Calcium signals participate in a large variety of physiological processes. In many instances, they involve calcium entry through inositol 1,4,5-trisphosphate (IP3) receptors (IP3Rs), which are usually organized in clusters. Recent high-resolution optical experiments by Smith & Parker have provided new information on Ca2+ release from clustered IP3Rs. In the present paper, we use the model recently introduced by Solovey & Ponce Dawson to determine how the distribution of the number of IP3Rs that become open during a localized release event may change by the presence of Ca2+ buffers, substances that react with Ca2+, altering its concentration and transport properties. We then discuss how buffer properties could be extracted from the observation of local signals.
Calcium signals participate in a large variety of physiological processes (Berridge et al. 1998). At basal conditions, free cytosolic [Ca2+] is very low (approx. 100 nM). It is much higher in the extracellular medium and in internal reservoirs, such as the endoplasmic reticulum (ER). Ca2+ entry in the cytosol through specific channels that become open upon stimulation is the basic component of Ca2+ signals. This change in cytosolic [Ca2+] is then translated into different end responses depending on the cell type and on the resulting spatio-temporal distribution of [Ca2+]. Thus, it is of interest to determine how different factors affect this distribution.
One of the Ca2+ channels involved in intracellular Ca2+ signals is the inositol 1,4,5-trisphosphate (IP3) receptor (IP3R). It is located on the surface of intracellular membranes, particularly, of the ER. IP3Rs are biphasically regulated by Ca2+. For this reason, kinetic models of the receptor assume that there is at least one activating and one inhibitory Ca2+-binding site, so that Ca2+ binding to the first one induces channel opening (provided that IP3 is also bound to the IP3R) while binding to the second one induces channel closing (De Young & Keizer 1992; Fraiman & Ponce Dawson 2004; Shuai et al. 2007). The affinity for Ca2+ of the activating site is larger than that of the inhibitory site. In this way, Ca2+-induced Ca2+ release (CICR) occurs in which the Ca2+ ions released through an open channel induce the opening of other nearby channels with IP3 bound.
IP3Rs are not uniformly distributed on the membrane of the ER. They are organized in clusters separated by a few micrometres (Yao et al. 1995; Smith & Parker 2009). The combination of this inhomogeneity and of CICR gives rise to a large variety of intracellular Ca2+ signals, which go from local ones such as ‘blips’ (Ca2+ release through a single IP3R) and ‘puffs’ (Ca2+ release through several IP3Rs in a cluster) to waves that propagate throughout the cell (Sun et al. 1998). These signals have been observed using fluorescence microscopy and Ca2+-sensitive dyes (Bootman et al. 1997; Callamaras et al. 1998; Sun et al. 1998), and various models have been presented to interpret the observations (Swillens et al. 1999; Shuai et al. 2006; Swaminathan et al. 2009; Thul et al. 2009; Bruno et al. 2010). The use of total internal reflection fluorescence (TIRF) microscopy and a fast charge-coupled device camera in intact mammalian cells represents an experimental breakthrough that is giving more direct information on intracluster properties (Smith & Parker 2009). In particular, the distribution of the number of channels that open during puffs can be obtained without much processing. By keeping data that come from clusters of similar size, it is possible to get rid of cluster-to-cluster variability and obtain a distribution that provides an insight into the intracluster Ca2+ dynamics during puffs. We have recently presented a simple model with which we could reproduce the distribution reported in Smith & Parker (2009) and interpret its shape in terms of the competition of two stochastic processes: IP3 binding and Ca2+-mediated interchannel coupling (Solovey & Ponce Dawson 2010).
Interchannel coupling is mediated by the Ca2+ released through an open IP3R that subsequently diffuses to a neighbouring channel. This interaction may be affected by the presence of Ca2+ buffers that change the effective mobility and concentration of free Ca2+ ions (Allbritton et al. 1992). Cells contain a wide variety of these substances, particularly, Ca2+-binding proteins, which are often selectively expressed in specific subpopulations or at certain stages during the cell life. Exogenous buffers, on the other hand, can be used as an experimental tool to perturb Ca2+ signals at time and distance scales inaccessible to direct visualization (Dargan & Parker 2003). In this paper, we use the model introduced in Solovey & Ponce Dawson (2010) to analyse how the addition of exogenous buffers may affect the distribution of the number of channels that open during puffs. This study shows how the applicability and limitations of the model of Solovey & Ponce Dawson (2010) may be tested experimentally. It also shows how information on the effect of Ca2+ buffers on the intracluster dynamics may be extracted in experiments from the observed puff-size distribution.
The organization of the paper is as follows. In §2, we explain the main features of the model introduced in Solovey & Ponce Dawson (2010). In §3, we study numerically how the addition of different amounts of exogenous buffers affects the communication between channels, and we determine how the parameters of the model of Solovey & Ponce Dawson (2010) should be varied accordingly. Based on this, in §4, we study how the puff-size distribution given by the model of Solovey & Ponce Dawson (2010) changes with the addition of buffers. The conclusions are summarized in §5.
2. The model
In this section, we summarize the main features of the model introduced in Solovey & Ponce Dawson (2010) to describe the distribution of puff sizes reported in Smith & Parker (2009). Based on previous analyses (Bruno et al. 2010), the model assumes that a cluster occupies a circle of fixed radius, R=250 nm, and that N IP3Rs are randomly distributed inside it with uniform probability. The model has been developed for data collected from similar-type clusters, so that the value of N is also fixed. In the model, each IP3R of a cluster has a probability p of having IP3 bound, and if an IP3R becomes open it induces the opening of all other available IP3Rs (i.e. with IP3 bound) within a distance rinf of it, giving rise to a puff. These newly opened IP3Rs in turn trigger the opening of new IP3Rs with IP3 bound that are within the distance rinf from an open one. This scheme gives rise to a cascade of openings that stops when there are no more available IP3Rs within the radius of influence (i.e. the distance rinf) of any open IP3R. This implies that each puff is characterized by two random variables: the number of available IP3Rs, Np, and the number of open IP3Rs, n. The values that Np can take on depend on N and p. The latter is proportional to [IP3]. The values that n can take on depend on Np and on the spatial extent of CICR represented by rinf. The probability that an available IP3R makes a transition to the open state is an increasing function of the [Ca2+] it senses. The [Ca2+] profile around an open IP3R, on the other hand, is a decreasing function of the distance to the Ca2+ source that depends on the Ca2+ current and on the factors that interfere with Ca2+ transport (e.g. buffers). Having a fixed value of rinf carries the assumption that the Ca2+ profile around an open IP3R is the same regardless of how many of them are simultaneously open in the cluster (see Solovey et al. (2008) for a discussion on this).
Given a cluster, the model generates a sequence of puffs by determining, for each of them, which of the N IP3Rs are available, which one is the first to become open and then applying the rule that all available IP3Rs within a distance rinf of an open one become open. The probability, Pn, of having a puff with n open channels can then be written as 2.1 where PA(Np) is the probability that there are Np available IP3Rs before the occurrence of the puff and Po(n/Np) is the conditional probability that n channels open given that there are Np available IP3Rs. Equation (2.1) highlights the two stochastic components that shape Pn: IP3 binding (described by PA(Np)) and Ca2+-mediated interchannel coupling (described by Po(n/Np)). The relative weight of both factors depends on the relationship between two typical length scales of the problem: the radius of influence, rinf, and the mean distance between available IP3Rs, D, which is a random variable that is given by . If, for most events, the values of Np are such that rinf/D is very large, then the opening of any IP3R of the cluster leads to the opening of all available IP3Rs. In such a case, Po(n/Np)≈δnNp and Pn≈PA(n), which is a binomial. On the other hand, if rinf/D is small for most events, then Pn is concentrated near n=1, regardless of how many available IP3Rs there are in each realization. We refer to these two extreme cases as IP3 and Ca2+ limited, respectively. In between these extreme cases, one or the other behaviour may be favoured depending on the value of Np, i.e. on the realization. In such a case, the dominant stochastic component of Pn depends on the value of n.
3. Intracluster Ca2+ distribution in the presence of different buffers
In this section, we study how the addition of exogenous Ca2+ buffers affects the Ca2+-mediated channel–channel interaction. The aim is to determine how rinf should be varied in the model to account for the presence of these added buffers. To this end, we simulate the reaction–diffusion system that models the dynamics of cytosolic Ca2+ as described in Solovey et al. (2008) in the presence of one open IP3R. An open IP3R is considered to be a point source that releases a constant Ca2+ current that was estimated from experimental data to be 0.1 pA (Bruno et al. 2010). The reaction–diffusion system (Solovey et al. 2008) also includes a cytosolic Ca2+ indicator dye and an exogenous buffer, either ethylene glycol tetraacetic acid (EGTA) or bis(2-aminophenoxy)ethane tetraacetic acid (BAPTA). The Ca2+ indicator represents the dye usually used in fluorescent microscopy experiments and exogenous buffers are used in experiments to prevent the initiation of Ca2+ waves. We repeat the simulations for different amounts of the exogenous buffer and compare the [Ca2+] distributions obtained. The simulated equations are 3.1 where F is the Ca2+ dye used in the experiments, B is the exogenous buffer (B=EGTA or BAPTA), σ is proportional to the Ca2+ current and E is an immobile species that accounts for the effect of all endogenous buffers. In equations (3.1), the reaction terms are of the form with X=E, F, B and [X]T is the total concentration of the corresponding species. The parameter values are as in Solovey et al. (2008): DCa=220 μm2 s−1, DF=15 μm2 s−1, kEon=400 μM−1 s−1, kEoff=500 s−1, [E]T=300 μM, kFon=150 μM−1 s−1, kFoff=300 s−1 and [F]T=25 μM. For B=EGTA we consider kBon=5 μM−1 s−1 and kBoff=0.75 s−1 and for B=BAPTA, kBon=600 μM−1 s−1 and kBoff=100 s−1. We perform the simulations using a forward Euler method in time and finite differences in space with a second-order expression for the Laplacian. We use spherical coordinates, and the simulation domain is a 2 μm radius sphere. The boundary conditions are no-flux at r=2 μm. We perform simulations for [B]T between 100 μM and 1 mM for both EGTA and BAPTA.
Clusters have been estimated to be 400×400 nm in size (Shuai et al. 2006; Smith & Parker 2009; Bruno et al. 2010) while typical interchannel distances are assumed to be around 20 nm (Ur-Rahman et al. 2009). The model described in §2 reproduces the observations of Smith & Parker (2009) for rinf=0.25 μm. Thus, for exogenous buffers to alter interchannel communication they need to be fast enough (Zeller et al. 2009) so that they act over time scales shorter than 0.3 ms (the typical time it takes for Ca2+ to diffuse over a 0.25 μm distance). EGTA is a slow buffer while BAPTA is fast. Their different effects on the intracluster Ca2+ dynamics are evident in figure 1 where we have plotted [Ca2+] at a distance r=0.05 μm (figure 1a) and r=0.25 μm (figure 1b) from the open IP3R in the presence of EGTA (dot-dashed line) or BAPTA (solid line) as functions of the corresponding exogenous total buffer concentration. We observe that [Ca2+] at both distances is very insensitive to the addition of EGTA ([Ca2+](0.05 μm) ≈ 3.1 μM and [Ca2+](0.25 μm)≈0.3 μM for all [EGTA]T) while it is significantly altered by the presence of BAPTA. Considering that the probability per unit time that an available IP3R makes a transition to the open state is proportional to [Ca2+], then the ratio of [Ca2+] at a given distance in the presence of BAPTA and in the presence of EGTA gives an estimate of the factor by which rinf should be multiplied in the presence of one or the other buffer. This ratio ranges between 0.66 and 0.15 at r=0.05 μm and between 0.45 and 0.2 at r=0.25 μm. Taking into account that rinf=0.25 μm is the value at which the model of Solovey & Ponce Dawson (2010) reproduces the observations of Smith & Parker (2009), which were obtained in the presence of moderate amounts of EGTA and given that [Ca2+](0.25 μm)≈0.3 μM for all the values of [EGTA]T considered here, we can also estimate rinf in the presence of BAPTA as the distance from the 0.1 pA source at which [Ca2+]=0.3 μM for the simulations with BAPTA. We show this distance as a function of [BAPTA]T in figure 1c. The ratio of this distance to 0.25 μm varies between 0.58 and 0.23 for the values of [BAPTA]T considered.
4. The effect of Ca2+ buffers on Pn
The studies of the previous section illustrated in figure 1 consistently indicate that the value rinf=0.25 μm estimated in Solovey & Ponce Dawson (2010) should be decreased by a factor between 0.6 and 0.2 to account for the presence of BAPTA with total concentrations between 100 μM and 1 mM. We show in figure 2 the distribution, Pn, obtained with the model for rinf=0.25 μm (figure 2a), rinf=0.16 μm (figure 2b) and rinf=0.07 μm (figure 2c), which correspond to situations with EGTA, [BAPTA]T=100 μM and [BAPTA]T=500 μM, respectively. We observe in this figure that the addition of 100 μM BAPTA already introduces a noticeable change in the observed distribution of puff sizes, Pn. The distribution for [BAPTA]T=500 μM is highly concentrated around n=1 (with Pn=0 for n≤4), a situation that is even more pronounced for [BAPTA]T=1 mM (data not shown).
We have combined the results of numerical simulations of intracellular Ca2+ dynamics with the model introduced in Solovey & Ponce Dawson (2010) to determine how the distribution of the number of IP3Rs that become open during Ca2+ puffs may change by the presence of different Ca2+ buffers. In particular, we have determined that the addition of 100 μM of a fast buffer such as BAPTA may introduce noticeable effects on the observed distribution. Adding such a fast buffer could decrease the amplitude of the smallest puffs below the threshold of detection. However, this is not likely to happen for 100 μM. Thus, we suggest to perform experiments as in Smith & Parker (2009) but in the presence of different amounts of BAPTA (up to 100 μM) to probe the model of Solovey & Ponce Dawson (2010) and to extract information on how the spatial extent of CICR varies with this buffer.
This research was supported by UBA (UBACyT X145), ANPCyT (PICT 17-21001), Santa Fe Institute and CONICET (PIP 112-200801-01612).
One contribution of 13 to a Theme Issue ‘Complex dynamics of life at different scales: from genomic to global environmental issues’.
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