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Edited by: Guillermo A. Cecchi, IBM Watson Research Center, USA

Reviewed by: Markus A. Dahlem, Technische Universitaet Berlin, Germany; Irina Baran, Carol Davila “University of Medicine and Pharmacy,” Romania

*Correspondence: Guillermo Solovey, Laboratory of Mathematical Physics, The Rockefeller University, 1230 York Avenue, New York, NY 10065, USA. e-mail:

This article was submitted to Frontiers in Computational Physiology and Medicine, a specialty of Frontiers in Physiology.

This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.

Mean field models are often useful approximations to biological systems, but sometimes, they can yield misleading results. In this work, we compare mean field approaches with stochastic models of intracellular calcium release. In particular, we concentrate on calcium signals generated by the concerted opening of several clustered channels (calcium puffs). To this end we simulate calcium puffs numerically and then try to reproduce features of the resulting calcium distribution using mean field models were all the channels open and close simultaneously. We show that an unrealistic non-linear relationship between the current and the number of open channels is needed to reproduce the simulated puffs. Furthermore, a single channel current which is five times smaller than the one of the stochastic simulations is also needed. Our study sheds light on the importance of the stochastic kinetics of the calcium release channel activity to estimate the release fluxes.

An important problem in different areas of science is to relate phenomena that occur at different scales. In biology, an observed behavior at a certain level of organization is often the result of interactions at a lower level, with very different physical and temporal scales. Calcium (Ca^{2+}) signals are an extraordinary example of a multiple scale biological system suitable for being studied with different experimental techniques, each of which focus on a particular temporal and spatial scale. Therefore, it is important to understand the interactions of processes occurring at different scales and how to infer properties at unobserved scales from experimental observations. In this respect, mathematics is invaluable (Cohen,

Ca^{2+} ions are important signaling elements that regulate a large variety of cellular functions such as fertilization, proliferation, development, learning and memory, contraction and secretion (Berridge et al., ^{2+} as a intracellular cell messenger is possible because cells have mechanisms to control efficiently and precisely the distribution of Ca^{2+} in space and time. Within cells, Ca^{2+} is kept at high concentrations in stores such as the endoplasmic reticulum (ER). Release of Ca^{2+} from the ER occurs through specialized channels among which inositol triphosphate (IP_{3}) receptors play a most relevant role (Foskett et al., _{3}Rs are spatially organized in clusters located on the membrane of the ER and separated by a few microns (Yao et al., _{3}Rs depends on both the IP_{3} and cytosolic Ca^{2+}concentration ([Ca^{2+}]; Taylor, ^{2+} in both facilitating and inhibiting the opening of IP_{3}Rs. For relatively low [Ca^{2+}], the Ca^{2+} released by one channel increases the open probability of neighboring channels, whereas at high [Ca^{2+}] it favors a closed state of the channel (Iino, _{3}Rs on cytosolic Ca^{2+} creates communication between channels. As a result of the spatial organization of channels and the process of Ca^{2+}-induced Ca^{2+}-release (CICR), cytosolic Ca^{2+} signals display a hierarchical spatiotemporal organization which in some cells (e.g., oocytes) have length scales that span over six orders of magnitude (Callamaras and Parker, ^{2+} signals go from local ones such as “blips” (Ca^{2+} release through a single IP_{3}R) and “puffs” (Ca^{2+} release through several IP_{3}Rs in a cluster; Sun et al., ^{2+} diffusion between clusters (Yao and Parker, ^{2+} sensitive dyes (Bootman et al., _{3}Rs or the detailed Ca^{2+} dynamics within a cluster of channels in most cell types. The spatial resolution of conventional light microscopy is limited by diffraction. This determines that features below the 250-nm length scales are unobservable in typical confocal images. This limit can be beaten using super-resolution techniques. Techniques of this type have been applied to Ca^{2+} signals using total internal reflection fluorescence (TIRF) and a fast CCD (Smith and Parker, _{3}Rs are close enough to the plasma membrane. In particular, it does not work in oocytes. A vast body of literature on Ca^{2+} signalsin this cell type exists which is still awaiting a clarification of the underlying Ca^{2+} and IP_{3}R dynamics. Oocytes, on the other hand, are intact cells and can be used to explore changes in the morphology of the signals with maturation or with depth (Callamaras and Parker,

As with experiments, there are different modeling approaches that differ on the spatial or time scale they try to resolve. Some models rely on the hypothesis that channels are in such close contact that the concentration of Ca^{2+} can be considered homogeneous throughout the cluster (Shuai and Jung, ^{2+} dynamics since Ca^{2+} diffusion within the cluster can be neglected. Other models inferred basic puff properties using a deterministic dynamics of spatially localized channels within a cluster combined with an analysis of fluorescence experiments (Shuai et al.,

We have recently analyzed experimental calcium puffs observed in _{3}Rs during a puff is a non-linear function of the number of open channels: linear when the number is small (between 1 and ∼20 channels) and proportional to the square root of the number of channels for larger puffs. This type of scaling could be attributed to local Ca^{2+} depletion inside the ER. In fact, in Thul and Falcke (^{2+} release from the ER and compared them with signal mass measurements in ^{2+} dynamics, they found that the release current was approximately proportional to the square root of the number of open channels. In view of these results, the change of scaling could be explained as follows (Bruno et al., ^{2+} current scales as the square root of the number of channels. Thul and Falcke (^{2+} dynamics has been emphasized (Solovey et al.,

In this work we compare mean field and stochastic puff models, with the latter including the description of individual channel openings and closings. In particular we show that the stochastic model reproduces the experimental observations of Bruno et al. (_{3}Rs at any given time. It is the average Ca^{2+} current released during the whole puff duration that scales non-linearly with the number of open or available channels, as does the (instantaneous) total current in the mean field models of Bruno et al. (_{3}R current is needed than the actual one. Therefore, using mean field models to derive the underlying channel current from an image can lead to an underestimation of its value. The present study shows the limitations of mean field models and the importance of including a detailed description of the intra-cluster dynamics in order to infer realistic single channel properties from collective signals such as puffs.

The organization of the paper is as follows. In Section

We consider two kinds of models that differ mainly on the way we deal with the channel temporal dynamics. In ^{2+} through a few tens of IP_{3}Rs within a cluster; (b) the size of the Ca^{2+} release region is ∼500 nm^{2}, independent of the number of channels. As we describe in Section _{3} binding and unbinding. Thus, by available channels we mean IP_{3}Rs with IP_{3} bound.

Clusters consist of _{p}_{p}

The stochastic transitions between open and closed states of the channels occur at time scales much faster than the time scale that the experiments can resolve. For this reason, inferred values of the puff calcium current should be taken as averaged values. In contrast, in the simulations we know exactly how many channels are open with a resolution of ∼0.1 ms, much better than the resolution with which we could infer the Ca^{2+} currents from the experimental records in Bruno et al. (

The number of open channels, _{p}^{2+} current _{ch}_{ch}_{p}_{ch}^{2+} current is

with _{ch}_{ch}^{2+} current is given by:

with _{01} = 0.017 pA, _{02} = 0.08 pA, and _{pt}

The single channel current is 0.1 pA, regardless of the number of open channels. We use this value for the current because it agrees with previous estimates of the type 1 IP_{3}R current in the ER membrane of _{3}Rs (Vais et al.,

The transition between the open and closed conformations of the channels is stochastic. We use two simple IP_{3}R kinetic models which differ solely on the behavior of the mean open time as a function of [Ca^{2+}]. We call these models ^{2+}]) and ^{2+}]). We use these models to test whether a Ca^{2+} dependent or a Ca^{2+} independent open to closed transition leaves a detectable signature on puffs or not. Both models assume that the channel can be in three states, closed (C), open (O), or inhibited (I). The stochastic transitions between states occur according to the following schemes:

These models do not consider the kinetics of IP_{3} binding or unbinding to the IP_{3}R. According to most IP_{3}R kinetic models, these processes occur so fast that the fraction of IP_{3}Rs with IP_{3} bound can be assumed to be fixed and given by an equilibrium relationship (Young and Keizer, _{3}R with IP_{3} bound. In addition, the kinetic schemes do not consider either that the channels can reopen once they enter the inhibited states. This simplification is reasonable to investigate basic puff properties. Namely, the time an IP_{3}R remains inhibited under the conditions for which puffs are observed has been estimated as ∼2 s (Fraiman et al., _{3}R open probability obtained in reconstituted bilayer experiments (Bezprozvanny et al., ^{+} as the channel carrier (Mak et al., _{3}Rs remain in a native membrane. However, the Ca^{2+} concentrations that are used on the luminal side of the channel in the patch clamp experiments are much lower than the ones encountered under physiological conditions. As discussed in Fraiman and Ponce Dawson (

The values of the parameters of the kinetic models we consider in this paper are listed in Table _{CO}_{3}R model (Young and Keizer, _{3}R is 10 ms (a typical IP_{3}R open duration; Mak et al., ^{2+}]_{mouth}^{2+}] at the mouth of an isolated open channel. We use [Ca^{2+}]_{mouth}_{ch}

_{3}R models described in Section

Parameter | Value | Units |
---|---|---|

_{CO} |
20 | μM^{−1} s^{−1} |

_{OI} |
100 | s^{−1} |

μM^{−1} s^{−1} |
||

[Ca^{2+}]_{mouth} |
41.77 | μM |

We calculate the duration of the Ca^{2+} release during a puff as:

where 1_{{n(t)>0}} = 1 if _{M}^{2+} release duration corresponds to the total time with at least one open channel.

We calculate the average Ca^{2+} current released as:

It corresponds to the current of a puff that releases the same total amount of Ca^{2+} but at a constant rate during the whole duration of the Ca^{2+} release. It is equivalent to a case where all channels of the cluster open and close simultaneously and release the same total amount of Ca^{2+} during the same time as the original puff.

We solve numerically the set of coupled reaction-diffusion equations that result from the diffusion of cytosolic Ca^{2+} in the presence of localized Ca^{2+} channels. The details of the methods to simulate puffs are described in Solovey et al. (^{2+}, we consider the following species: an immobile endogenous buffer, a cytosolic fluorescent Ca^{2+} indicator (

In the case of the mean field models, we use a fine spatial grid where individual channels are located. We use clusters of 1, 3, 5, 10, and 15 channels for the linear mean field model and 1, 3, 5, 10, 15, 20, 25, 30, 40, and 50 channels for the simulations with the non-linear mean field model. The release duration was set to 18 ms, the mean duration of Ca^{2+} release inferred from experiments in Bruno et al. (

_{3}Rs _{3}Rs ^{2+} current is ∼0.25 pA. The number of channels are 3 in the linear mean field and 17 in the non-linear mean field model. In

To generate puffs with the stochastic model we use a simplified method also introduced in Solovey et al. (^{2+}] distribution within the cluster but describe its dynamics with a quasi-stationary approximation. In order to determine the contribution of each open channel to the [Ca^{2+}] distribution within the cluster, we first analyze this distribution when there is only one channel in the same cytosolic environment. The total [Ca^{2+}] within the cluster when there are several open channels is then approximated by a linear combination of the contributions due to each individual open channel. This detailed description of [Ca^{2+}] within the cluster is used to determine the openings and closings of the various channels. In this way we obtain the total Ca^{2+} current that is released from the cluster as a function of time. The spatiotemporal [Ca^{2+}] distribution outside the cluster is determined using a coarser grid in which the cluster is represented by a point source whose current is proportional to the number of open channels determined before. A reaction-diffusion system is solved on this coarser grid. We simulated 10 puffs for a cluster of a given number of channels that we vary between 10 and 50.

The fluorescence ^{2+} ions bound to the fluorescent dye (

where

In order to gain a more intuitive understanding of the behaviors encountered with the numerical simulations, we also analyze a cartoon-like model that is amenable to analytic calculations. We consider the IP_{3}R kinetic models introduced in Section ^{2+} dynamics is simplified in two ways. First, at time zero all _{p}^{2+} dynamics, specially in the ^{2+}. To further simplify the dynamics of the ^{2+}] which is proportional to the number of open channels ^{2+}] at time ^{2+}](^{2+}]_{mouth}^{2+}]_{mouth}_{p}_{p}

The two versions of the semi analytic model can be considered as extreme cases regarding the channels “coupling strength” (DeRemigio and Smith, _{OI}^{2+}.

The mean puff duration can be computed analytically for both models using Equation _{M}_{M}_{p}

for models

The average released current, _{ave}

so that:

for models

Simulations can display the time course of puffs at different scales: from the single channel activity to the fluorescence signal. Useful parameters that characterize puffs are the amplitude (_{f}^{2+} release. However, in puffs simulated with the mean field model the current is steady and the fluorescence increases during the total duration of Ca^{2+} release (Figures

The examples of Figures _{3}R model is not significant. There is a rapid recruitment of IP_{3}Rs when Ca^{2+} release starts. This initial phase is followed by a usually larger falling phase in which channels close one by one, in agreement with Smith et al. (

The amplitude of the puffs simulated with the non-linear mean field model is similar to the amplitudes of the puffs obtained with the stochastic model. The number of channels is also similar. However, the linear mean field model reproduces the same amplitude as the stochastic puffs but with significant fewer channels. The examples displayed in Figure

Interestingly, the ratio of puff amplitudes is similar in all models when the number of channels is duplicated, as it is shown in Figure

As illustrated in Figures ^{2+}] elevations is more effective as the distance between IP_{3}Rs decreases. Approximately 80% of the puffs simulated using the stochastic model are such that all available channels of the cluster participate although only approximately half of them were simultaneously open (data not shown). In the remaining ∼20% of the puffs, only ∼2 channels did not open during the 150-ms of the simulation time. This is more likely to occur in small clusters (〈N_{p} ∼ 13〉) where the spacing between IP_{3}Rs is relatively large and the coordination between channels due to CICR is less effective.

It is important to note that the average current and fluorescence time course of several stochastic simulations does not converge to the mean field counterpart. The mean field model assumes that all channels open and close simultaneously, therefore the release current is constant and the fluorescence increases during the puff.

Observed puff amplitudes do not grow linearly with the maximum underlying current estimated from the observations (Bruno et al., ^{2+} ions that are released during a puff. Nevertheless, this is not the case. Calibration experiments showed that dye saturation occurs at much larger amplitudes (A ∼ _{max}_{0} = 48) than those of the puffs analyzed in Bruno et al. (^{2+}-bound dye concentration is never larger than 35 μM (for _{p}^{2+}-bound dye concentration within the volume corresponding to the pixel of the image. This processes depend on the kinetic rates of the reaction, the Ca^{2+} current, and the rate at which the Ca^{2+} bound dye leaves the volume due to diffusion.

_{3}Rs follow the

Although puff amplitudes depend non-linearly on the underlying maximum currents, both for the simulations and for the experiments, the mean values of the puff current can be approximated by a function of the mean puff amplitudes with a relatively small dispersion. This indicates that, even if the interaction with the dye acts as a non-linear filter, the puff amplitude could eventually be used to estimate the underlying Ca^{2+} current.

We have analyzed in the previous Section the relationship between puff amplitude and the Ca^{2+} current that underlies the signal. We have shown, in particular, that these quantities depend non-linearly on one another in mean field models (linear and non-linear), stochastic models (independently of the IP_{3}R kinetic scheme), and experiments. Even though the amplitude–current relationship obtained in all these cases is similar, we have observed that the number of channels involved in the linear and the non-linear mean field models are very different. In order to analyze this aspect in more detail, we now explore the dependence between number of channels and puff amplitude.

We show in Figure _{p}_{p}^{2+}-dye reaction introduces between Ca^{2+} current and fluorescence, there is an additional source of non-linearity when going from the number of channels to puff amplitude. This additional source of non-linearity is related to the temporal coarse-graining imposed by the experimental temporal resolution acting on data that is characterized by different time courses depending on the size of the puffs. As shown in Figure _{3}Rs within the cluster. All models we use in the simulations assume that the spatial cluster size is constant regardless of the number of channels that they contain. Thus, the simulated puffs that involve more open channels and have larger amplitudes, occur in clusters where the mean distance between available channels is shorter. This implies that CICR occurs on a faster time scale and the peak current is achieved within a shorter time. This non-linearity is absent in the linear mean field model.

_{p}_{3}R kinetic model. ^{2+} current (Equation _{01}_{p}_{p}_{p}_{01} = 0.019 pA and _{02} = 0.105 pA. _{01} should not identified as the actual single IP_{3}R current but as an effective value that follows from the assumption that all channels open and close simultaneously.

In order to compare the new non-linearity that arises in the simulations of the stochastic puff model with the one that is intrinsic to the non-linear mean field model we compute the average current (_{ave}_{p}^{2+} during the same time as the original puff). Therefore, it is equivalent to replacing the stochastic puff by one that could have been obtained with a mean field model. We found that the derivative of _{ave}_{p}_{p}_{p}_{p}_{ave}_{01}_{p}_{p}_{p}_{p}_{01} = 0.019 pA, which is five times smaller than the actual single channel current used in the simulations. Interestingly, it agrees with the value that the non-linear mean field model needed to reproduce the experimental observations (see Equation

The fact that the effective single channel current is smaller than the actual one can be understood by considering two extreme (hypothetical) puffs with _{p}_{r}_{p}_{r}_{ave}_{ch}N_{p}t_{r}_{r}I_{ch}N_{p}_{ch}_{ave}_{ave}_{ch}_{p}_{eff}_{ave}_{p}_{ch}

In order to get an intuitive understanding of why the slope between _{ave}_{p}_{p}_{ave}_{p}_{3}Rs because they lead to different results. Although it is clearer for _{p}_{ave}_{p}_{ave}_{p}_{3}R models for _{p}_{eff}_{ave}_{p}_{p}_{eff}

_{p}_{3}R kinetic model. _{p}_{3}R kinetic model. In

The semi-analytical model allows us to go one step further in the analysis. We can use the approximation 8 to estimate _{ave}_{n}_{ave}_{ch}_{p}_{n}_{3}R kinetic models that we considered, _{n}_{n}^{2} respectively. Approximating _{p}_{ave}_{p}_{D}_{p}_{ave}_{p}_{p}_{ave}_{p}_{p}_{ave}

Intracellular signals that involve Ca^{2+} release from the ER through clustered IP_{3}R–Ca^{2+} channels are crucial in a variety of cell types. A complete understanding of the processes involved in such signals requires a combination of experiments and mathematical modeling. Given the multiple time and spatial scales involved, mean field approaches are frequently used in models and data analysis. In a sense, data from fluorescence microscopy experiments is intrinsically of mean field type since it is averaged over the finite space and time resolution of the experimental method. In this work we have compared mean field approaches to stochastic models of Ca^{2+} puffs, a type of localized signal that involves Ca^{2+} release from several IP_{3}Rs in a cluster and which has been observed in intact cells using optical techniques. We found that mean field models only match more accurate stochastic models under two unrealistic assumption: (a) a non-linear relationship between the current and the number of open channels and (b) a single channel current five times lower.

We have used two mean field models that we presented in a previous work (Bruno et al., _{3}Rs of the cluster open and close simultaneously. They differ on how the current depended on the number of channels: linear (Equation

The first observation is that the amplitude of puffs is non-linearly related to the released current, as it was shown in Bruno et al. (

Recent experimental results concluded that the total calcium current released through a cluster of IP_{3}Rs during a puff is a non-linear function of the number of open channels (Bruno et al., ^{2+} depletion inside the ER (Thul and Falcke, _{01} approximately equal to the single channel current of the non-linear mean field model (∼0.02 pA). This results implies that the mean field assumption also underestimates the single channel current by a factor of 5 (Figure

We investigated the differences between stochastic and mean field approaches analytically as well. For this purpose, we used a simplified stochastic model to describe the dynamics of clustered IP_{3}Rs (semi-analytical model). Results using this model confirmed that the assumption that channels open and close simultaneously leads to an underestimation of the single IP_{3}R channel current and that the relationship between the average current and the number of available channels is non-linear (Figure

In a previous work (Bruno et al.,

There is some controversy on how IP_{3}Rs behave (Bezprozvanny et al., _{3}R kinetic models (Young and Keizer, ^{2+} or not. Therefore, in the case of the stochastic models, we did simulations using two simple kinetic models of the IP_{3}R that only differed on whether the transition from the open to the closed state of the channel was Ca^{2+} dependent or not (^{2+}] sensed by each channel is additive and proportional to the number of open channels, as if all IP_{3}Rs were in the same spatial location. In contrast, when using the ^{2+} released by one channel has no influence on the other open channels. The real situation is somewhere in between both, as it is shown in Figure _{3}R models. We therefore conclude that this single channel property (a Ca^{2+} dependent or a Ca^{2+} independent open to closed transition) does not leave a detectable signature on puffs.

Ca^{2+} puffs are the result of a complicated underlying spatiotemporal dynamics. The relationship between puff amplitude and the underlying Ca^{2+} current can only be understood in terms of the interplay between the intensity and the dynamics of the release. The dynamics of the release, on the other hand, is strongly dependent on the spatial organization of the channels that become open. The combination of all these factors and the experimental limitations can lead to (unexpected) non-linear relationships. Modeling efforts that do not include the variability of the individual channel openings and closings provide averaged descriptions that are still valid. In particular, they can be used as building blocks of mathematical models of more global signals such as waves. However, if one is willing to understand the intra-cluster dynamics or to extract information on single channel kinetics from puff observations, a stochastic description is unavoidable.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We describe here the derivation of the formulas for the mean duration of the puff in the semi-analytical model, Equation

eq: stoch.metodos.semianalitico.Iavemedio

In this model we consider that the puff starts with _{p}_{p}_{p}^{2+} release, Equation

In Equation _{n}_{n's}^{−1} (which may depend on _{3}R model). The number of simultaneously open channels reduces to

The minimum between _{n}^{−1}. Finally, the mean Ca^{2+} release duration of a cluster of _{p}

Equation _{i}_{OI}_{3}Rs are coupled by Ca^{2+}, so that Ca^{2+} release from one channel affects the inhibiting probability of the rest opened channels. Different levels of coupling were considered in Groff and Smith (^{2+}] sensed by an open channel is [Ca^{2+}]_{mouth}^{2+}]_{mouth}^{2+}] value at the mouth of one isolated open channel. In most of this paper, we work with two extreme cases, ^{2+}]_{mouth}_{d}_{OI}n_{d}^{2+}] is additive.

The variance of the puff duration can be obtained using Equation _{n}

It is also possible to calculate the distribution function of _{r}_{r}_{p}_{n}_{p}_{n}^{−1}], following (Cox,

where

The average current released during a puff, Equation _{ave}

where _{n}_{p}_{n}^{−1} (in the case of the _{i}_{OI}_{d}_{OI} n_{n}_{p}_{ave}_{p}

The approximate solutions in Equation _{p}_{p}

We acknowledge useful conversations with Ian Parker and Luciana Bruno. Guillermo Solovey would also like to acknowledge Marcelo Magnasco and the staff of the Center for Studies in Physics and Biology (The Rockefeller University). Daniel Fraiman and Silvina Ponce Dawson are members of the Carrera del Investigador Científico (CONICET). This research was supported by UBA (UBACyT X145) and CONICET (PIP 11220080101612).