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Release Currents of IP₃ Receptor Channel Clusters and Concentration Profiles

Hahn Meitner Institut, 14109 Berlin, Germany

Correspondence: Address reprint requests to M. Falcke, E-mail: falcke{at}hmi.de.

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS AND PARAMETERS RESULTS DISCUSSION REFERENCES

 
We simulate currents and concentration profiles generated by Ca²⁺ release from the endoplasmic reticulum (ER) to the cytosol through IP₃ receptor channel clusters. Clusters are described as conducting pores in the lumenal membrane with a diameter from 6 nm to 36 nm. The endoplasmic reticulum is modeled as a disc with a radius of 1–12 µm and an inner height of 28 nm. We adapt the dependence of the currents on the trans Ca²⁺ concentration (intralumenal) measured in lipid bilayer experiments to the cellular geometry. Simulated currents are compared with signal mass measurements in Xenopus oocytes. We find that release currents depend linearly on the concentration of free Ca²⁺ in the lumen. The release current is approximately proportional to the square root of the number of open channels in a cluster. Cytosolic concentrations at the location of the cluster range from 25 µM to 170 µM. Concentration increase due to puffs in a distance of a few micrometers from the puff site is found to be in the nanomolar range. Release currents decay biexponentially with timescales of <1 s and a few seconds. Concentration profiles decay with timescales of 0.125–0.250 s upon termination of release.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS AND PARAMETERS RESULTS DISCUSSION REFERENCES

 
Many cells control the cytosolic Ca²⁺ concentration by release and uptake of Ca²⁺ from intracellular stores like the endo- and sarcoplasmic reticulum. Ca²⁺ flows down its electrochemical gradient upon opening of channels on the membrane of these stores. One of these channels is the inositol 1,4,5-trisphosphate receptor channel (IP₃R), which is found in many different cells (Berridge, 1993, 1997; Berridge et al., 1998). Because the opening probability of the IP₃R depends on the cytosolic Ca²⁺ concentration, channels can communicate with each other via the released Ca²⁺. Thus complex spatiotemporal patterns are formed. (Ridgway et al., 1977; Lechleiter et al., 1991; Nathanson et al., 1994; D'Andrea and Vittur, 1995; Marchant et al., 1999; Marchant and Parker, 2001; Bootman et al., 1997, 2001). This extensive behavior of intracellular Ca²⁺ dynamics has caused a great amount of modeling studies (Dupont and Goldbeter, 1993, 1994; Borghans et al., 1997; DeYoung and Keizer, 1992; Wagner and Keizer, 1994; Atri et al., 1993, Sneyd et al., 1993; Sneyd and Sherrat, 1997; Sneyd and Dufour, 2002; Falcke et al., 1999a,b, 2000a; Bär et al., 2000; Falcke, 2003a,b).

However, despite the fact that release through a single IP₃ receptor channel or a closely packed group of channels is in the core of the dynamics, it has attracted less attention. Although several studies on the concentration dynamics close to open ryanodine receptor channels exist, little has been done for the IP₃R (Melzer et al., 1984, 1987; Blatter et al., 1997; Smith et al., 1998; Izu et al., 2001; Pratusevitch and Balke, 1996; Mejia-Alvarez et al., 1999; Rios et al., 1999; Gonzalez et al., 2000) but see Swillens et al. (1998) for IP₃R. The major differences between release through ryanodine receptor channels and IP₃Rs are the currents and timescales. Currents between 0.35 and 3.3 pA are observed in experiments mimicking release from the sarcoplasmic reticulum and elemental events last typically 10 ms (Rios et al., 1999; Mejia-Alvarez et al., 1999). Currents through a single IP₃R are estimated to be in the range from 0.1 to 0.5 pA and elemental events last a few hundred milliseconds (Parker et al., 1996a; Sun et al., 1998). Hence, peak concentrations, the spread of free Ca²⁺ in the cytosol and the development of the store content will be different from release by ryanodine receptor channels.

The existing simulations and analytic calculations demonstrate that huge concentration gradients occur around release sites (Smith et al., 2001). Furthermore, the arrangement of channels in small groups or as single channels on the ER membrane creates spatially discrete release sites (Sun et al., 1998; Marchant and Parker, 2001; Mak et al., 2000; Mak and Foskett 1997, 1998; Wilson et al., 1998). The typical distance between these release sites is a few micrometers (Marchant and Parker, 2001; Thomas et al., 1998) and therefore larger than the diffusion length of free Ca²⁺ in the cytosol of 0.4–1.3 µm (Wang and Thompson, 1995). Hence, concentration values can differ by 2–3 orders of magnitude between the locations of neighboring channels or channel clusters. Several theoretical studies have shown that the consequences of the discreteness and the concentration gradients can reach from the loss of the ability to oscillate (Sneyd and Sherrat, 1997; Falcke, 2003b) to the termination of wave propagation (Sneyd and Sherrat, 1997; Falcke et al., 2000b; Coombes, 2001). It is therefore important to know at least approximately the concentration values, gradients, and dynamics at the mouth of a releasing channel and at a distance of a few micrometers. The purpose of this study is to provide a quantitative idea of these release characteristics to support further modeling of intracellular Ca²⁺ dynamics. That goal is different from other studies aiming at the interpretation and evaluation of specific experiments, e.g., by drawing conclusions from dye concentration profiles with respect to the profile of free Ca²⁺ (Melzer et al., 1984, 1987; Blatter et al., 1997; Smith et al., 1998; Pratusevitch and Balke, 1996; Mejia-Alvarez et al., 1999; Rios et al., 1999; Gonzalez et al., 2000). The study at hand contains much information that is experimentally relevant as well, especially because we tried to model as close to experimental findings as possible.

We will consider the process of Ca²⁺ release through an open channel. The dynamics of the channel state (open, closed, or inhibited) is not the subject of this study. Because the channel current through the IP₃R in vivo is not very well known, we will start from lipid bilayer experiments. We will transfer their results on the dependence of channel currents on lumenal concentrations to in vivo geometries and concentrations. The following section introduces the model equations and explains the choice of parameter values. The subsequent section will present simulation results. We find that the release currents depend linearly on the concentration of free Ca²⁺ in the lumen but are less sensitive to lumenal buffer concentrations, total lumenal Ca²⁺ content, or lumenal diffusion coefficients for ranges of values that can be expected to hold in vivo. The release current is approximately proportional to the square root of the number of open channels in a cluster. Cytosolic concentrations at the location of the cluster range from 25 µM to 170 µM. Concentration increase due to open clusters in a distance of a few micrometers is found to be in the nanomolar range on the timescale of puff duration. Concentration profiles built up by release decay on a timescale of 0.125–0.250 s. Release currents decay biexponentially with timescales of <1 s and a few seconds.

	METHODS AND PARAMETERS

TOP ABSTRACT INTRODUCTION METHODS AND PARAMETERS RESULTS DISCUSSION REFERENCES

 
We simulate release of Ca²⁺ in a cylindrical volume divided by the lumenal membrane perpendicular to the cylinder axis. The smaller part represents the ER and the larger part the cytosol. The channel is a pore in the center of the ER membrane with radius R_s (see Fig. 1). The initial condition is the stationary Ca²⁺ distribution resulting from the pumps and the leak flux P_l. No flux boundary conditions were applied at the outer surface of the cylinder. We chose cylindrical coordinates for our simulations with the positive z-direction pointing from top to bottom in Fig. 1.

View larger version (22K): [in this window] [in a new window]   FIGURE 1  Volume within which release was simulated. Rotational symmetry allows us to reduce the integration to a plane cutting radially into the volume as shown.

The cytosolic free Ca²⁺ concentration c.

The free Ca²⁺ concentration in the endoplasmic reticulum E.

The concentration of stationary buffer b_s in the cytosol with Ca²⁺ bound.

The concentration of mobile buffer b_m in the cytosol with Ca²⁺ bound.

The concentration of stationary buffer b_Es in the endoplasmic reticulum with Ca²⁺ bound.

The concentration of mobile buffer b_Em in the endoplasmic reticulum with Ca²⁺ bound.

The reaction-diffusion equations in the cytosol are:

The equations include diffusion of free Ca²⁺ c (D²c), diffusion of mobile buffer b_m (D_m²b_m), and the reactions of stationary buffer b_s and mobile buffer b_m with free Ca²⁺ ( (B_i – b_i)c – b_i, B_i total buffer concentration, i = s,m).

Analogously, the dynamics in the endoplasmic reticulum are:

The flux J through the membrane separating ER and cytosol is given by:

(1)

(2)

Here, the values of E and c have to be taken at the membrane (position z_m). R denotes the cluster radius, P_l the coefficient of the leak flux density, and P_P the maximal pump strength. The constants in Eq. 1 will be determined in the next section. The currents are incorporated into the volume dynamics by setting the boundary condition at the ER membrane like:

(3)

Fitting the single channel flux
The expression for the flux through an open channel (Eq. 1) is fitted to data by Bezprozvanny and Ehrlich (1994). Bezprozvanny and Ehrlich measured single channel currents of channels reconstituted into planar lipid bilayers with Ba²⁺ as charge carrier in dependence on the trans (lumenal) concentration. The constants of Eq. 1 are fitted by simulation of fluxes in these experiments. To perform a fit to the data, we need to mimic the spatial set up of the diffusion problem for a single channel. The current through a single channel is determined, among other parameters, by the geometry of the channel pore. The size of the channel pore is in the range from 10 Ų (Hille, 2001) to 40 Ų (Lindsay and Williams, 1991; Lindsay et al., 1991), if we assume it to be similar to the pore of the ryanodine receptor channel. However, we are interested in concentration profiles on the length scale of a few micrometers. We need to find a compromise in terms of length scales between the geometry of the channel pore and the micrometer length scale to reach a computationally treatable model. Here, we use results of Mejía-Alvarez et al. (1999) who performed similar simulations for ryanodine receptor channels. They estimated the size of the channel sponge on the lumenal side of the ryanodine receptor channel. The channel sponge is the volume within which negative charges surrounding the channel pore reside. The radius of this sponge was calculated to be the radius of the Donnan equilibrium potential created by the negative charges (Mejia-Alvarez et al., 1999). It was estimated to be 5–10 nm (Mejia-Alvarez et al., 1999). Therefore, we consider the channel to be a conducting pore in the membrane of approximately this radius. In most simulations we use 6 nm (allowing for a numerical grid mesh size of 2 nm).

The trans chamber in the simulations used to fit the dependence of the current on lumenal concentrations has a radius of 12 µm and a thickness of 6.24 µm. The cis chamber is set up with the same radius and is 5.22 µm thick. These measures are sufficient to exclude geometric restrictions, i.e., could serve as a model of a chamber much larger than (12 µm)² x 5.22 µm. The diffusion coefficient for free Ca²⁺ is set to 600 µm² s^–1 corresponding to the value in water (Kushmerick and Podolsky, 1969). The cis chamber contains 4 mM of mobile buffer ( (µMs)^–1, s^–1, D_m = 30 µm² s^–1) and 100 µM of stationary buffer ( (µMs)^–1, s^–1). However, the buffers have essentially no effect on the current values.

As suggested by the data of Bezprozvanny and Ehrlich (1994), we take a saturating barrier model of a conducting pore with one ion binding site for the flux (Keener and Sneyd, 1998). The general expression of the flux through such a pore is given by Eq. 1. Note, that only four of the five constants in this equation are independent. To fit them to the data by Bezprozvanny and Ehrlich we simulate the current I in bilayer experiments. The current is equal to the integral of the flux density over the pore cross section (F Faraday's constant, r radial coordinate, z_m axial coordinate of the membrane, R_s single channel radius). We performed simulations for the homogeneous initial values E = 10, 20, 30, 40 mM, which were also used in the experiments. In these simulations for the fitting procedure, the expression J = P_ch(E – c) was used for the channel flux density. Adjusting P_ch for each of the four lumenal Ca²⁺ concentrations in such a way as to reproduce the measured currents allows us to read off the values of the lumenal and cytosolic free Ca²⁺ concentration at the channel and to determine the constants in Eq. 1: = 9.3954 µms^–1, = 1.1497, ß = 1.1949 10^–3, = 1.1444 10^–7 µM^–1, and = 1.1556 10^–7 µM^–1.

Local depletion of Ca²⁺ due to the channel flux does not occur in this geometry and with these concentration values and diffusion coefficients. That can be concluded from the values of E at the channel E = 9.5, 19.3, 29.2, and 39.2 mM belonging to the initial concentrations 10, 20, 30, 40 mM. Hence, the concentration at the channel is well approximated by concentration values in the bulk of the chamber far away from the channel in lipid bilayer experiments. We will see below that this does not apply to in vivo situations. The results for the constants of Eq. 1 imply a half maximum value (ß/) of 10.44 mM for E, which is close to the value of 9 mM given in Bezprozvanny and Ehrlich (1994). The difference E – c remained large in this lipid bilayer simulation. Hence, there is little effect of Ca²⁺ on the cis (cytosolic) side on channel flux density in the lipid bilayer experiments. We will see below, that the concentration values on the lumenal and cytosolic side can be close to each other and the feedback by cytosolic Ca²⁺ becomes relevant in different geometries and for different concentrations in the lumenal part.

Other parameter values
According to Alberts et al. (1994; p. 580), the lumenal space of a flattened cistern of the endoplasmic reticulum of a liver cell is 20–30-nm wide. The diameter of tubes belonging to the network part of the ER is up to 60 nm. We will consider the case of release from cisternae with a thickness of 28 nm and tubular networks. Tubular networks are not described by implementing their real geometry in the simulations. We still use the same cylindrical geometry. However, diffusional transport is reduced due to the tubular shape of the network compared to unobstructed volume and we have to incorporate this reduction. Here, we take advantage of results by Ölveczky and Verkman, who showed that the turtuosity of such a tubular network can be accounted for by reducing the diffusion coefficient by 40–60% (Ölveczky and Verkman, 1998). Hence, we model tubular networks by a disc with a height of 60 nm and a reduction of lumenal diffusion coefficients to one-half of the values for cisternae.

The endogenous cytosolic buffer capacity (B_s/K_s) of immobile buffer was chosen to be 40 according to findings by Zhou and Neher (1993). In agreement with the same experiments, that buffer has a rather large dissociation constant of 2 µM.

The diffusion coefficient of free Ca²⁺ in the cytosol was measured by Allbritton et al. as 223 µm² s^–1 (Allbritton et al., 1992). The diffusion coefficient of Ca²⁺ in water is 600 µm² s^–1 (Kushmerick and Podolsky, 1969). It was found, that the diffusion coefficients of substances in the cytosol, which are believed not to be bound in the cytosol, are approximately one-half of the value in water (Kushmerick and Podolsky, 1969). This observation supports the value measured by Allbritton et al. (1992) and hence we adopt that value as the diffusion coefficient of free Ca²⁺ in the cytosol. It is difficult to estimate the diffusion coefficient of free Ca²⁺ in the ER. In most of the simulations, we use the same value as in the cytosol. However, one could assume that similar to the reduction of diffusion when going from water to the cytosol by one-half, a further reduction occurs when going from the cytosol to the lumen of the endoplasmic reticulum. Hence, we will show results with diffusion coefficients of free Ca²⁺ in the ER reduced by one-half with respect to value in the cytosol, too.

The results of the measurements of the total Ca²⁺ content of the ER depend on the method used. Direct determination by electron microscopy techniques finds a range of 5–50 mM. The highest values occur in the terminal cisternae of the sarcoplasmic reticulum (Meldolesi and Pozzan, 1998). Indirect measurements based on fluorescent indicators in the cytosol find 5–10 mM (Meldolesi and Pozzan, 1998).

Most of the lumenal Ca²⁺ is bound to buffer. The buffer protein calsequestrin is predominantly found in muscle cells. It binds Ca²⁺ with high capacity (50 ions per molecule) and a K_d of 1 mM (Meldolesi and Pozzan, 1998). Calreticulin is mostly found in nonmuscle cells. It binds 20–50 Ca²⁺ ions with low affinity (K_d = 0.3–2.0 mM) (Michalak et al., 1992). Changing the lumenal concentration of calreticulin has complex consequences for IP₃-induced Ca²⁺ release (Camacho and Lechleiter, 1995; Roderick et al., 1998; John et al., 1998).

The concentration of free Ca²⁺ in the endoplasmic reticulum is especially important because it is the driving force for release. It is typically in the range of a few hundred µM and may reach up to 1 mM (Meldolesi and Pozzan, 1998). A few examples of specific measurements are: in rat embryo R6 fibroblasts 500 µM (Foyouzi-Youssefi et al., 2000), in rat myotubes sarcoplasmic reticulum after 6 days in culture 300 µM (Robert et al., 1998), in BHK21 fibroblasts 539 ± 92 µM (Hofer and Schulz, 1996), in HeLa cells 60–400 µM (Miyawaki et al., 1997). More values can be found in Meldolesi and Pozzan (1998).

We translate these findings into parameter values for buffers in the endoplasmic reticulum. We use the total concentration of buffers to control the total Ca²⁺ content of the ER and the buffer dissociation constant to set the concentration of free Ca²⁺. We will show examples with a total content in the range from 5.24 mM to 67.87 mM and a free concentration from 127 µM to 715 µM. Similarly, diffusion coefficients of mobile buffers in the lumen of the ER could only be guessed. Consequently, we vary this parameter from 0 to 30 µm² s^–1.

Numerical simulations use a 4th order Runge Kutta algorithm (Press et al., 1992). Spatial discretization was 2 nm for single channel simulations, nm and 4 nm for larger clusters. The time discretization is chosen between 25% and 90% of the stability criteria (Press et al., 1992).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS AND PARAMETERS RESULTS DISCUSSION REFERENCES

 
All results presented in this section are obtained in the cell-like geometry. Thus, the lumenal height is set to 28 nm when we refer to cisternae in the ER and 60 nm for the tubular ER. Additionally we reduce the diffusion coefficient in the latter case. We begin the presentation with simulation results for single channels. The concentration in a vicinity of a few nanometers of the channel mouth rises within microseconds upon opening of a channel. On the same timescale of microseconds, the concentration of free lumenal Ca²⁺ at the channel vestibule drops to create the large gradients necessary to transport Ca²⁺ to the channel. All ensuing slow relaxations cause concentration changes typically an order of magnitude smaller than this initial fast rise. This very fast initial timescale appears in the current as well. It quickly drops within the first microseconds to a level that then changes on the timescale of milliseconds.

A channel opens and closes rapidly once it is activated. Typical dwell times are in the range of a few milliseconds (see, e.g., Mak and Foskett, 1998; Mak et al., 2001)). The concentrations of fast buffers and free Ca²⁺ within a vicinity of <1 µm follow or are influenced by this rapid opening and closing of the channel. That is demonstrated in Fig. 2. The concentrations immediately at the channel react with a sharp rise or drop upon channel opening and closing.

View larger version (53K): [in this window] [in a new window]   FIGURE 2  Concentration values of cytosolic free Ca2+ at the channel mouth (top left) and in a distance of 1 µm and 2 µm (top right), lumenal free Ca2+ at the channel vestibule (bottom left), and cytosolic mobile buffer with Ca2+ bound at the channel mouth (bottom right). Open and closed times are 5 ms each. For parameter values, see Set 2 in Table 1. The amplitude of the current pulses is 0.0785 pA and decreases by 1% only in the first 400 ms (data not shown).

When more than one channel in a cluster is activated, the channels open and close on a timescale of a few milliseconds. This leads to an average number of permanently open channels. Therefore we simulate clusters with a given number N_O of open channels. Channels in a cluster are assumed to be closely packed with a distance of 12 nm (Swillens et al., 1999). Release through the different channels of a cluster cannot be considered as independent, if the width of the concentration profile formed in the lumen equals approximately or is larger than channel spacing. Our simulations showed that a single channel release current of 0.041 pA causes a profile of free Ca²⁺ in the lumen with a full width at half-depth of 10 nm after 15 ms. Hence, the total release current of a cluster is not the sum of the single channel currents but channels influence each other. That interaction of release through the different channels of a cluster is the reason for the relation between the number of open channels and current we will demonstrate below. Swillens and Dupont found that release through several closely packed individual channels can be very well approximated by release through a single membrane area of corresponding size (Swillens et al., 1999). Based on these observations we model a cluster with several open channels by setting the area of the conducting pore proportional to with R_s being the single channel radius. The radius of the pore is then called cluster radius R and is determined by We will simulate release for the typical duration of a puff of 400 ms in most of the examples. The section entitled "Long timescales, higher cluster density" will consider longer simulations.

The currents resulting from different numbers of open channels in a cluster for a high filling state of the ER are shown in Fig. 3. They remain essentially constant after an initial relaxation on timescales from a few milliseconds to a few tens of milliseconds for all numbers of open channels. The current increases by about a factor of 10 while going from 1 to 21.77 open channels.

View larger version (10K): [in this window] [in a new window]   FIGURE 3  Dependence of currents on time. The lines show results for 1, 2, 4, 11.11, and 21.77 open channels from lower to higher values. For parameter values, see Set 1 in Table 1. Release through a single channel was simulated for 0.115 s only.

View larger version (14K): [in this window] [in a new window]   FIGURE 4  Dependence of current on cluster radius at t = 0.1 s. () Set 1 except hER = 0.06 µm, DE = 23 µm2 s–1, DEm = 3.1 µm2 s–1. () Set 1, except BEs = 100 mM, BEm = 0, (X) Set 1 except E = 336 µM, total lumenal content 5.24 mM, BEs = 100 mM, BEm = 0. () Set 1 except total lumenal content 7.43 mM, hER = 0.06 µm, DE = 110 µm2 s–1, DEm = 16.95 µm2 s–1, BEs = 100 mM, BEm = 0. () Set 1 except E = 127.8 µM, total lumenal content 2.80 mM, BEs = 100 mM, BEm = 0. (*) Set 1 except E = 715 µM, total lumenal content 7.43 mM, BEs = 100 mM, BEm = 0.

View larger version (21K): [in this window] [in a new window]   FIGURE 5  Dependence of concentration of cytosolic Ca2+ on time at different distances r from the channel cluster. Lines show results for 1, 2, 4, 11.11, and 21.77 open channels from lower to higher values. For parameter values, see Set 1 in Table 1. Release through a single channel was simulated for 0.115 s only.

View larger version (20K): [in this window] [in a new window]   FIGURE 7  Dependence of release on the fraction of lumenal mobile buffer. (Top left) Concentration of free cytosolic Ca2+ at the cluster in dependence on the number of open channels. (Top right) Current values. (Bottom left) Concentration of free lumenal Ca2+ at the cluster. (Bottom right) Concentration of free cytosolic Ca2+ in a distance of 4.8 µm from the cluster. (All panels) Values were taken after the channels were open for 0.4 s. In case of a single open channel in the cluster, the values were obtained after 0.115 s. For parameter values, see Set 1 in Table 1 except the concentration of lumenal buffers. They are: BEs = 50 mM, BEm = 50 mM (); BEs = 75 mM, BEm = 25 mM (X); BEs = 87.5 mM, BEm = 12.5 mM (); and BEs = 100 mM, BEm = 0 mM (). The values marked by (+) were obtained with DE = 110 µm2 s–1 and BEs = 100 mM, BEm = 0mM.

View larger version (9K): [in this window] [in a new window]   FIGURE 8  Release with parameters mimicking a tubular ER. (Left panel) Concentration of free Ca2+ at the cluster in dependence on the number of open channels. (X and +) Set 2 in Table 1, lumenal concentration (X), cytosolic concentration (+); ( and ) Set 2 in Table 1 except DE = 63 µm2 s–1, DEm = 8.475 µm2 s–1, lumenal concentration (), cytosolic concentration (). (Right panel) Currents for the same parameters as used in the simulations of the left panel. The symbols are chosen accordingly.

View larger version (9K): [in this window] [in a new window]   FIGURE 10  Release with different initial concentrations of free lumenal Ca2+. (Left panel) Ca2+ concentration at the location of the cluster: E = 715.56 µM (), total concentration of Ca2+ in the ER 67.87 mM; E = 336.59 µM (X), total concentration of Ca2+ in the ER 49.36 mM; and E = 127.80 µM (), total concentration of Ca2+ in the ER 26.87 mM. (Right panel) Currents for the same sets of parameters as in the right panel and for two different numbers of open channels. Matching colors have same concentration of free and total Ca2+ in the ER. (For each color) Smaller current values two open channels, larger values 21.77 open channels. Parameter values not mentioned here are those of Set 1 in Table 1.

View larger version (23K): [in this window] [in a new window]   FIGURE 6  Dependence of concentration of cytosolic mobile buffer with Ca2+ bound on time belonging to the simulations shown in Fig. 5. The lines show results for 1, 2, 4, 11.11, and 21.77 open channels from lower to higher values. For parameter values, see Set 1 in Table 1. Release through a single channel was simulated for 0.115 s only.

Fig. 9 shows the results for different diffusion coefficients of lumenal free Ca²⁺. Decreasing the lumenal diffusion coefficient from the cytosolic value of 223 µm² s^–1 down to 23 µm² s^–1 lowers the current by about a factor of 3, but reducing it to 110 µm² s^–1 decreases the current to 80% of the value with the cytosolic diffusion coefficient only. Hence, if diffusion coefficients in the ER should be one-half of the cytosolic diffusion coefficients only (as discussed as a possibility above), that does not have a major impact on the release currents for not too large a number of open channels in a cluster.

View larger version (10K): [in this window] [in a new window]   FIGURE 9  Dependence of the release current with four open channels at t = 0.1 s on the diffusion coefficient of lumenal free Ca2+ DE. hER = 0.028 µm, concentration of free Ca2+ in the ER 715 µM, total lumenal concentration 67.87 mM, BEm = 0, BEs = 100 mM. For other parameter values see Set 1 of Table 1.

View larger version (12K): [in this window] [in a new window]   FIGURE 11  Dependence of currents at t = 0.1 s on the bulk concentration of free lumenal Ca2+. Lumenal buffer concentrations are BEs = 100 mM, BEm = 0 (solid lines), and BEs = 10 mM, BEm = 0 (dashed lines). Small current values are obtained with one open channel and large values with 36 open channels.

Cytosolic buffers, spread of released Ca²⁺ in the cytosol
In general, cytosolic concentrations of free Ca²⁺ around a cluster with open channels drop by orders of magnitude on the length of 1–2 µm already. That is illustrated by including concentration values at different distances from a cluster in Figs. 2, 5, 7, and 12. Although free Ca²⁺ reaches concentrations of several tens of micromolar at the cluster, concentration increases by a few hundred nanomolar only in a distance of 1 µm (Fig. 12) and by several nanomolar at 2.4 µm (Figs. 5 and 12).

View larger version (22K): [in this window] [in a new window]   FIGURE 12  Ca2+ concentration at different distances from the cluster for different concentrations of mobile buffer. Bm = 0 µM (), Bm = 5 µM (X), Bm = 15 µM (), Bm = 30 µM (), Bm = 40 µM () all = 700 (µMs)–1, Km = 0.247 µM, and Dm = 40 µm2 s–1. The currents at 0.4 s belonging to the numbers of open channels for which data are shown are: 0.041 pA (at 0.115 s), 0.071 pA, 0.119 pA, 0.218 pA, 0.299 pA, and 0.366 pA. For parameter values, see Set 1 in Table 1 except the total concentration of Ca2+ in the ER. It is 49 mM resulting in a resting concentration of free Ca2+ in the ER of 336.7 µM, BEs = 100 mM, and BEm = 0.

The spread of released Ca²⁺ depends very sensitively on buffer concentrations. Hence, we can only try to determine limiting values of concentrations a few micrometers away from the cluster. We will consider the impact of concentration of fast mobile buffer, buffer binding rates for fixed dissociation constant, variation of dissociation constants of fast buffer, and buffer diffusion.

The major effect of increasing buffer binding rates is a stronger localization of the Ca²⁺ profile and larger peak values of the profile of buffer with Ca²⁺ bound. Furthermore, the rate of rise of concentrations in a fixed distance from the cluster strongly depends on buffer rates. Two observations seem remarkable here. First, the increase of diffusional spread while going from (µMs)^–1 to (µMs)^–1 is comparable to the effect of the much larger step from (µMs)^–1 to That agrees with results obtained for the stationary profiles, where major changes occurred in that range of binding constants, too (Falcke, 2003a). Hence, buffers become slow from that point of view for k⁺B smaller than 800 s^–1. Second, facilitated diffusion of Ca²⁺ by mobile buffer is more effective with fast buffers than with slow ones. It increases the concentration compared to the case without mobile buffer beyond a certain distance from the cluster. However, that increase is in the range of a few nanomolar only.

The results for different concentrations of mobile buffer are shown in Fig. 12. Note that the increase in free Ca²⁺ concentration at r = 4.5 µm and t = 0.4 s without any mobile buffer reaches 11 nM only. Rather low concentrations of fast mobile buffer have an impact already. On length scales up to 2.5 µm, the use of 5 µM fast mobile buffer reduces the spread of free Ca²⁺ considerably. However, buffer assists spread of free Ca²⁺ by facilitated diffusion at 5 µM and 15 µM at distances larger than 4 µm. There is essentially no difference between the results for B_m = 30 µM and B_m = 40 µM. The dissociation constant of mobile buffer has a major impact on spatial spread of free Ca²⁺ in areas, where the free Ca²⁺ concentration has approximately the value of the dissociation constant or is smaller than it. Decreasing the dissociation constant reduces the spread of free Ca²⁺.

Varying the diffusion constant of mobile buffer D_m from 20 µm² s^–1 to 70 µm² s^–1 reduces the spread of free cytosolic Ca²⁺ on length scales up to 2.5 µm. The faster diffusion of mobile buffer is, the faster Ca²⁺ bound buffer can be exchanged for Ca²⁺ free buffer and the more Ca²⁺ is buffered close to the cluster. However, increasing D_m had little effect on the spread of free cytosolic Ca²⁺ to a distance of 4.5 µm. That suggests facilitated diffusion to become important on larger length scales only for large D_m.

It is often easier to measure signal mass in experiments than local rise times or concentrations. The release current can be estimated from signal mass measurements, if the fraction of Ca²⁺ binding to mobile buffer is known. The fraction of released Ca²⁺ binding to mobile buffer does essentially not depend on the current but on the buffer concentration. It increases from 0.2 to 0.7 when the buffer concentration is raised from 5 µM to 40 µM.

It was hypothesized by Marchant and Parker (2001) that the built-up of pacemaker Ca²⁺ has a role in wave initiation during long period wave nucleation in Xenopus oocytes. Pacemaker Ca²⁺ results from an incomplete decay of the concentration increase due to the previous puff at the time when another puff occurs. To obtain an estimate of the decay of cytosolic free Ca²⁺ upon closing of the last open channel in a cluster we simulated this event (see Fig. 13, Table 2). Cytosolic concentrations decay on timescales of a few tens of milliseconds at the cluster and a few hundreds of milliseconds in a distance of 2.5 µm away from the cluster (see Table 2).

View larger version (13K): [in this window] [in a new window]   FIGURE 13  Simulation of the decay of cytosolic free Ca2+ c and mobile buffer with Ca2+ bound bm upon termination of release after the cluster was open for 0.4 s. Note the logarithmic scale for the concentrations. Bottom group of lines c, top group of lines bm. (Solid lines) Four channels open, concentration at the cluster; (dashed lines) 36 channels open, concentration at the cluster; and (dotted lines) 36 channels open, concentration 4.8 µm away from the cluster.

The decay of currents within a few seconds raises the question for the timescales of this process. We simulated release for 10 s for two different ER radii and two different concentrations of total Ca²⁺ in the lumen. The experiments that can serve for comparison are superfusion experiments by Marchant and Taylor (1998). The superfusion medium is replaced every 80 ms. To be compatible with these experiments—at least as far as parameter values are known and our modeling framework allows—we set cytosolic concentration values to resting level every 80 ms and collect the amount released in 80-ms bins. The results can be fitted very well to a sum of two exponentials and a constant. The relative amplitudes of the exponentials and timescales are shown in Figs. 14 and 15. Half-times and relative amplitudes are in the range observed experimentally in hepatocytes (Marchant and Taylor, 1998; Dufour et al., 1997). The half-times decrease with increasing number of open channels, increasing cluster density, and with decreasing lumenal content. The same qualitative behavior holds for the fraction of the amplitude of the exponential with the smaller half-time. The ER with lower content can maintain release at numbers of open channels presumably typical for oscillations for 4 s at low cluster density and 2 s for high cluster density (estimated by twice the longer t_1/2). The ER with high content can maintain release 2–3 times longer. Hence, depletion of the ER may occur during Ca²⁺ oscillations with release phases of several seconds for low and intermediate lumenal content.

View larger version (27K): [in this window] [in a new window]   FIGURE 14  Analysis of the temporal decay of release from the ER with high initial lumenal content. Amounts released within 80 ms were sampled in bins. The result was fitted to The symbols of the amplitude curves match the symbols of the corresponding t1/2. X marks the contribution of the constant term. (Left) Radius of the lumenal compartment is 1 µm corresponding to an average cluster distance of 2 µm. (Right) Radius of the lumenal compartment is 2 µm corresponding to an average cluster distance of 4 µm. The upper row shows the t1/2, the lower row the relative amplitude. Initial concentration of free lumenal Ca2+ E = 715.56 µM, initial total concentration of Ca2+ in the ER 67.87 mM, Pp = Pl = 0.

View larger version (25K): [in this window] [in a new window]   FIGURE 15  Analysis of the temporal decay of release from the ER with low initial lumenal content. Amounts released within 80 ms were sampled in bins. The result was fitted to The symbols of the amplitude curves match the symbols of the corresponding t1/2. X marks the contribution of the constant term. (Left) Radius of the lumenal compartment is 1 µm corresponding to an average cluster distance of 2 µm. (Right) Radius of the lumenal compartment is 2 µm corresponding to an average cluster distance of 4 µm. The upper row shows the t1/2, the lower row the relative amplitude. Initial concentration of free lumenal Ca2+ E = 336.6 µM, initial total concentration of Ca2+ in the ER 5.24 mM, Pp = Pl = 0.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS AND PARAMETERS RESULTS DISCUSSION REFERENCES

We find that large gradients build up in the lumen during release, which are necessary to transport Ca²⁺ to the channel. Thus transport properties of the ER were expected to be a major factor in determining the release current. However, release currents change by 30% only within a realistic range of diffusion coefficients of free Ca²⁺ in the ER. Neither were they very sensitive to the fraction of mobile buffer.

The proportionality of the release current to with m 2 is important for interpreting experiments of the IP₃ dependence of release currents. As an example we consider results by Parker et al. (1996b) on the increase of the initial release rate with IP₃ in local measurements. The initial rate raises like 120 (µMs)^–1 [IP₃]/([IP₃] + 600 nM). If we assume that IP₃ needs to bind in a noncooperative way to two subunits for the channel to open, the number of open channels N_O is proportional to and hence the current to Note that our results cannot be directly transferred to global measurements because global release might increase by opening of channels situated not close to an open channel.

Currents in our simulations are in the range from 0.015 pA (E = 127 µM, one open channel) to 0.8 pA (E = 715 µM, 36 open channels). We can compare this to signal mass released in Xenopus oocytes, because signal mass was found to be the most reliable measurement in characterization of puffs (Sun et al., 1998). According to Sun et al. (1998), between 0.004 pC and 0.22 pC are released during puffs (most of them release <0.12 pC). The amount of Ca²⁺ released in 0.4 s in our simulations spans the range from 0.006 pC to 0.32 pC. Hence, it covers the range measured in Xenopus oocytes. We can compare more specifically the signal mass for five open channels and different values of free lumenal concentrations. The number of five open channels is typical for a puff (Sun et al., 1998). That is 0.021 pC at E = 127 µM and 0.055 pC at E = 336 µM. Sun et al. assume that one-half of the released Ca²⁺ binds to the dye. However, we find that 70% bind to the dye at a dye concentration of 40 µM. Hence, most of the puffs reported in Sun et al. (1998) are actually below 0.0857 pC. If we further assume puffs with a large signal mass to last 600 ms instead of 400 ms (see Sun et al., 1998), we need to multiply the simulated values with a factor 1.5. Finally, we find that the range of signal mass in our simulations agrees with the range found for puffs in Xenopus oocytes, if the concentration of free lumenal Ca²⁺ is between 127 µM and 336 µM. That is in agreement with recent measurements of free lumenal Ca²⁺ in Xenopus oocytes (Falcke et al., 2003).

The increase of cytosolic Ca²⁺ in a distance of 4.5 µm from an open channel after 0.4 s of release is rather small. It stays below 15 nM in the examples we have presented. If the resting concentration is 100 nM (40 nM) and three Ca²⁺ ions need to bind for the channel to open, then a rise of 15 nM increases the opening probability per unit time by a factor of 1.52 (2.6). According to recent measurements, a resting level of 100 nM appears more realistic in Xenopus oocyte (Falcke et al., 2003). The rather small increase in the open probability caused by releasing clusters in the vicinity of a closed cluster coincides with the finding that a single puff cannot initiate a wave (Marchant and Parker, 2001). If there are three releasing clusters in the vicinity of a closed cluster, the open probability already increases by a factor of 3.05 (resting level 100 nM).

The timescale of the decay of concentration profiles upon the termination of release can be set in relation to puff frequencies just below the wave initiation threshold (Marchant et al., 1999). Marchant et al. measured the distribution of time intervals between puffs. The distribution has a peak for 1.5–2 s and falls to half the peak value in the bin 0–0.5 s (Marchant et al., 1999). Hence, an impact of local pacemaker Ca²⁺ on puff behavior would require that the IP₃Rs are able to sense concentration differences in the nM range and that the puff frequency is 3 s^–1. These considerations apply, if diffusion dissipates release profiles completely. Pacemaker Ca²⁺ can of course still occur as global concentration increase in which case its dynamics is determined by other processes than diffusion.

Buffers had essentially no effect on the peak concentration values of free cytosolic Ca²⁺ at the channel cluster in our simulations. This is important in assessing experiments using buffers to eliminate feedback of Ca²⁺ on the channel dynamics. Typically, high concentrations of EGTA are used. The full width at half-maximum is 10.7 nm only with 2 mM buffer concentration in our simulations. However, the half-maximum is still a concentration of a few tens of micromolar of free Ca²⁺ typically, which is capable of exerting a feedback on channel behavior.

Most simplified models up to date couple regulation of the channel to the average bulk concentration. However, the concentration experienced by the regulatory binding sites of the IP₃R is 2–3 orders of magnitude larger than the average bulk concentration. Most models of channel dynamics assume dissociation constants for binding of Ca²⁺ to the activating binding site in the submicromolar range. Furthermore, lipid bilayer experiments using cations different from Ca²⁺ as the ion conducted by the IP₃R—hence allowing for separate control of the Ca²⁺ concentration on the cis side—indicate that these dissociation constants are realistic indeed (Ramos-Franco et al., 1998; Mak et al., 1998, 1999, 2001). Additionally, if channels are able to sense an open cluster 2–5 µm away or pacemaker Ca²⁺, they need to be able to sense concentration changes in the nanomolar range. These findings and considerations entail a very high opening probability for all IP₃-bound channels of a cluster as soon as the first channel of a cluster opens, because the dissociation constant of the activating Ca²⁺ binding site is much smaller than the concentrations at the cluster location. Together with the large spatial gradients occurring around an open cluster, that will have consequences for the existence and stability of asymptotic solutions of deterministic cluster models. We will discuss that topic in an upcoming report.

The timescales and the amplitudes of the exponential functions describing the decay of currents in simulations over 10 s (Figs. 14 and 15) are in the range of but not equal to experimentally determined values (Marchant and Taylor, 1998; Dufour et al., 1997). These values are obtained in experiments designed to evaluate timescales of channel dynamics. The smallest t_1/2 we find are larger than the experimental values when a high lumenal content is used in the simulation but smaller when we apply a low lumenal content (Marchant and Taylor, 1998). An analysis based on better knowledge of lumenal parameters would be necessary to assess the meaning of these findings for experimental results. However, conclusions on channel state dynamics cannot be drawn from the mere existence of more than one timescale in current decay.

The geometry we have chosen for our simulations applies best to cisternae of the ER because it allows for diffusion into half-space only. The radius of tubes is assumed to be 30 nm (Alberts et al., 1994). The gradient of profiles created by small currents (<0.1 pA) decreases by 80% on this length scale and clusters creating this current have a radius much smaller than the tube radius. Hence, for small clusters, the tube will essentially act like the membrane of a cisternae. The Ca²⁺ gradient created by currents in the range of 0.3 pA decreases by 50% on the first 30 nm. Hence, diffusion "around" the tube will have an impact. However, the reduced peak value of cytosolic Ca²⁺ due to diffusion increases the difference E – c in Eq. 1, which increases the current and hence counteracts the decrease of cytosolic Ca²⁺. Therefore, we assume that our simulations are a good approximation for Ca²⁺ concentration profiles close to a cluster on a tube of the ER as well. The results, which will depend on whether cisternae or tubular ER is modeled, are the timescales of the decay of currents.

In summary, our results provide information on the relation of currents to the number of open channels and current values to the concentration of free lumenal Ca²⁺, which assist in the analysis of experimental results. We provide ranges of values for concentrations and concentration gradients on which modeling can be based.

Submitted on July 8, 2003; accepted for publication December 9, 2003.

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