Figure 1.
The structure of the Siekmann model.
The model is comprised of two modes. One is the drive mode containing three closed states
,
,
and one open state
. The other is the park mode which includes one closed state
and one open state
.
are rates of state-transitions between two adjacent states and
and
are transitions between the two modes [7].
Figure 2.
oscillations in ASMC in lung slices are generated by a stochastic mechanism.
A: experimental spiking in ASMC in lung slices, stimulated with 50 nM MCh. In the upper panel we filter out baseline noise by using a low threshold of 1.42 (relative fluorescence intensity) and then choose samples with amplitude larger than 1.75. The ISI calculated from the upper panel is shown in the lower panel. B: relationship between the standard deviation and the mean of experimental ISIs. Data obtained from 14 ASMC in 5 mouse lung slices. The relationship is approximately linear with a slope of 0.66, which implies that an inhomogeneous Poisson process governs the generation of oscillations. The dashed line indicates where the coefficient of variation (CV) is 1 (as it is for a pure Poisson process). Variation in ISI is mainly caused by both use of different doses of MCh and different sensitivities of different cells to MCh. Error bars indicate the standard errors of the means (SEM).
Figure 3.
A 2-state open/closed model quantitatively reproduces the 6-state model.
A: histograms of interspike interval (ISI) distribution for both the 6-state and the simplified models. The ISI is defined to be the waiting time between successive spikes. Each histogram contain an equal number of samples (180). B: comparison of average ISI, average peak value of (
in the model) and average spike duration. All distributions were computed at a constant
.
Figure 4.
More detailed comparisons between the 2-state and the 6-state models, and a comparison to experimental data.
As a function of concentration (
), the two models give the same ISI (A), peak
(C) and spike duration (E). These results agree qualitatively with experimental data, as shown in panels B, D and F respectively. Quantitative comparisons are generally not possible as the relationship between
concentration and agonist concentration is not known. Error bars represent
. Data for each MCh concentration are obtained from at least three different cells from at least two different lung slices.
Figure 5.
Stochastic and deterministic simulations exhibit similar dynamic properties.
A: simulated stochastic (upper panel) or deterministic (lower panel) oscillations at
. B: a typical stochastic solution projected on the
plane. The average
represents the average value of
over the 20
. Statistics (
) of the initiation point (blue square), the peak (red square) and termination point (green square) are shown in the inset. 116 samples are obtained by applying a low threshold of
and a high threshold of
to
. C: a typical periodic solution of the deterministic model (black curve), plotted in the
phase space. The arrow indicates the direction of movement.
is the slowest variable so that its variation during an oscillation is very small. This allows to treat
as a constant (
in this case) and study the dynamics of the model in the
phase space. The color surface is the surface where
(called the critical manifold). The white N-shaped curve is the intersection of the critical manifold and the surface
. D: projection of the periodic solution to the
plane. The red N-shaped curve is the projection to the
plane of the white curve shown in C. The evolution of the deterministic solution exhibits three different time scales separated by green circles (labelled by a, b and c) and indicated by arrows (triple arrow: fastest; double arrow: intermediate; single arrow: slowest).
Figure 6.
Comparison of parameter-dependent frequency changes in the stochastic and deterministic models.
All curves are computed at
except in panel A, which uses a variety of
. Other parameters are set at their default values given in Table 1. A: as
increases,
oscillations in both models increase in frequency. B: as
influx increases (modeled by an increase in receptor-operated calcium channel flux coefficient
), so does the oscillation frequency in both models. C: as
efflux increases (modeled by an increase in plasma pump expression
), oscillation frequency decreases. D: as SERCA pump expression,
, increases, so does oscillation frequency. E: as total buffer concentration,
, increases, oscillation frequency decreases.
Figure 8.
Schematic diagram of the model.
represents cytoplasmic
concentration, excluding a small local
(whose concentration is denoted by
) close to the
release site (i.e., an
cluster). Upon coordinated openings of the
, SR
(
) is first released into the local domain (
) to cause a rapid increase in
. High local
then diffuses to the rest of the cytoplasm (
), and is eventually pumped back to the SR (
).
Figure 7.
Dependence of calcium oscillation baseline on calcium influx and SERCA expression.
A: increasing influx (described by ) increases the average trough of
oscillations. B: decreasing SERCA expression (described by
) increases the average trough of
oscillations. All curves are computed at
.
Table 1.
Parameter values of the stochastic calcium model.
Table 2.
Parameter values of the model.
Figure 9.
Stationary data and fits of and
.
Stationary transition rates of and
,
and
, as functions of
concentration were estimated and fitted for two
,
(A) and
(B). Circles and squares represent the means of
and
distributions computed by MCMC simulation [7]. Note that MCMC failed to determine the values of
and
at
for
, as the
was almost in the drive mode for these cases. The corresponding fitting curves (solid for
; dashed for
) are produced using Eqs. 7–12.
Figure 10.
Open probability curves for various .
is equal to the sum of probabilities of the
in
and
. Three representative curves correspond to
,
and
(from bottom to top) respectively. Data (average open probability) are from [5].