Biomedical Engineering Reference
In-Depth Information
x
,
T
=
(ln )
r
M
[11]
j
jk
k
=
1
where
r
is a random variable drawn from a uniform distribution on the interval
[0,1] and M
jk
is the transition rate from state
j
to state
k
. The sum is over the
x
pathways out of state
j
. The resulting dwell time
T
j
is an exponentially distrib-
x
uted random variable with parameter
. At the end of the dwell time,
M
=
M
jk
k
=
1
the new state of the channel is determined by assigning random numbers to a
portion of the interval [0,1] based on the probabilities of changing to neighbor-
ing states. These probabilities are equal to the rate constant for a particular tran-
sition divided by the sum of the rate constants for all possible transitions. Once
the new state is determined, another random number is used to calculate the
dwell time in the new state. At an instantaneous voltage step, channels remain in
their current state, but the dwell times are recalculated.
Extensive experimental data were required to fully determine the model
parameters. The majority of data were taken from human SCN5A-encoded Na
channels. Experimental data obtained at temperatures of 13( and 21(C were
used to constrain the model, and the ability of the model to predict data collected
at 17(C was tested. Constraining data included: (a) ionic currents in response to
voltage-clamp; (b) gating charge accumulation; (c) steady-state inactivation
curve; (d) rate of tail current relaxation; (e) time course of recovery from inacti-
vation; and (f) single-channel open times. A cost function defined as the squared
error between simulated and experimental data (including both whole-cell cur-
rent and single-channel data) was minimized to determine an optimal model
parameter set. A simulated annealing algorithm (30) was needed to perform this
minimization, as the cost function exhibited many local minima. The resulting
model was able to reproduce a broad range of membrane current data (28), and
Figures 2B-E demonstrates the ability of the model to predict channel/current
properties at 17(C as well as single-channel data not included in the fitting
process. A similar methodology has been used to develop quantitative models of
other myocyte membrane currents, most notably
I
Kr
,
I
Ks
,
I
CaL
, and
I
to1
(12,31-33).
2.4. Model Components: Intracellular Ion Concentration Changes
We illustrate the process of modeling time-varying changes of intracellular
ion concentration with reference to the common pool model architecture shown
in Figure 1B. In the common pool model formulation, there are four distinct Ca
2+
compartments (the cytosol, subspace, NSR, and JSR) and one Na
+
and potassium
(K
+
) compartment (the cytosol). Note that in present common pool myocyte
models, the cytosolic concentrations of both Na
+
,K
+
and Ca
2+
are assumed to be
