Biomedical Engineering Reference
In-Depth Information
tioned before, when the channel leaves state i it moves into state j with probability
−
ii
. Thus, if we divide the non-diagonal elements of each row of the ma-
trix
Q
by minus the diagonal element on that row, we get a new matrix whose rows
represent conditional probability distributions of the next state to be entered. The
diagonal elements must be zero.
In this model we get the matrix
q
ij
/
q
−
$
%
00
.
0164
0
0
.
9836
0
0
.
0013
0
0
.
9987
0
0
00
.
7895
0
0
.
2015
0
(5.5)
.
.
.
0
0073
0
0
0242
0
0
9685
0
0
0
1
0
Thus, for example, a channel in state 4 (AR) has a probability of 0.0073 of opening
(move to state1), probability 0.0242 of binding a second agonist molecule but has a
very high probability of 0.9585 of losing its agonist molecule (move to state 5). In
contrast, a doubly occupied channel
A
2
R
(state 3) has a probability of 0.7895 of
opening (move to
A
2
R
∗
, state 2) rather than losing one of its agonist molecules.
There are many similar models that can be constructed in this way, but the above
will suffice for this chapter. Some models can get quite large: Rothberg and Magleby
[68] have considered a 50-state model for a calcium-activated potassium channel.
Ball [20] has studied a model based on molecular structure that has 128 states of
which 4 are open: by exploiting assumptions of symmetry this effectively reduces to
a model with 3 open states and 69 closed states - still quite big!
Although we shall see that it is possible to eliminate some models for a particular
mechanism on the basis of observable characteristics, a certain amount of indeter-
minacy arises because we cannot see everything that a channel does (we cannot see
which individual state the channel is in, only if it is open or closed). It is possible
that two or more distinct models may give rise to the same observable features under
fixed conditions, see for example [38, 40, 55]. However, further discrimination be-
tween models is possible by observing the same channel under different conditions,
changing voltages or agonist concentration.
5.4
Transition probabilities, macroscopic currents
and noise
In order to predict macroscopic currents and the behaviour of noise measurements,
we first need to study some transition probabilities for a single channel.
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