Biomedical Engineering Reference
In-Depth Information
where
I
Na
(
t
) is the Na current,
N
is the number of Na channels,
G
Na
is single-
channel conductance,
P
open
(
t
) is the probability of occupying the open states (O
1
+ O
2
),
V
(
t
) is membrane potential, and
E
Na
(
t
) is the reversal potential for Na
given by the Nernst equation.
The number of coupled differential equations, and hence the number of pa-
rameters that need to be constrained for the model, may be reduced through ap-
plication of two fundamental principles. First, the state occupancy probabilities
for a Markov chain model must sum to one. Second, there are several loops in
the model that must satisfy the principle of microscopic reversibility. Micro-
scopic reversibility is derived from the law of conservation of energy and states
that the product of rate constants when traversing a loop clockwise must be
equal to the product when traversing the same loop counterclockwise (24). For
the closed-closed-inactivated loops, satisfying microscopic reversibility requires
that the transitions among the closed-inactivated states be scaled by
a
, the same
factor used to scale the transitions between rows. Microscopic reversibility is
preserved around the closed-open-inactivated loop by isolating the
H
,
S
, and
z
terms in the product and satisfying each term separately using the following
equations:
%
H
=
%
H
+
%
H
+
%
H
+
%
H
+
8ln
RT
a
% % %
, [5]
H
H
H
HH
H
on
EE
cf
E
cn
of
%
S
=
%
S
+
%
S
+
%
S
+
%%%%,
S
S
S
S
[6]
HH
H
on
EE
cf
E
cn
of
z
=+++
z
z
z
z
.
z
[7]
HH
H
on
E
of
EE
Similarly, microscopic reversibility is preserved around the closed-open-open
loop using the following equations for
H
I
,
S
I
, and
z
I
:
%
HHHHHH
I
=
%
+
%
+
%%%,
[8]
H
F
O
E
X
%
S
=
%
S
+
%
S
+
%%%,
S
S
S
[9]
I
H
F
O
E
X
z
=+
z
z
z
[10]
.
I
H E O
These microscopic reversibility constraints thus reduce the dimension of the
parameter estimation problem, as transition rates HH and I are fully constrained.
The model of Figure 2A may also be viewed as a Markov chain description
of single channel behavior. Single-channel gating may be simulated using the
method of Clay and DeFelice (29). In this method, the length of time a channel
stays in its current state (i.e., its dwell time, denoted as
T
j
) is calculated accord-
ing to the formula
