonist molecule. The states have been numbered
to facilitate later mathematical representation. State 1 is the open state in which an
agonist molecule is bound to a receptor on the channel; in state 2 a molecule is bound
but the channel is shut; in state 3 the channel is shut and its receptor is unoccupied.
The rate constants are shown next to each possible transition.
A single channel makes transitions between its states in a random fashion and the
transition rates will determine the probability distributions that describe the occu-
pancy times of the various states and the states into which the transitions take place.
In the CK mechanism, for example, the shutting rate,
, of an open channel must be
interpreted in a probabilistic way: roughly, we can say that the probability of an open
channel shutting in the next small interval of time
t
, is approximately
t
. More
precisely, we can interpret the transition rate as
α
=
lim
t
→
0
Prob
(
channel shut at
t
+
∆
t
|
channel open at
t
)
/
∆
t
Thus the transition rate is thought of as the limit of a conditional probability over a
small time interval. Notice that this is supposed to be the same at whatever time
t
we start timing our interval, and also to be independent of what has happened earlier,
i.e., it depends only on the present (time t) state of the channel. This is a fundamental
characteristic of our type of random process (a homogeneous Markov process).
More generally, we can define any transition rate in this way. Denote by
q
ij
the
transition rate from state
i
to state
j
. Then, for
j
not equal to
i
,
q
ij
=
lim
0
Prob
(
channel in state
j
at time
t
+
∆
t
|
channel in state
i
at time
t
)/
t
t
→
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