Biomedical Engineering Reference
In-Depth Information
that may be critically state-dependent, such as how ionic channels interact with
drugs and toxins (26,27). Accordingly, much recent effort in modeling of car-
diac ionic currents has focused on development of biophysically detailed
Markov chain models of channel gating. We will therefore illustrate the generic
concepts involved in modeling of ion channel function and membrane currents
by reviewing our recent efforts to model the cardiac sodium (Na
+
) channel (28).
This model is able to reproduce and predict a wide range of single channel and
whole-cell current properties (28), and the ways in which this model is formu-
lated and constrained is illustrative of modern approaches to ion channel and
current modeling.
The structure of the model is shown in Figure 2A. The channel can oc-
cupy any of 13 states. The top row of states corresponds to zero to four voltage
sensors being activated (C
0
through C
4
) plus an additional conformational
change required for opening (C
4
O
1
and C
4
O
2
). The bottom row of states
corresponds to channel inactivation. Affinity of the inactivation particle binding
site is hypothesized to increase by a scaling factor (
a
) as the channel activates
and to decrease by the same factor as the channel deactivates. Closed-closed
and closed-open (horizontal) transitions are voltage dependent and closed-
inactivated (vertical0 transitions are voltage independent. Transition rates are of
a form given by Eyring rate theory (24), and include explicit temperature de-
pendence:
¬
kT
h
%
H
%
Sz
V
-
M
M
M
M
=
exp
+
+
®
,
[1]
-
-
-
RT
R
RT
where
k
is the Boltzmann constant,
T
is the absolute temperature,
h
is the Planck
constant,
R
is the gas constant,
F
is Faraday's constant,
H
M
is the change in
enthalpy,
S
M
is the change in entropy,
z
M
is the effective valence (i.e., the charge
moved times the fractional distance the charge is moved through the membrane),
and
V
is the membrane potential in volts.
The probability of occupying any particular channel state is described by a
set of ordinary differential equations, written in matrix notation as
s
P
(t)
=
WP
(t),
[2]
s
t
where
P
(
t
) is a vector state occupancy probabilities, and
W
is the state transition
matrix.
W
is in general a function of voltage and time. For the voltage-clamp
conditions generally used to constrain ion current models,
W
is piecewise time-
independent, thus Eq. [2] has the analytic solution
P
(
t
) = exp(
W
t
)
P
(0).
[3]
