Citation: Chen X, Kang J, Fu C, Tan W (2013) Modeling Calcium Wave Based on Anomalous Subdiffusion of Calcium Sparks in Cardiac Myocytes. PLoS ONE 8(3):
e57093.
https://doi.org/10.1371/journal.pone.0057093
Editor: Yulia Komarova,
University of Illinois at Chicago, United States of America
Received: October 28, 2012; Accepted: January 17, 2013; Published: March 6, 2013
Copyright: © 2013 Chen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 11272014 and No. 10825208), and National Key Basic Research Program of China (Grant No. 2013CB531200). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Nomenclature
spatial coordinates, 
m
time, ms
fractional order of the spatial derivative.
, 
diffusion coefficients along 
-axis and 
-axis, 
free 
concentration, 
M
resting 
concentration, 
M
Ca-bound fluo-3 concentration, 
M
total fluo-3 concentration, 
M
Ca-bound endogenous buffer concentration, 
M
total endogenous buffer concentration, 
M
, 
forward rate constants for dye and endogenous buffer reactions, 
, 
reverse rate constants for dye and endogenous buffer reactions, 
, 
spatial separation of CRUs along 
-axis and 
-axis, 
m
current through the CRU, pA
Faraday’s constant, 
SR pump Michaelis constant, 
M
maximum SR pump rate, 
SR pump Hill coefficient
CRU Hill coefficient
molar flux of a clustered RyR channel, 
open time of CRU, ms
stochastic switching function equaling either 0 or 1
sensitivity parameter, 
M
probability of 
spark occurrence,/calcium release unit/ms
maximum probability of 
spark occurrence,/calcium release unit/ms
wave velocity along 
-axis, 
wave velocity along 
-axis, 
In the endoplasmic or sarcoplasmic reticulum(SR) of cardiac cells, there stores plenty of 
, the concentration of which is 2–3 orders of magnitude greater than that in the cytosol. During the excitation-contraction(EC) coupling process, triggered by L-type 
channels, 
is released from SR through ryanodine receptors(RyRs) on the z-lines [1]–[4], where RyR is one kind of 
release units(CRUs). This event is called “
spark”. 
-induced 
release(CICR) makes RyRs fire in succession such that 
concentration rises [1], [5], the process of which is called calcium transient. Physiologically, calcium homeostasis is important for the contraction and relaxation of the heart muscle. However, in some pathological conditions, spontaneous propagating wave of 
may occur, which is called “calcium wave”. The occurrence of calcium wave can affect the heart’s normal function, and may induce some disease, such as ventricular arrhythmias [6].
The model of 
spark using Fick’s Law failed to reproduce the full-width at half maximum(FWHM) of experimental results for 
sparks. Simulated results for 
spark based on Fick’s Law presented a lower FWHM (∼1.0 μm), which was only half the width of experimental result (∼2.0 μm). Izu et al. [7] tried to increase the current through RyR to get larger FWHM, however, the spark amplitude also increased (∼10 times), which is far beyond experimental results and physiological conditions. In contrast, the results obtained with the anomalous subdiffusion model of 
spark were found to be in close agreement with the experimental ones so that the “FWHM Paradox” was successfully explained [8]–[10]. Therefore, it is confirmed that diffusion of 
in cytoplasm obeys no longer Fick’s Law, but the anomalous subdiffusion.
A 
wave is formed from propagation of 
sparks. According to the results for 
sparks, 
wave should also obey the anomalous subdiffusion. However, all previous work on 
waves were based on Fick’s Law [11]–[14]. Anisotropic 
diffusion was studied by Girard et al. [11]. Keizer and Smith [12] investigated 
waves under stochastic firing of CRUs. Izu et al. [13] combined large CRU currents [7], stochastic firing of CRUs, asymmetric distribution of CRUs and anisotropic 
diffusion to investigate the propagation of 
waves. Lu et al. [14] studied the effect of rogue RyRs on 
waves in ventricular myocytes with heart failure.
In this work, we develop a mathematical 2D model based on anomalous subdiffusion of 
sparks. The anomalous subdiffusion model is used to study 
waves propagation from an initial firing of one CRU at a corner or in the middle of the considered region. We reproduce wave velocities of experimental results using a small current through CRUs which is close to the physiological conditions. The phenomenon of calcium waves collision is also investigated. With physiological 
currents(2pA) through CRUs, an initial firing of four adjacent CRUs is shown to form a 
wave. Furthermore, study on how the system becomes unstable is also performed by changing the transverse distance of CRUs.
Methods
0.1 Anomalous Diffusion Model for Calcium Waves
Figure 1 shows a 2-dimensional schematic of a cardiac myocyte(establishing line resources along 
-axis [13]) which contains plenty of CRUs. The regular intervals of CRUs are 
along 
-axis and 
along 
-axis. The governing equation for 
waves based on the anomalous subdiffusion model can be expressed as
(1)where 
is the free 
concentration; 
and 
are diffusion coefficients for anisotropic diffusion with 
and 
[15]; 
and 
are fluxes due to 
fluorescent indicator dye and endogenous stationary buffers; 
is pumping rate of SR 
-ATPase, and 
is a SR leak that is to balance 
; The expressions of 
, 
, 
, and 
are
(2)
(3)
(4)
(5)
(6)
(7)where 
identifies each of buffer species. 
and 
represent total concentration of the indicator and buffers, respectively. 
and 
are concentration of the 
-bound complexes. 
, 
, 
and 
are reaction kinetics. 
is the affinity constant for SR pumps, 
the Hill constant, and 
the maximum rate. SR leak is used to balance 
in resting state.
is the flux of 
release from CRU, the expression of which is the same as that of Izu et al. [13],
(8)where 
is a molar flux of a clustered RyR channel(
is current through the CRU and F is Faraday’s constant), and 
the Dirac delta function, S a stochastic function which controls the firing of the CRU, and 
the firing time. Within a time interval 
, the probability that the CRU fires is 
, where 
with 
the maximum probability of 
spark occurrence, 
the sensitivity parameter and 
the Hill coefficient.
The anomalous space diffusion is model used in Eq.(1), where 
is the Riemann-Liouville operators which is defined as
(9)
(10)where 
is an integer with 
(
). In Eq.(1), when 
, anomalous space superdiffusion occurs, while anomalous space subdiffusion occurs when 
. Particularly, our model reduces to Fick’s Law when 
. According to Li et al. [10], calcium sparks follow the anomalous space subdiffusion of 
, so we only consider the space subdiffusion in the following sections.
Results and Discussion
Modeling a 
wave from a Single 
Spark
waves have been shown to be initiated and sustained by 
sparks [17]. Under pathological conditions, 
sparks fire spontaneously and stochastically, so whether a single spark can trigger a 
wave is important to the stability of cardiac myocytes [4]. Simulations based on Fick’s Law reveal that large currents through CRUs and high calcium concentrations are needed to trigger a 
wave [13]. In this work, based on the anomalous subdiffusion model, we find the current which can trigger a normal 
wave initiating from a single 
spark at the corner of considered region.
According to Li et al. [10], calcium sparks follow the anomalous space subdiffusion of 
, so this subdiffusion order is also taken in our simulation. In our model, initial source is a 10ms opening of one CRU, the longitudinal intervals are 
. When we take 
, the longitudinal wave velocity (
initiating from the corner) is in good agreement with the experimental result(
[17]).
Figure 2 shows 
waves propagating on a discrete rectangle lattice initiating from a 10ms opening of the CRU at point (2,0.8). The snapshots are the 
concentration distribution at 10, 30, 50, 70, 90, 110, 130, 150, 170, 190 and 200ms (left to right, top to bottom). From image to image, we can see CRUs fire stochastically by turns while 
concentration wave propagates to the points of CRUs. At the beginning, CRUs fire one by one, and the amplitude(maximum of 
concentration in a region) is not very large; but with the increase of time, some of CRUs fire simultaneously in a short time so that 
sparks influence each other, and the amplitudes of 
sparks become larger and larger. For example, the CRU at (2,1.6) fires at 
, the CRU at (2,2.4) fires at 
; at 
, the CRUs at (4,0.8) and (4,2.4) fires simultaneously, in a short time interval, at 
the CRU at (2,4.8) fires. In image 10, sparks occur at (18,4.0), (18,4.8), (18,5.6), (18,6.4), (18,7.2), (18,8.0), (18,8.8) and (18,9.6) in rapid succession, which may trigger a calcium transient. Image 11 shows that the boundary limits the propagation of 
wave, but in an actual cardiac myocyte with size 
, calcium transient will be observed. In addition, though the space intervals of CRUs along y-axis are more compact than along x-axis, transverse wave velocity 
is smaller than that along x-axis 
(because the diffusion coefficient along 
-axis 
is smaller than that along 
-axis 
).
Our numerical results show that a single 
spark can trigger a normal 
wave under the pathological condition of 
which is consistent with the experimental results(longitudinal wave velocity 
). Physiological current through CRUs is about 2pA. However, the current may be increased (but not so large as 20pA [7], [13]) by external or internal factors, such as some disease or some electroneurographic signals. Our model present that a spontaneous 
spark can form a 
wave. The physical reason is, subdiffusion of 
is slower than that for Fick’s diffusion. When an event of 
spark occurs, high value of 
concentration may stay in a larger region around the firing CRU (for one 
spark, FWHM is 
for subdiffusion and 
for Fick’s diffusion ), the firing probability of adjacent CRUs becomes higher. Then the fire-diffuse-fire process can be initiated and sustained, a 
wave can propagate. So a smaller current and fewer sparks are needed to form a 
wave with our model than that using Fick’ Law.
Because of the large FWHM for one spark due to anomalous subdiffusion, one firing CRU will trigger a 
wave. Then we prohibit the event of another spontaneous spark so that it will not affect the initial 
wave. In other words, when local 
concentration is larger than resting 
concentration, 
sparks may occur. So the “wall” of high 
concentration spreads from the left corner to the top of cardiac myocyte. Image 12 shows the sequence of CRU firing along 
(the longitudinal linescan). The horizontal axis denotes time 
(from left to right,200ms), and the vertical axis denotes spatial coordinate 
(20
). Except for initial two sparks, CRUs fire at nearly regular intervals, and the “wall” of high 
concentration is nearly a straight line. Here, from the initial spark to the second spark, it takes more time than those of the subsequent sparks, which is different from the results by Izu et al. [13](in their simulation, the transverse linescan is adopted, but the qualitative profile must be the same). It is because the subsequent sparks are triggered by two or more adjacent sparks, and the longitudinal wave velocity approaches to a constant value, but it takes more time to trigger the next CRUs from the initial signal spark.
Effect of the Anomalous Subdiffusion Order
The anomalous diffusion order 
, which determines the diffusion mode of 
waves, was shown to affect the wave velocity considerably in last subsection(comparing with Fick’s Law). When 
, a large value of 
means a wild spread of initial concentration, but with the increase of time, the remanent concentration at initial point will be smaller due to the wild spread of calcium concentration. So the anomalous diffusion order 
affects not only the wave velocity, but also the amplitude of each CRU, and further the average amplitude of a 
wave. Here, 
is taken to be 2.00, 2.05, 2.15 and 2.25 in order to figure out whether the amplitudes of 
waves will change obviously with the variation of wave velocities. The initial condition is still a 10ms opening of the CRU at the point (2,0.8).
Table 3 presents the effect of anomalous fractional order 
on the longitudinal wave velocity 
and the average amplitude, respectively. Comparing with the results based on Fick’s law, velocities of 
waves increase considerably when anomalous subdiffusion order 
becomes bigger. For 
, the wave velocity(
) is almost twice as big as that based on Fick’s law(
, 
). It is because FWHM along 
-axis for 
(
) is almost twice as large as that for 
(
). Here, in 
waves, FWHM for one CRU is affected by adjacent sparks, so it is a little bigger than FWHM of a single 
spark(
[10]). In contrast to wave velocity, the variation of amplitudes is not very considerable. For 
, amplitude is 81% as that for 
. The physical reason is that although for 
, FWHM along 
-axis is twice as big as that for 
, the full duration at half maximum(FDHM) along 
-axis for one spark still has a obvious decrease. So when the total release of 
concentration is almost the same, under the expansion of spatial affection and the decrease of temporal continuity, amplitude of 
waves does not decreases obviously. In addition, wave velocities and amplitudes do not vary linearly with 
. When 
is larger, the effect of subdiffusion on 
waves is greater. It is because when the variation of 
is small, the other parameters, such as the speed of diffusion 
and the release strength of sparks 
, play important roles in 
waves. When 
becomes bigger, replacing the primary position of the diffusion speed and release strength, diffusion mode affects 
waves significantly(wave velocities).
Effect of Initial Location
Propagation of a 
wave from a corner of the cardiac myocyte has been studied. It is found that the reflecting boundaries increase the amplitude of the initial spark, then further promote the propagation of the 
wave. In order to figure out the boundaries determine the propagation of the 
wave or just affect the wave velocity, we change the location of the initial 
spark and study how the reflecting boundaries affect 
waves. In general, the process in which more than two sparks firing together, then several 
waves propagating, meeting and dissipating is very common in cardiac myocytes. This event is called 
waves collision, and it was observed in experiments [17]. We will discuss in the following the interaction of several 
waves.
As shown in Fig. 3, it takes only 120ms for a 
wave initiating from a 10ms opening of the CRU at a middle point (10,9.6) to propagates to the left and the bottom boundaries. Triggering from the middle of the region, the “walking distance” of a 
wave becomes shorter, and it will spread more quickly to the boundary. Due to the shorter “walking distance”, less sparks will occur simultaneously, and it will not make the 
wave develop sufficiently; but trigger from the corner, while the 
wave spreads wildly, large amount of sparks will fire together in a small region(Figure 2, right corner of Image 10). Comparing with the 
wave initiating from a corner without the effect of the boundaries, the initial concentration of the region will be smaller, then the probability of CRUs firing will be lower, so the events of 
sparks are more stochastic and irregular. Affected by the absence of the reflecting boundaries and the shorter “walking distance”, longitudinal wave velocity 
reduces to 
, and 
reduces to 
. So 
waves are easier to occur at the boundaries of cardiac myocytes, whcih can be compared with the experimental results [17], [18], [19]. Initiation of Free 
waves [18] and spontaneous 
waves [19] is kinetically favored near the boundaries, and the waves initialing from the boundaries are also easier to propagate. In ref. [17], though the results are obtained by line-scan, initiation of the waves is always near the endpoints of the line, and the waves are always triggered near the boundaries of cardiac myocytes. However, the amplitude is smaller than that of the wave from the corner though the change is not obvious. It is because the initial condition is an opening of only one spark, the reflecting effects of the boundaries are not sufficiently obvious. So the reflecting boundaries can increase the propagating probability of 
waves, though they are not the crucial factor for the propagation of 
waves. In contrast, the anomalous subdiffusion mode of 
concentration is the decisive factor for whether the 
wave can be formed by a single 
spark.
Figure 4 presents the event of two 
waves collision(3D images). The initial condition is 10ms opening of CRUs at points (2,9.6) and (18,9.6), and they will form two 
waves. Two 
waves will meet at the middle as shown in Image 4, at 
. Several sparks fire at the same time, and local 
concentration reaches a peak value. With increasing time, 
concentration will return to a lower value under the effect of buffers and pump, and no sparks will occur because of the CRUs’ “refractory period”. The 2D linescan image shows the process of two 
waves collision and vanishment(Fig. 8 in [17]), but it cannot show how the propagating direction changes. When 
concentration reaches a peak value in the middle, CRUs along 
-axis(
) are closed, but CRUs along 
-axis(
) have never been opened before. Therefore, 
waves can propagate along the line of 
. Finally, all CRUs are closed and will not reopen.
Modeling 
Waves Under a Physiological Current
To reproduce the feature of calcium waves found in experiments(primary result is wave velocity), a large current through CRU has been used in the former subsection. However, physiological value of 
is about 2pA. So in the following discussion, 
is adopted to study how many adjacent normal 
sparks can trigger a 
wave and find out the longitudinal interval of CRUs which could make a single 
spark trigger a normal 
wave. Because of the small value of 
, wave velocity and amplitude will be smaller. In order to make 
waves spread all over the region, the computation time is prolonged to 500ms. To diminish the effect of the reflecting boundaries, the initial location is chosen at the middle of the region.
Figure 5a shows whether a 
wave can be triggered by one spark at the middle of the region for 
. When 
, the wave only spreads through half the region, and the events of 
sparks are almost isolate. Under physiological conditions, even considering anomalous subdiffusion, high value of 
concentration may stay in a larger region around the firing CRU. For a small current, the amplitude of one spark is still small, the 
wave cannot propagate to the whole region, so the cardiac myocyte is stable when a normal spontaneous spark occurs.
The initial number of firing sparks is changed to study how many adjacent normal 
sparks can trigger a 
wave. The result is shown in Fig. 5b. It can be seen that four CRUs firing simultaneously at the middle will form a “weak” 
wave in the region. At 
, the wave reaches the top, bottom and right boundaries, and several sparks can be found at the same time. However, three adjacent normal 
sparks can only form a local 
wave. So with the computation time of 500ms, for 
, 
and 
, four adjacent CRUs firing together is the critical initial condition to trigger a 
wave. However, wave velocity and amplitude here is very small(
), and the 
concentration of the whole region is much smaller than that in Figs. 2, 3, 4.
In Figure 6a, the longitudinal interval is changed. For the case of 
, it takes only 110ms for the wave to reach the left boundary. With the simultaneous firing of several CRUs, an obvious 
concentration wave is observed. Although the amplitude is smaller, longitudinal wave velocity(
) is comparative with the case of 
, 
. The physical reason for such a significant change which happens by changing 
to 
is that FWHM for 
is about 
, and if the interval between two CRUs reduces to 
, the half maximum value of a spark can “reach” adjacent CRUs easily. In addition, less interval makes more CRUs fire together, and the wave will be easier to propagate.
Our results have revealed that two factors(
and number of firing CRUs) can both make a 
wave propagate. But which the effect is more significant? Figure 6 shows when 
, three 
waves trigger from one, four, and nine initial adjacent 
sparks, respectively. From 6a to 6c, both longitudinal wave velocity and amplitude become larger (
, amplitudes are 
), but the difference is not obvious as that between Figs. 5a and 6a. So the longitudinal interval of CRUs affects 
waves more significantly than the number of firing CRUs. If the longitudinal interval of CRUs becomes smaller due to some reasons, such as cardiac myocytes deformation, 
waves will easily occur, then cardiac myocytes will be unstable.
