Citation: Rüdiger S, Jung P, Shuai J-W (2012) Termination of Ca2+ Release for Clustered IP3R Channels. PLoS Comput Biol 8(5):
e1002485.
https://doi.org/10.1371/journal.pcbi.1002485
Editor: Stanislav Shvartsman, Princeton University, United States of America
Received: August 26, 2011; Accepted: March 7, 2012; Published: May 31, 2012
Copyright: © 2012 Ruediger 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 China National Funds for Distinguished Young Scientists under grant 11125419, the National Science Foundation of China under Grant 30970970 and National Institute of Health under grant 5RO1GM065830 for Shuai. 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
is a universal messenger of eukaryotic cells and regulates various cellular processes such as morphology, enzyme activity, and gene expression [1], [2]. In many cells, 
signaling is achieved by modulation of cytosolic 
concentrations arising from exchange of 
with the endoplasmic reticulum (ER). SERCA pumps produce steep concentration gradients across the ER membrane by moving 
from the cytosol into the ER. Liberation of 
from the ER occurs through inositol 1,4,5-trisphosphate receptor ( 
R ) channels in the ER membrane.
Rs regulate 
transport in response to binding of 
and 
to receptor sites on the cytosolic domain of channels [3]. While binding of the second messenger 
generally promotes the opening of channels, the dependence on cytosolic [
] is biphasic. Small increases in [
] compared to rest level concentrations increase the open probability of 
R channels. The dissociation constant, 
, of the responsible binding site is at sub-µM scale. This stimulating 
binding gives rise to a self-amplifying mechanism called 
induced 
release (CICR): 
released by one or several channels diffuses in the cytosol and increases the open probability of neighboring channels by binding to their activating binding sites. As the level of 
rises further, inhibitory binding of 
dominates. Consequently, the open probability decreases significantly as 
levels reach values comparable to an inhibitory dissociation constant, 
, at several tens of µM . Together, the activating and inhibiting binding processes allow for cooperative openings and closings of receptor channels.
Elevations of 
concentrations appear in two basic patterns that reflect the spatial organization of 
R channels [4]. In many cells, 
R channels are distributed in clusters on the ER membrane. It is often found that CICR synchronizes channels within clusters, resulting in patterns of localized and short-lived events called puffs [5]. In this regime, 
release does not spread to neighboring clusters, which are typically separated by a few µm . Recent studies emphasize the role of sub-cellular 
rises for physiological function [6]–[8].
In a different patterning mode, activity of channels from many clusters synchronizes to form cell-wide oscillations [9], [10] or, in larger cells, waves [11]. Global release, lasting for up to several tens of seconds, is triggered by random initiation events (i.e., random openings of a few channels) and carried through the cytosol by CICR between nearby clusters. It is important to note that waves do not only last longer because of the long time it takes to travel through the cell. After a wave has passed, elevations in [
] persist for several seconds. In marked contrast, the calcium signal during a puff vanishes within around 100 ms [12]. This discrepancy and the related question how calcium signals are terminated provide the background for this publication.
Because of their spatial localization, puffs have been studied with theoretical methods by considering a single cluster and ignoring the coupling to channels outside of the cluster. Recently, we and other groups have modeled puffs by a stochastic gating scheme for clustered channels coupled to evolution equations for local 
concentration within the cluster microdomain [13]–[18]. The principal dynamics during a puff is sketched in the left panels of Fig. 1 and can be understood qualitatively by comparison of 
levels with the dissociation constants 
. In an initial phase, random opening of a single channel in the cluster drives local [
] to values above the dissociation constant for activation: [
] 
(dotted line). Fast intra-cluster CICR then causes further channels to open (left top panel, green line). Subsequently, 
concentration within the cluster microdomain increases strongly. The local 
concentration reaches values above the dissociation constant of inhibiting binding sites: [
]
(left bottom panel, dashed line), which leads to inhibition (red line) and eventually to closing of channels.
Since the kinetic rates of the inhibitory sites of 
Rs are large, the duration of puffs is short, matching the experimental observations [17]. Furthermore, the modeling predicts that subsequently channels loose 
from the inhibition sites rapidly, usually within a time scale comparable to the puff duration. Thus the refractory time after a puff is normally small and channels are available for re-opening shortly after a puff ends [19].
Despite the impressive agreement of this modeling picture with experimental findings, the shortness of the refractory period raises an important question: If channels are inhibited only for tens or hundreds of milliseconds, calcium ions that have been released during the previous puff could re-open the channels shortly by binding again to activating binding sites. The residual [
] , caused by the 
transiently remaining in the microdomain after the last open channel closes, could then be sufficiently large to re-open channels (Fig. 1, right panel). If such re-openings occur repeatedly over an extended period, release may last much longer than the experimentally measured duration of puffs. Using a standard model for 
gating coupled to a simple law for [
] evolution, we first show in this paper that indeed large residual 
microdomains can generate release at a long time-scale. Because the duration of those events reflects the temporal evolution in the wave regime, from here on we refer to this dynamical regime as waves.
The 
microdomains and the dynamical evolution of residual [
] after channel closing was recently studied by us with numerical simulations of the detailed reaction-diffusion equations [20]. Importantly, we found that 
levels can be decaying much slower than previously assumed. This result suggests that residual [
] and the mechanism we describe are relevant for the dynamics of 
waves. This also implies that residual 
could be a decisive factor for the observation of waves or puffs under different physiological conditions. In this paper we focus on the puff-wave transition under physiological changes related to 
binding proteins.
It is well-known that 
microdomains are very sensitive to certain mobile 
binding and buffering proteins. The buffer proteins alter the distribution of 
released from channels by reducing the spatial extent of microdomains [21]. Often this leads to strong effects on the dynamics of 
release [12], [21]. However, it was also found that a change of the microdomain extent is small for those buffers that bind 
slowly. It therefore was a surprise that slow buffers such as EGTA and parvalbumin can dramatically change the appearance of 
release. While for normal conditions, experiments under stimulating levels of 
show a wave regime, in presence of slow buffers waves dissolve and puffs appear [12], [22]. To explain this behavior we will here address a lesser known aspect of mobile buffers, which is that the temporal relaxation response of the microdomain ( i.e., the residual 
) is affected.
Our simulations in [20] have shown that binding of 
to slow buffers indeed lowers the amplitude of residual 
. This effect can be substantial even though the effect on the 
evolution during the puff is small (Fig. 1, left bottom panel). This result suggests that the short duration of puffs, under buffered conditions, is caused by the fast decay of residual 
concentrations. Here we therefore propose that the appearance of puffs in cells loaded with EGTA can be understood from the fact that residual 
amplitudes are driven close to rest levels by binding to the buffer. Relating the action of buffer to a parameter of our [
] model, we show that indeed the modulation of 
microdomain collapse can explain the puff-wave transition under control of buffer concentration.
This effect entails a final question that we would like to address here: If residual 
causes sustained channel openings during waves, what then terminates the waves? While we find that puffs are terminated by 
inhibition [13], we also find that the unbinding of 
is responsible for termination of waves. Thus 
plays a role not only in stimulation of 
signals and in modulating their amplitude, but dynamics of 
is also important for the termination of 
waves. Taken together, this means that wave-like release requires binding of 
to the receptor for stimulation, slow decay of residual 
for sustaining the activity for several seconds, and finally unbinding of 
for termination of the wave. Interestingly, the dynamical 
binding/unbinding mechanism that we describe rests on the sensitivity of 
to 
[3]. This sensitivity is a well-established property of the 
gating to the receptor, but was hitherto neglected in theoretical and numerical studies of 
release.
Model
Single channel gating
Equilibrium gating behavior of single channels has been investigated experimentally by patch-clamping [3], [23]. Briefly, experimental results are the following: At small cytosolic 
concentrations, an increase in [
] increases the probability of opening, 
. For larger 
concentrations, an increase in [
] leads to decreases in the open probability (Fig. 2A). Furthermore, binding of the second messenger 
to the receptor increases the open probability. Fig. 2A shows that for higher levels of 
the 
curves are shifted upwards.
Several models have been proposed to discuss the open/close dynamics of a single 
R channel [24], [25]. Here we invoke the DeYoung-Keizer (DYK) model [24], which comprises the three basic binding processes of the receptor in its standard form. An 
R consists of four identical subunits, each containing three binding sites: an activating site for 
, an inhibiting 
site, and an 
binding site. The three binding sites allow for 8 different states for each subunit, which can be mapped onto triples (
) of each subunit. The index 
indicates the state of the IP
site, 
the one of the activating 
site and 
the state of the inhibiting 
site. An index is 1 if a 
ion or 
is bound and 0 if not. Following earlier work we require at least three of the four subunits to be in the state (110) for the channel to open [26].
The 8 states allow for 24 different transitions, which can be associated with the edges of a cube (see Fig. 2B). Transitions between the respective subunit states are governed by rate constants, some of which are concentration dependent. There are five pairs of binding/unbinding rates, one for activation and two for inhibition and 
binding, respectively. Each pair is given by the binding rate 
and the unbinding rate 
. The ratio, 
, is the dissociation constant, 
. In Fig. 2B, 
denotes the 
concentration and 
denotes the 
concentration in the cytosol.
We now discuss the properties of the 
R channel model where we have fitted parameters 
and 
to experimental data (see Supporting Text S1). Solid lines in Fig. 2A show the open probability for a DYK model [27] where parameters were fitted against nuclear patch clamp data taken from [28], [29].
The bell-like shape of the open probability curve is reflected in the different dissociation constants for activation (
) and inhibition (
and 
). Specifically, the open probability will be large for [
] above 
and below 
. Furthermore, in experiments the exact position of the peak open probability depends on the 
concentration. Often, in the discussion of dynamical models, the 
binding process is neglected, or it is absorbed into the activation and inhibition processes by assuming quasi-equilibrium.
The fact that for increasing 
concentration the location of the right tail of the bell curve is sensitive to [
] is reflected in the DYK model by the existence of two dissociation constants for inhibition. Accordingly, 
, which is the dissociation constant for inhibition at large [
] , should be much larger than 
, which is the dissociation constant at small [
] . Indeed, after fitting the DYK model we use 
µM while 
0.111 µM . In contrast, the activation threshold, quantified by 
, appears to remain constant (in our estimate at 0.25 µM , see Supporting Text S1).
Note that the 
dependence of 
and 
also imposes a corresponding dependence of 
affinity on the binding state of the inhibiting binding site. For all binding/unbinding loops given in Fig. 2B, the thermodynamic constraint requires that in equilibrium the dissociation constants satisfy the detailed balance conditions. Clearly, for a loop involving the activation process, the scheme shown in Fig. 2B is satisfying this condition. However, for a loop involving inhibition and 
binding we obtain:
(1)Thus it follows that the 
dependence of inhibition requires 
, i.e., the 
binding depends on the inhibiting site. Because of 
, we then have 
. The fact that 
will play a crucial role in the interpretation of our results below.
Clusters of 
R channels and inhomogeneous 
distribution
A major factor in the shaping of 
signals is the clustered distribution of 
channels on the membrane of the ER. Fluorescence visualization has shown that 
is released through clusters of channels that comprise up to a few tens of channels. Early experimental studies showed that clusters occupy domains of much less than one micrometer in diameter but could not resolve their inner structure. This lead to a general model of a cluster as a small patch on the ER membrane, in which channels are distributed homogeneously and tightly packed. If channels in the cluster open, released calcium creates a microdomain, which is characterized by a large calcium concentration and, in this modeling picture, an approximately homogeneous distribution of calcium within the area. Subsequent progress in visualization, however, called this virtual domain picture into question [30] and prompted the inhomogeneous modeling which we will adapt in this study. A further strong hint on the inhomogeneous distribution of 
within the cluster originated from our detailed comparison of stochastic simulations with experimental puff data from Smith and Parker [5]. If one assumes a homogeneous distribution of calcium within the cluster microdomain, as was often done in modeling approaches, the experimental amplitude distribution and lifetime of puffs cannot be fitted correctly. However, under the assumption of a non-homogeneous distribution of calcium, we were able to obtain a very accurate fitting of experimental data [17]. Here we further follow recent experimental results in assuming the typical peak open numbers to be 5–10 channels [31].
In our paper [20] we numerically simulated detailed models for 
R clusters. Three-dimensional reaction-diffusion equations were used to calculate the evolution of [
] as well as that of relevant buffer proteins. These equations were coupled to a stochastic algorithm for the gating state of each channel. 
influx through open channels was modeled by disk-shaped two-dimensional source areas on the surface of the simulation domain (see Fig. 3A). The radius of each channel's source area was set to 6 nm. The flux through the source area was adjusted to a current of 0.1 pA per channel in the open state. The size of the simulation domain was large enough (at the order of several µm ) so that the released 
does not significantly alter global 
concentrations far away from the open channels. Channels were placed in regular lattices on the domain surface with distance on the order of several tens to hundreds of nm. A main result was that 
is distributed very inhomogeneously in the cluster microdomain even for small distance of channels. While the concentration directly at the open pore is larger than 100 µM , we found much smaller concentrations in the space between channels, even if measured directly at the membrane surface between two open channels. We concluded that the influence that an open channels exerts to closed channels occurs at a much smaller 
concentration than the feedback that the released 
exerts to its own source channel (see Fig. 3B).
To reduce the complexity of this direct numerical approach we introduced a concept of scale separation. While the local concentration at the pore of open channels defines a relatively fixed and large value (denoted 
hereafter), the “coupling” concentration, 
, at any closed channel in the cluster is a much smaller quantity (Fig. 3B). In an approximation step we replaced this quantity by a suitable 
average concentration for the microdomain [17], [20], which is independent on the position of the closed and open channels within the cluster. Our simulations also showed that this quantity, which describes a typical concentration at closed channels within the microdomain, strongly depends on the number of open channels, 
. Therefore, and taking into account a small rest level concentration 
, we obtain:
(2)where 
is a function of the number of open channels with 
. This ansatz assumes that the [
] at any closed channel in the cluster follows the number of 
open channels in a quasi-stationary way. In fact, earlier simulations have shown that local increases in the concentration quickly relax to a stationary profile after channel opening [20], [32], [33]. In this situation, the stationary approximation in Eq. 2 is justified. Below we will complement it with a simple description of temporal relaxation of 
after channel closing.
We now summarize the static [
] part of the modeling features for 
R clusters:
released from a channel creates a nanodomain near the channel pore, which is spatially small but exhibits 
at very high concentration. An open channel is exposed directly to this nanodomain. Consequently, the 
concentration governing further gating transitions of the open channel is large. We denote the corresponding concentration value by
(3)and choose 
µM for our simulations [33].- If a channel is closed, the local concentration at that channel originating from
released by 
other open channels is described well by a linear relation [17]
(4)This quantity provides a typical concentration within the cluster microdomain. If channels are open (
), 
is much larger than the 
concentration a few μm away from the cluster. But it is also much smaller than the local value 
at the pore of an open channel. In [17] we concluded from detailed comparison with experimental results that the scale-separation is needed for agreement. We here use 
µM and 
µM , where 
denotes the resting level concentration of 
.
We will assume a cluster that consists of 20 channels. The values of 
can be determined by two different methods. If one knows the full time-dependent evolution of the 
distribution during puffs, 
and 
and their dependence on the number of open channels can be determined by averaging the local [
] at all open channels or all closed channels, respectively [17], [20]. They can, however, also be introduced as phenomenological parameters. In [17] we have determined realistic values and have shown that this latter approach yields a very accurate description of experimentally observed puffs. The parameter value for the coupling concentration 
corresponds to a cluster with radius of around 300 nm. As will be seen below, this setup leads to typical numbers of open channels of around 5–10, consistent with experiments in neuroblastoma cells.
Non-stationary microdomains and mobile buffers
A further important factor in 
signaling is the role of 
buffers. It is well-known that mobile buffers can strongly reduce the size of microdomains and thereby also decrease the amount of 
that reaches a target by diffusion. Here, however, we would like to focus on a property of buffers, which was much less studied so far, namely the influence of buffers on the microdomain collapse after channel closing, i.e., the residual 
.
In [20] we have studied the effect of mobile and immobile buffers on microdomains. By detailed numerical simulations of three-dimensional reaction-diffusion equations we characterized the alteration of open channel domains by buffers. Most importantly, we found that mobile buffers reduce the amount of residual 
. This happens because mobile buffers may bind free 
and support the transport of 
away from the microdomain. They, therefore, speed up the collapse of residual [
] .
We also described in [20] which buffers can reduce the spatial extension of a domain around an open channel. It happens only if the buffer is mobile and has fast reaction kinetics. As described in [20] the fast mobile buffer plays a crucial role in reducing the coupling of channels within a cluster. Within our modeling approach this would result in a decrease of coupling constant 
in Eq. 4 owing to the presence of fast mobile buffer. However, we are here interested in the role of slow mobile buffers, such as EGTA, which have little effect on the spatial extent of microdomain. Therefore, in the remainder of this paper, we neglect any changes of spatial appearance. We will therefore fix the coupling parameter 
and exclusively study the effect of microdomain [
] decay caused by slow mobile buffers.
It remains to cast the temporal evolution law for the 
microdomain after changes in the open channel number. As has been described earlier, after opening of a channel, the equilibration of [
] within the microdomain occurs very fast [20], [32]. Therefore, we assume instantaneous [
] equilibration according to Eq. 4 after an increase in open channel number 
. However, after channel closing, residual 
remains and we here use an approximation of this microdomain collapse with an exponential equilibration:
(5)Here 
is given by Eq. 4. 
is the rate of equilibration. Obviously, this simple law does not reflect the full complexities in the microdomain collapse but it does here serve as the most simple model for qualitative analysis [20], [34].
Results
Dependence of release dynamics on 
A first analysis concerns the behavior of release events under changes of [
] using a decay rate of 
s
. This small value of 
and the resulting slow temporal collapse of the 
microdomain, can be viewed as a control “experiment” without slow mobile buffer. Fig. 5 shows the typical time evolution of the number of open channels and the microdomain concentration 
for two different values of 
concentration. In both cases, well-defined release events (puffs or wave-like bursts) are apparent.
To analyze their properties statistically, we now define a release event based on the evolution of the number of open channels. An event begins with the transition from 0 to 1 open channel and ends when the last channel closes, i.e., when the number of channels is at zero again. However, to account for the noisy gating behavior sometimes observed in the wake of a release event, we consider the transition from 1 to 0 open channels not to be the end of the release event, if a re-opening of a channel occurs within a time span of 500 ms. The event is then considered terminated at the closing of the last channel if no further opening occurs within 500 ms. In Fig. 6 we characterize traces such as those in Fig. 5A,B accordingly for a range of values of [
] . Here we have evaluated the typical lifetime of release events as well as the duration between consecutive events (interpuff interval, IPI). Fig. 6 shows that for 
s
both mean lifetime and mean interpuff interval generally increase with [
] .
This result can be related to the transition from puffs to waves under increase of cytosolic 
concentration. Within a stochastic theory of 
release, wave generation can be understood in the following way. Waves begin by the random opening of one or a few clusters [36]. The released calcium spreads to neighboring clusters and because of their 
dependent probability to open, the initial event triggers a wave. Increase of 
supports the propagation of release activity since the duration and/or amplitude of the local trigger event increases and the excitability of neighboring clusters can increase. In this sense our result in Figs. 6A principally allows to interpret the increase of lifetime as a transition from local to global release. When the [
] parameter is increased, the duration of the cluster release prolongs. Therefore, for large [
] cluster cooperativity is likely, while for small [
] puffs are too short to allow propagation of activity. Here, however, we will not pursue this line of thought but instead focus on the effect of a change in decay rate 
.
Dependence of release dynamics on decay rate
Fig. 6 also shows that the lifetime and IPI of release events strongly depends on the decay rate 
. For large 
s
the mean lifetime is much smaller than for 
s
. We will now consider this dependence and its origin in detail. The 
concentration will be set to 
μM in the following. This value is larger than 
, so that in rest state the 
R subunits are normally occupied by IP
and reside in the upper plane of the DYK cube (Fig. 2B).
Fig. 7 shows the typical evolution of open channel numbers for different values of the decay rate 
. For large decay rate, 
s
we find frequent short spikes that resemble the experimentally observed puffs lasting less than 1 second. However, for smaller 
, long events occur, which typically consist of one high initial spike and a burst of openings of decreasing amplitude (wave regime). Lifetime of wave-like release becomes increasingly longer as the decay rate decreases (Fig. 7).
We will next analyze the behavior of channel gating that leads to the different dynamics of puffs and waves. Starting with the case of puffs (fast microdomain collapse), Fig. 8A shows the evolution of the number of open channels and Figs. 8B–D present the three quantities characterizing the inhibition and refractory behavior. Figs. 8B and C show the numbers of subunits in the particular inhibited states 111 and 011, respectively. The state 111 is obtained by direct transition from the open state, 110, while the local 
is large. Fig. 8B clearly shows the fast onset of inhibition, which leads to the closing of all channels within about 200 ms. Because of the fast time scale of inhibition, not only is there a fast onset of inhibition, but, as seen in Fig. 8B, there is also only a short refractory time. The latter is caused by a fast rate of de-inhibition, 
, which is around 2 s
. This implies that the channels can be re-activated almost immediately after the termination of the last puff. However, because of the fast decay of 
concentration for large 
, calcium ions are normally too dilute to cause immediate re-openings.
We conclude that for (short) puffs the dynamics involves 
activation and 
inhibition of subunits. This result confirms earlier models of 
dynamics, which relate the termination of 
puffs to the binding of 
to inhibiting binding site. This dynamics therefore does not need and does not incorporate the third component of the DYK model, which is the binding or unbinding of 
. Fig. 8C shows that the number of subunits in state 011 (those subunits that have not bound 
but have 
bound to the activating and inhibiting sites) does not change much during a puff. Fig. 8D shows the number of channels, 
, that have bound 
to three or four subunits. Note that 
in is the number of channels that are principally available for opening during a puff. Typically only a fraction of the 20 channels is available for opening (this effect will be discussed below.) Thus, Fig. 8D indicates no changes in 
occupation during a puff. Taken together, our observations indicate that a typical trajectory of the subunits during puffs is a cycle that involves 
activation and inhibition only. It thus only involves the upper horizontal plane of the DYK cube (see Fig. 9A).
We now discuss the behavior during the wave-like events occurring for smaller decay rate 
(persistent 
microdomains). Similarly to the case of short puffs, rapid inhibition occurs, which involves units transferring to the 111 state (Fig. 8F). However, because of the large transient microdomain [
] , immediately after termination of an opening and restoration of subunits to the de-inhibited state, a further opening of channels occurs. This behavior leads to the perpetual cycling of some of the subunits around the upper plane of the DYK cube, explaining the occurrence of long bursts of openings shown in Fig. 8E.
One may think that the existence of longer signals for persistent 
microdomains is simply a consequence of subunits cycling the activation/inhibition loop. However, we will now explain that the four states of the upper DYK cube are not sufficient to obtain the well-defined bursts shown in Fig. 8E. This relates to the fact that subunits cycling along the upper loop need a further mechanism if they are to finally terminate the 
release. Since inhibition is already part of the cycling that builds the bursts, another mechanism is needed to end the bursts. Surprisingly, our model suggests that the dissociation of 
from subunits provides this mechanism.
First, Fig. 8G shows that during the course of a wave, more subunits accumulate in the state 011. The number of subunits in this state increases for this particular burst from 22 to 34, that is 12 subunits dissociate 
during the burst. Note that the rates of 
binding/unbinding are smaller than those of the other processes involved during a puff (see Supporting Text S1). Further, many of the channels have only three of the subunits bound to 
, so that a further dissociated subunit means that the channel is not available for opening. Accordingly, the number of available channels drops from 8 to 3 during the burst event (Fig. 8H).
We found that the subunits undergo the 
dissociation while being in the state 111. The mechanism of 
dissociation therefore depends on the time that a subunit spends in the 111 state. This means that the 
dissociation increases with the duration of 
release. Fig. 9C shows that 
dissociation indeed occurs for long release events. It thus constitutes a second inactivation process that is needed to terminate the wave-like release. Fig. 9B illustrates the mechanism, which involves gradual 
dissociation of subunits while cycling the 
activation/inhibition loop.
The 
unbinding mechanism relies on the fact that the dissociation constants of 
binding are very different for subunits with and without 
bound to the inhibiting site. Stimulating levels of 
concentration ( i.e., 0.07 µM in our simulations) are well above the value 
(
µM ), so that many subunits have bound 
. However, inhibited subunits possess a much higher dissociation constant 
(
µM ), such that loss of 
can occur while subunits populate the inhibited states. Therefore, the termination of waves due to 
dissociation is possible only when the 
binding depends on inhibition. This leads to the conclusion that wave termination is related to the shift of peak open probability with 
concentration (see Fig. 2B).
Behavior of lifetime and interpuff interval and comparison to experiments
The effect described above lead to strong changes in the statistical properties of release events depending on the decay rate 
. This concerns first of all the distribution of release lifetime. For fast collapse of the 
microdomain, most events are of short duration. As an example, Fig. 10A shows the lifetime distribution for 
s
(dark gray bars). Most puffs in this case last between 100 and 300 ms. If, however, the 
value is substantially lowered, most events last considerably longer than 1 s (light gray bars). Clearly, this behavior follows from the slow decay of microdomain 
values and the related re-opening of channels.
Fig. 10B shows the dependence of average event lifetime on 
. In the chosen range of 
the event lifetime varies over one order of magnitude, which is what was observed in [12],[31]. Our results resemble well the experimental findings for dependence of 
release spikes if 
is associated to the level of buffer concentration.
Interpuff intervals (IPI) have been experimentally determined as well. For puffs of small amplitude, Fraiman et al. found a distribution of IPIs peaked between 1 and 2 seconds and strongly decaying for larger intervals [37]. As can be seen in Fig. 10C, simulations of our model for 
s
yield a similar distribution with rare occurrence of IPIs with more than a few seconds (dark gray bars). If 
is much smaller than this value, the distribution is much flatter. Correspondingly, the smaller the 
the larger is the mean IPI (Fig. 10D).
The mean IPIs found in our simulations agree quantitatively with experimental values. In SH-SY5Y cells loaded with EGTA, puffs occur with a frequency of about 0.23 Hz, roughly corresponding to an IPI of four seconds [31]. In the absence of EGTA, the experimental IPIs increase to about six seconds, again agreeing with our mean IPIs for small 
s
. These numbers demonstrate the realistic representation of experimental results in [31] by our modeling approach.
In view of the mechanism proposed in this paper, the increase of IPI for small 
appears to be counter-intuitive. If residual 
is large, one would naively expect that a higher activation probability and thus an earlier appearance of the next wave results. Therefore, for small 
one could expect shorter IPIs. Here, however, we find for small 
that the next wave occurs typically several seconds later. In fact, this delay is not the result of simply a longer lifetime of release events. Figs. 10E,F present the time of dormancy between two consecutive release events, i.e., the length of intervals in which no channel is open. The increase of dormancy interval for persistent residual microdomains (small 
) indicates the presence of a refractory mechanism that suppresses the probability of wave generation for a few seconds. We briefly would like to discuss the origin of this behavior.
We first remind that for large 
a very short refractory period after the termination of a puff was noticed (see Fig. 8B). The refractory period is the time where channels are inhibited or otherwise not available for re-opening. This time is of the order of at most a few hundreds ms. This means that the refractory period cannot account for the mean IPI for large 
, which is at around 4 s. Instead, most of the IPI for large 
is related to a waiting time for a first “trigger” channel to open [38].
The small 
case, however, provides a different scenario. Clearly, the large number of 
unbinding subunits decreases the probability of re-opening. The loss of 
thus amounts to a kind of refractory period. The chance of triggering a wave is much smaller after long-lasting events where many 
molecules are lost (Fig. 9C), mainly because the number of channels available for opening is smaller. Only after this refractory period, channels have re-bound the lost 
and regain their potential to trigger a wave. It is interesting to note that this 
refractoriness should generally reduce 
excitability. It may therefore provide a new mechanistic explanation for the smaller excitability of cells for the first few seconds after waves [36].
Discussion
The spatio-temporal patterns of 
signals are known to depend on many factors, including the 
concentration and the presence of buffer proteins. We have here discussed the role of “hidden” quantities in the shaping of 
signals and argued that they may serve as mediators of signal modulation. We have shown that their dynamics can explain features of 
puffs and waves in a consistent way.
Most importantly, our analysis reveals that changes in the collapse of 
microdomains can have a profound effect on release duration. In this paper, we have represented this collapse of 
microdomain by the rate 
, which describes the decay of residual 
in the microdomain after closing of channels. The most significant result is that for slow collapse (small 
) one generally finds long release reminiscent to temporal evolution during global waves, while for fast collapse much shorter events – puffs – appear. For slow collapse, residual free 
re-activates channels by binding to their activating binding sites soon after channel closing. Often, re-activation leads to re-opening of channels because the refractory time of open channels is relatively short, i.e., inhibited subunits unbind 
from their inhibited sites shortly after the channel has closed and are therefore available for re-opening. This property distinguishes the 
- 
system from other excitable systems where refractoriness lasts much longer than the actual spike [39].
Buffers and the puff-wave transition
The effect that we describe provides a novel explanation of the action of EGTA buffer on 
release. Experiments with various exogenous buffers have clearly demonstrated the strong effect of buffers on dynamics of 
release [12], [40]. Dargan et al. [12], [41] studied how the 
signal depends on buffers injected into Xenopus oocytes. In these experiments, the amount of 
in the cytosol after stimulation by 
was measured by fluorescence recordings using an additional dye buffer. For EGTA buffer the response to the 
stimulation was sharply shortened compared to the release in untreated cells. In contrast, BAPTA-injection leads to a more homogeneous and prolonged release.
By relating large values of 
with the action of EGTA on calcium domains our results imply a shortening effect of EGTA on release of calcium. It is important to note that the release duration here refers to the proper open state of channels in a cluster and not to indirect effects due to a dye-buffer competition, which may additionally shorten traces of puffs in cells loaded with EGTA [12], [42]. To our knowledge, our work is the first theoretical study to show the consequences of EGTA and similar slow buffers on channel gating dynamics within a cluster.
How realistic is our assumption that the presence of EGTA speeds up the collapse of 
microdomains? Earlier work on full reaction-diffusion equations for buffer and 
has shown that the collapse of free 
concentration at the channel pore after closing of the channel strongly depends on the mobile buffer present in the system [20]. Our detailed three-dimensional simulations have shown, for instance, that in the presence of EGTA a free 
concentration of 0.5 μM is reached after less than 1 ms, while in the absence of EGTA it takes about 6 ms. This observation serves as motivation for the simplified decay dynamics for free 
concentration in our model.
Having argued that the effect of EGTA is to accelerate the collapse of 
microdomain, it remains to discuss why, experimentally, the presence of BAPTA does not lead to puffs and shortening of release. The difference between the cases of EGTA and BAPTA can be attributed to the different kinetic rates of 
binding/unbinding. EGTA is a buffer that binds 
relatively slow, while BAPTA reacts around 100 times faster. This implies that BAPTA can interupt the communication between channels during a puff [20], while EGTA is too slow to bind substantial amounts of 
during a puff. In other words, BAPTA reduces the peak levels of microdomain [
] in Fig. 1 and residual [
] , while EGTA only diminishes residual [
] . We therefore speculate that the fast intra-cluster action of BAPTA leads to a very different dynamics, for instance by reducing the initial phase of a puff or reducing inhibition. Detailed analysis of this problem will be performed in the future.
With respect to physiological conditions in vivo, our results suggest that the duration of 
-evoked 
signals is a highly variable quantity. Experimentally the lifetime depends not only on 
concentration and buffer content but was also shown to be sensitive for instance to changes in temperature [43]. The mobile buffer, however, appears to be special in that it allows control not only of the duration of release events but also of channel cooperativity. The concentration of cell specific buffer such as the 
binding protein parvalbumin is one example showing the importance for tuning the duration of local 
-evoked 
signals. Parvalbumin is a slow 
binding protein, which is known to have an important physiological role in muscle and neuronal cells. John et al. have shown that parvalbumin, similar to EGTA, inhibits repetitive 
waves and evokes release at discrete release sites [22]. Buffer concentration determines the sensitivity and cooperativity of 
action and confers a threshold for the ability of the cell to transition from a local to a global mode of 
signaling. It is therefore possible that cell-specific expression of parvalbumin and potentially other buffers may serve to shape intracellular 
puff and wave signals for specific physiological roles by the mechanism described in this paper.
Acknowledgments
We would like to acknowledge many useful conversations with Ian Parker.
Author Contributions
Analyzed the data: SR PJ JWS. Wrote the paper: SR PJ JWS. Conceived model: SR JWS Wrote computer code: SR JWS.
References
- 1.
Berridge M, Lipp P, Bootman M (2000) The versatility and universality of calcium signalling. Nat Rev Mol Cell Biol 1: 11–22.
- 2.
Berridge M, Bootman M, Roderick H (2003) Calcium signalling: dynamics, homeostasis and remodelling. Nat Rev Mol Cell Biol 4: 517–529.
- 3.
Foskett J, White C, Cheung K, Mak D (2007) Inositol Trisphosphate Receptor Ca2+ Release Channels. Physiol Rev 87: 593.
- 4.
Dupont G, Combettes L, Leybaert L (2007) Calcium dynamics: spatio-temporal organization from the subcellular to the organ level. Int Rev Cyt 261: 193–245.
- 5.
Smith I, Parker I (2009) Imaging the quantal substructure of single IP3R channel activity during Ca2+ puffs in intact mammalian cells. Proc Natl Acad Sci U S A 106: 6404.
- 6.
Wei C, Wang X, Chen M, Ouyang K, Song L, et al. (2008) Calcium ickers steer cell migration. Nature 457: 901–905.
- 7.
Di Capite J, Ng S, Parekh A (2009) Decoding of cytoplasmic Ca2+ oscillations through the spatial signature drives gene expression. Curr Biol 19: 853–858.
- 8.
Parekh A (2010) Decoding cytosolic ca2+ oscillations. Trends Biochem Sci 36: 78.
- 9.
Berridge M (1990) Calcium oscillations. J Biol Chem 265: 9583–9586.
- 10.
Rooney T, Sass E, Thomas A (1989) Characterization of cytosolic calcium oscillations induced by phenylephrine and vasopressin in single fura-2-loaded hepatocytes. J Biol Chem 264: 17131–17141.
- 11.
Lechleiter J, Girard S, Peralta E, Clapham D (1991) Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252: 123–126.
- 12.
Dargan S, Parker I (2003) Buffer kinetics shape the spatiotemporal patterns of IP3-evoked Ca2+ signals. J Physiol 553: 775–788.
- 13.
Swillens S, Dupont G, Champeil P (1999) From calcium blips to calcium puffs: Theoretical analysis of the requirements for interchannel communication. Proc Natl Acad Sci U S A 96: 13750–13755.
- 14.
Shuai J, Jung P (2003) Optimal ion channel clustering for intracellular calcium signaling. Proc Natl Acad Sci U S A 100: 506–510.
- 15.
DeRemigio H, Groff J, Smith G (2008) Calcium release site ultrastructure and the dynamics of puffs and sparks. Math Med Biol 25: 65.
- 16.
Swaminathan D, Ullah G, Jung P (2009) A simple sequential-binding model for calcium puffs. Chaos 19: 037109.
- 17.
Rüdiger S, Shuai J, Sokolov I (2010) Law of mass action, detailed balance, and the modeling of calcium puffs. Phys Rev Lett 105: 048103.
- 18.
Bruno L, Solovey G, Ventura A, Dargan S, Dawson S (2010) Quantifying calcium uxes underlying calcium puffs in xenopus laevis oocytes. Cell Calcium 47: 273–286.
- 19.
Thurley K, Smith I, Tovey S, Taylor C, Parker I, et al. (2011) Timescales of ip3-evoked ca2+ spikes emerge from ca2+ puffs only at the cellular level. Biophys J 101: 2638–2644.
- 20.
Rüdiger S, Nagaiah C, Warnecke G, Shuai J (2010) Calcium Domains around Single and Clustered IP3 Receptors and Their Modulation by Buffers. Biophys J 99: 3–12.
- 21.
Neher E, Augustine G (1992) Calcium gradients and buffers in bovine chromaffin cells. J Physiol 450: 273–301.
- 22.
John L, Mosquera-Caro M, Camacho P, Lechleiter J (2001) Control of IP3-mediated Ca2+ puffs in xenopus laevis oocytes by the Ca2+-binding protein parvalbumin. J Physiol 535: 3.
- 23.
Taylor C, Tovey S (2010) IP3 receptors: Toward understanding their activation. Cold Spring Harb Perspect Biol 2: a004010.
- 24.
DeYoung G, Keizer J (1992) A single-pool inositol 1,4,5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration. Proc Natl Acad Sci U S A 89: 9895–9899.
- 25.
Sneyd J, Dufour J (2002) A dynamic model of the type-2 inositol trisphosphate receptor. Proc Natl Acad Sci U S A 99: 2398.
- 26.
Watras J, Bezprozvanny I, Ehrlich B (1991) Inositol 1,4,5-trisphosphate-gated channels in cerebellum: presence of multiple conductance states. J Neurosci 11: 3239–3245.
- 27.
Shuai J, Yang D, Pearson J, Rüdiger S (2009) An investigation of models of the IPR channel in Xenopus oocyte. Chaos 19: 037105.
- 28.
Mak D, Foskett J (1997) Single-channel kinetics, inactivation, and spatial distribution of inositol trisphophate (IP3) receptor in Xenopus oocyte nucleus. J Gen Physiol 109: 571–587.
- 29.
Mak D, McBride S, Foskett J (1998) Inositol 1,4,5-tris-phosphate activation of inositol trisphosphate receptor Ca2+ channel by ligand tuning of Ca2+ inhibition. Proc Natl Acad Sci U S A 95: 15821–15825.
- 30.
Demuro A, Parker I (2008) Multi-dimensional resolution of elementary Ca2+ signals by simultaneous multi-focal imaging. Cell Calcium 43: 367–374.
- 31.
Smith I, Wiltgen S, Parker I (2009) Localization of puff sites adjacent to the plasma membrane: Functional and spatial characterization of Ca2+ signaling in SH-SY5Y cells utilizing membranepermeant caged IP3. Cell Calcium 45: 65–76.
- 32.
Thul R, Falcke M (2004) Release currents of IP3 receptor channel clusters and concentration profiles. Biophys J 86: 2660–2673.
- 33.
Shuai J, Pearson J, Parker I (2008) Modeling Ca2+ Feedback on a Single Inositol 1, 4, 5-Trisphosphate Receptor and Its Modulation by Ca2+ Buffers. Biophys J 95: 3738.
- 34.
Mazzag B, Tignanelli C, Smith G (2005) The effct of residual Ca2+ on the stochastic gating of Ca2+-regulated Ca2+ channel models. J Theor Biol 235: 121–150.
- 35.
Tanimura A, Morita T, Nezu A, Shitara A, Hashimoto N, et al. (2009) Use of uorescence resonance energy transfer-based biosensors for the quantitative analysis of inositol 1, 4, 5-trisphosphate dynamics in calcium oscillations. J Biol Chem 284: 8910.
- 36.
Marchant J, Callamaras N, Parker I (1999) Initiation of IP3-mediated Ca2+ waves in Xenopus oocytes. EMBO J 18: 5285–5299.
- 37.
Fraiman D, Pando B, Dargan S, Parker I, Dawson S (2006) Analysis of puff dynamics in oocytes: interdependence of puff amplitude and interpuff interval. Biophys J 90: 3897–3907.
- 38.
Thul R, Falcke M (2006) Frequency of elemental events of intracellular Ca2+ dynamics. Phys Rev E 73: 61923.
- 39.
Izhikevich E (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. 441 p. The MIT press.
- 40.
Rintoul G, Baimbridge K (2003) Effects of calcium buffers and calbindin-D28k upon histamineinduced calcium oscillations and calcium waves in HeLa cells. Cell Calcium 34: 131–144.
- 41.
Dargan S, Schwaller B, Parker I (2004) Spatiotemporal patterning of IP3-mediated Ca2+ signals in Xenopus oocytes by Ca2+-binding proteins. J Physiol 556: 447–461.
- 42.
Zeller S, Rüdiger S, Engel H, Sneyd J, Warnecke G, et al. (2009) Modeling of the Modulation by Buffers of Ca2+ Release through Clusters of IP3 Receptors. Biophys J 97: 992–1002.
- 43.
Schipke C, Heidemann A, Skupin A, Peters O, Falcke M, et al. (2008) Temperature and nitric oxide control spontaneous calcium transients in astrocytes. Cell Calcium 43: 285–295.
