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The Ca²⁺ Dynamics of Isolated Mouse ß-Cells and Islets: Implications for Mathematical Models

Min Zhang ^*, Paula Goforth ^*, Richard Bertram , Arthur Sherman  and Leslie Satin ^*

^* Department of Pharmacology and Toxicology, Medical College of Virginia Campus, Virginia Commonwealth University, Richmond, Virginia; Department of Mathematics and Kasha Laboratory of Biophysics, Florida State University, Tallahassee, Florida; and Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland

Correspondence: Address reprint requests to Dr. L. S. Satin, Dept. of Pharmacology and Toxicology, Medical College of Virginia Campus, Virginia Commonwealth University, PO Box 980524, Richmond, VA 23298 USA. Tel.: 804-828-7823; E-mail: lsatin{at}hsc.vcu.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
[Ca²⁺]_i and electrical activity were compared in isolated ß-cells and islets using standard techniques. In islets, raising glucose caused a decrease in [Ca²⁺]_i followed by a plateau and then fast (2-3 min^-1), slow (0.2–0.8 min^-1), or a mixture of fast and slow [Ca²⁺]_i oscillations. In ß-cells, glucose transiently decreased and then increased [Ca²⁺]_i, but no islet-like oscillations occurred. Simultaneous recordings of [Ca²⁺]_i and electrical activity suggested that differences in [Ca²⁺]_i signaling are due to differences in islet versus ß-cell electrical activity. Whereas islets exhibited bursts of spikes on medium/slow plateaus, isolated ß-cells were depolarized and exhibited spiking, fast-bursting, or spikeless plateaus. These electrical patterns in turn produced distinct [Ca²⁺]_i patterns. Thus, although isolated ß-cells display several key features of islets, their oscillations were faster and more irregular. ß-cells could display islet-like [Ca²⁺]_i oscillations if their electrical activity was converted to a slower islet-like pattern using dynamic clamp. Islet and ß-cell [Ca²⁺]_i changes followed membrane potential, suggesting that electrical activity is mainly responsible for the [Ca²⁺] dynamics of ß-cells and islets. A recent model consisting of two slow feedback processes and passive endoplasmic reticulum Ca²⁺ release was able to account for islet [Ca²⁺]_i responses to glucose, islet oscillations, and conversion of single cell to islet-like [Ca²⁺]_i oscillations. With minimal parameter variation, the model could also account for the diverse behaviors of isolated ß-cells, suggesting that these behaviors reflect natural cell heterogeneity. These results support our recent model and point to the important role of ß-cell electrical events in controlling [Ca²⁺]_i over diverse time scales in islets.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
Although numerous studies have examined glucose-induced [Ca²⁺]_i signaling in isolated mouse or rat ß-cells (Gylfe, 1988; Wang and McDaniel, 1990; Herchuelz et al., 1991; Yada et al., 1992; Hellman et al., 1992; Dryselius et al., 1994; Larsson et al., 1996; Smith et al., 1997; Jonkers et al., 1999; Salgado et al., 2000) and islets (Jonkers et al., 1999; Valdeolmillos et al., 1989; Santos et al., 1991; Valdeolmillos et al., 1993; Gilon et al., 1994; Worley et al., 1994; Bergsten et al., 1994; Gilon and Henquin, 1995; Bergsten, 1995; Martin et al., 1995; Liu et al., 1998; Henquin et al., 1998; Barbosa et al., 1998), few have systematically compared [Ca²⁺]_i signaling in single ß-cells and intact islets under the same experimental conditions and in the same laboratory (Jonkers et al., 1999). In addition, most of what is currently known about [Ca²⁺]_i signaling in single ß-cells comes from studies of ob/ob mouse islets, which likely differ from those of normal mice. Furthermore, although the role of islet and ß-cell electrical activity in glucose-dependent [Ca²⁺]_i signaling and insulin secretion is relatively well established (Henquin et al., 1998; Satin, 2000; Ashcroft and Rorsman, 1989; Henquin, 1987), only a few studies have used simultaneous measurements of [Ca²⁺]_i and electrical activity to determine the specific role of ß-cell electrical events in the magnitude and timing of glucose-dependent [Ca²⁺]_i oscillations (Dryselius et al., 1994; Santos et al., 1991; Worley et al., 1994; Leech et al., 1994; Rorsman et al., 1992). Understanding detailed aspects of the [Ca²⁺]_i signaling of single mouse ß-cells and islets is important, since these characteristics can be used as constraints on existing as well as future theoretical models of islet bursting and [Ca²⁺]_i oscillations (e.g., Sherman, 1996; Satin and Smolen, 1994; Bertram et al., 2000; Goforth et al., 2002; Göpel et al., 1999; Kuznetsov et al., 2002).

In the present study, we compared [Ca²⁺]_i and electrical signaling in intact mouse islets and isolated ß-cells. Our aim was to test whether glucose-induced changes in [Ca²⁺]_i are primarily determined by the electrical firing patterns of ß-cells and islets, and to compare the experimental results with the predictions of a new model of islet bursting and [Ca²⁺]_i (Goforth et al., 2002; Bertram et al., 2000). In addition, as we recently showed that dispersed ß-cells exposed to 11.1 mM glucose exhibit heterogeneous electrical firing patterns that can be subdivided into three classes (Classes I–III) (Kinard et al., 1999), we examined whether these distinct electrical patterns in turn produce three corresponding patterns of [Ca²⁺]_i. If so, these different [Ca²⁺]_i signaling patterns could differentially support insulin secretion, since ß-cells release insulin in response to elevated [Ca²⁺]_i (Henquin et al., 1998; Satin, 2000; Henquin, 1987; Ämmälä et al., 1993).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
Culture of islets and islet ß-cells
Mouse islets were isolated from the pancreases of Swiss-Webster mice by collagenase digestion, as previously described (Khan et al., 2001; Hopkins et al., 1991). Islets were dispersed into single cells by shaking them in a low-calcium medium (Lernmark, 1974). Islets or ß-cells were placed on glass coverslips in 35-mm petri plates and cultured in RPMI-1640 medium with fetal bovine serum, L-glutamine, and penicillin/streptomycin (Gibco, Grand Island, NY). All cultures were kept at 37°C in an air/CO² incubator. Cells were fed every 2–3 days, and were kept in vitro for up to five days. In contrast, islets were cultured for 1–2 days. We did not see any systematic differences in the Ca²⁺ signaling of isolated ß-cells with time in culture, in contrast to previous reports (Gilon et al., 1994; Bergsten, 1995).

[Ca²⁺]_i measurements
Cultured islets or single ß-cells were loaded with the Ca²⁺-sensitive dye, fura-2/AM (Molecular Probes, Eugene, OR). 2 µmol/l fura-2-AM and 1 µl of 2.5% pluronic acid were added to cells in 35-mm culture dishes containing 1 ml of medium, and cells were incubated for 30 min at 37°C in the incubator. After loading, cells were washed once and then incubated in standard external solution for 20 min. [Ca²⁺]_i was measured by placing cells or islets in a small recording chamber mounted on the stage of an Olympus IX-50 inverted epifluorescence microscope (Olympus, Tokyo, Japan). Fura-2 was excited at 340/380 nm using a galvanometer-driven mirror that alternated light originating from a xenon light source between two prisms (HyperSwitch, IonOptix, Milton, MA). A photomultiplier and photon counting were used to quantify fura-2 emission at 510 nm (IonOptix). Fluorescence data were acquired and analyzed using IonWizard software (IonOptix).

[Ca²⁺]_i values were determined from the fluorescence ratio (R) of Ca²⁺-bound fura-2 (excited at 340 nm) to unbound fura-2 (excited at 380 nM). An in vitro calibration kit was used to convert R values to [Ca²⁺]_i levels in single ß-cell recordings (Molecular Probes). However, because of islet autofluorescence, the [Ca²⁺]_i levels of individual islets were determined using a standard equation (Grynkiewicz et al., 1985). To convert R to [Ca²⁺]_i using this equation, R_max and R_min were obtained by exposing islets to 10 µM ionomycin plus 3 mM Ca²⁺ or 1 mM EGTA plus 0 mM Ca, respectively, at the end of each experiment. The equilibrium constant for Ca²⁺ binding to fura-2 (K_d) was assumed to be 224 nM (Grynkiewicz et al., 1985). Due to uncertainties inherent in converting fura-2 ratios to [Ca²⁺]_i concentrations, the latter should be considered as estimates of [Ca²⁺]_i.

Electrophysiology
Cells or islets were superfused with a standard external solution containing (in mM): 115 NaCl, 3 CaCl₂, 5 KCl, 2 MgCl₂, 10 HEPES, 11.1 glucose (pH 7.2). The perforated patch-clamp technique was used to record islet or cell membrane potentials or ion currents. Pipettes were pulled from borosilicate glass using a two-stage horizontal puller (Sutter Instruments' P-97, Novato, CA). Pipette tips were then filled with a solution containing (in mM) 28.4 K₂SO₄, 63.7 KCl, 11.8 NaCl, 1 MgCl₂, 20.8 HEPES, 0.5 EGTA (pH 7.2), and 0.1 mg/ml amphotericin B. An Axopatch 200B patch-clamp amplifier (Axon Instruments, Union City, CA) was used in the standard tight-seal perforated patch-clamp mode to record membrane potentials or ion currents in current-clamp or voltage-clamp mode, respectively (Hamill et al., 1981). Pipette resistances ranged from 5 to 8 M using our internal solutions, whereas seal resistances ranged from 2 to 10 G.

Dynamic clamping was used to titrate artificial conductances into single cells to determine their effects on membrane electrical activity and [Ca²⁺]_i (Kinard et al., 1999; O'Neil et al., 1995). To implement dynamic clamp, membrane potential (V) was rapidly sampled via a 12-bit A/D-D/A board (Digidata 1200; Axon Instruments) in current clamp and then scaled appropriately. Artificial currents based on measured V were then calculated in software (DClamp; Dyna-Quest Technologies, Sudbury, MA) and scaled, and were sent out in real time to a patch-clamped cell. Membrane potential was then rapidly resampled and the process continued. See Eqs. 11– 13 below.

Solutions were applied to cells or islets using a gravity-driven perfusion system that allowed switching between multiple reservoirs and flow rates in excess of 1 ml/min. The experimental bath was exchanged within 30 s. All experiments were carried out at 32–35°C using a feedback controlled temperature regulation system (Cell Micro Controls, Virginia Beach, VA). All drugs were made up fresh daily from frozen stock solutions. Drugs and chemicals were obtained from Sigma Chemical (St. Louis, MO).

Data analysis and graphics were implemented using IGOR Pro software (WaveMetrics, Lake Oswego, OR) and statistics were performed using GraphPad Pro (GraphPad, San Diego, CA).

Classification of single ß-cell electrical patterns
The heterogeneous electrical activity of single ß-cells was sorted into three broad categories consisting of plateau cells, bursters, and spikers, based on characteristics such as spike amplitude, period, and the existence of plateau depolarizations. As we have previously found (Kinard et al., 1999), cell patterns could be sorted based on "activity fraction", the fraction of time the cell spent above a threshold level. Cell behaviors and thresholds in particular were very variable, however, so we used a related measure, "delta peak", determined as follows: all point histograms of voltage traces were constructed and fit by nonlinear least squares to a sum of two Gaussians (see Fig. 4 of Kinard et al., 1999, for examples). The delta peak was defined as the difference of the amplitudes of the two Gaussians, normalized by the larger peak: p = 100(L_a - R_a)/max(L_a, R_a), where L_a and R_a are the heights of the left and right peaks, reflecting the total duration of repolarized and depolarized periods, respectively. This procedure yielded scores varying from -100 to +100.

Modeling
Islet [Ca²⁺]_i oscillations were simulated using standard techniques and the recently published "calcium subspace" model (Goforth et al., 2002). This model is a member of the class of models we call "phantom bursters", where the interaction between two slow variables (S₁ and S₂) produces intermediate (and hence "phantom") kinetic behavior (Bertram et al., 2000). Electrical bursting in the subspace model results from the cyclic activation of a novel Ca²⁺-activated K⁺ current called K_slow (Göpel et al., 1999). As described in Goforth et al. (2002), K_slow is postulated to be activated by Ca²⁺ accumulation in a submembrane space (subspace) located between the endoplasmic reticulum (ER) and the plasma membrane. We assume that the submembrane portion of the ER and the plasma membrane are in close enough proximity to hinder the diffusion of Ca²⁺ between the subspace and the bulk cytosol and create a standing gradient of Ca²⁺ concentration. Although such a partially membrane-delimited compartment has not been identified in ß-cells, similar structures have been described in cardiac myocytes (Blaustein and Golovina, 2001) and in neurons (Bootman et al., 2001). The existence of a subspace thus remains a prediction of the model for the time being.

The faster phantom variable S₁ in this case is cytosolic [Ca²⁺], denoted c in the model equations, which mediates rapid changes in subspace [Ca²⁺], denoted c_SS, whereas the slower variable S₂ represents the [Ca²⁺] in the ER, denoted c_ER, which slowly accumulates and is passively released into the subspace. This model can reproduce the experimentally observed biphasic modulation of K_slow under voltage-clamp conditions by thapsigargin or insulin, which decrease ER Ca²⁺ pumping, as well as islet bursting in current clamp (Goforth et al., 2002). Although the specific parameters of the Ca²⁺ subspace model were chosen to highlight certain features of the observed data, the results obtained are general features of the model over a range of parameter values.

As the model is only slightly modified from that in Fig. 8 of Goforth et al. (2002), we describe here only changes in the model. The equation for membrane potential is

(1)

The three Ca²⁺ compartments are described by linear flux exchange equations:

(2)

(3)

(4)

The quantity that corresponds most closely to experimentally measured [Ca²⁺]_i is a weighted average of c and c_SS:

(5)

where V_SS is 40% of V_CYT.

To increase c_ER to the experimental range of 200–500 µM (Varadi and Rutter, 2002; Maechler et al., 1999; Tengholm et al., 2001), we have reduced the ER leak rate p_ER to 0.001 ms^-1 and increased the ER uptake rate k_SERCA to 0.2 ms^-1. The plasma membrane pump rate k_PMCA was adjusted to 0.2 ms^-1 to keep plateau fraction near 50%. In all other respects, the Ca²⁺ handling is identical to that in Fig. 8, Goforth et al. (2002).

View larger version (17K): [in this window] [in a new window]   FIGURE 8  Stochastic simulation of single-cell behavior (Eqs. 1–9). Parameters as in Fig. 2 and Modeling except gKATP, which is 42 pS, and gCa, which is 1500 pS for Class I firing (A), 1400 pS for Class II (B), and 1050 pS for Class III (C).

(6)

One consequence of this assumption is that inhibition of the SERCA pump eliminates medium and slow bursting in favor of fast spiking because it eliminates the only very slow negative feedback mechanism, K_slow. This is in agreement with experimental observations of Worley et al. (1994), Bertram et al. (1995), and Gilon et al. (1999), but does not account for the observations of Liu et al. (1998) and Miura et al. (1997) that slow oscillations can persist in thapsigargin. If g_KATP is treated as a slow negative feedback process (Magnus and Keizer, 1998; Miwa and Imai, 1999; Rolland et al., 2002) so that there are two S₂ variables, both sets of observations can be accommodated (Bertram and Sherman, unpublished simulations). Here, we have used the simpler model with very slow negative feedback only through K_slow, which is adequate to interpret the experimental data presented in this study.

View larger version (20K): [in this window] [in a new window]   FIGURE 2  Deterministic simulation of first phase transient and oscillations after addition of glucose (Eqs. 1–6, Methods). (A–C) A medium cell, representing a medium islet. Parameters are as in Modeling section, except gKATP, which is initially 1 nS and stepped down to 55 pS at T = 50 s to simulate the addition of glucose. For simplicity, no attempt is made to model the latency period or the phase 0 decrease of calcium. (D–F) A slow cell, representing a slow islet. Parameters as in A–C, except gKATP is stepped to 62 pS.

View larger version (20K): [in this window] [in a new window]   FIGURE 5  Deterministic simulation of steady-state bursting in the presence of glucose showing the three Ca2+ compartments (Eqs. 1–6, Methods). Parameters are as in Fig. 2 except gKATP, which is 60 pS throughout, and kPMCA, which is 0.18 ms-1. (A) Membrane potential. (B) Weighted average of [Ca2+]i and [Ca2+]SS, corresponding to measured cytosolic [Ca2+], which declines throughout the active phase. (C) Subspace [Ca2+], which rises throughout active phase and activates Kslow to terminate the burst. (D) Store [Ca2+], which rises slowly in active phase and falls slowly in silent phase.

(7)

where s satisfies the stochastic differential equation (Higham, 2001):

(8)

The intensity of the noise, , decreases as the square root of the number of ion channels (Fox, 1997):

(9)

We chose N_KATP = 500 and = 100 ms, values that produce reasonable visual agreement with experimental records. For simplicity, we present simulations in which noise was added only to the K_ATP channels; similar results were obtained by applying a similar approach to other channels. In the deterministic limit (large N_KATP), s approaches its mean value s₀ = /( + ß). For comparison with deterministic simulations, the mean value of g_KATP, g_KATP-max s₀, is reported in the figure legends.

View larger version (27K): [in this window] [in a new window]   FIGURE 10  Stochastic simulation of dynamic-clamp conversion (Eqs. 1–9). Deterministic parameters as in Fig. 5, except kPMCA = 0.2 ms-1. Stochastic parameters and dynamic-clamp parameters are as in Methods. Dynamic-clamp conductance is turned on at T = 20 s and maintained thereafter.

(10)

where W_n is a Gaussian random variable with mean 0 and variance t.

Deterministic simulations were carried out using the Gear-type algorithm CVODE. Source files for use with XPP or reimplementation of the simulations with other software are available at http://mrb.niddk.nih.gov/sherman.

Dynamic clamping
Dynamic-clamp current was applied using the same formulation as previously (Bertram et al., 2000):

(11)

(12)

(13)

We used K = 2 ms^-1 and G_max as indicated in the figure legends. In the experiments, an auxiliary computer read the instantaneous value of V from the cell and calculated the corresponding values of z and I_Clamp using Eqs. 12 and 13, and I_Clamp was injected into the cell in real time. In the simulations, I_Clamp was added to the right side of Eq. 1.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
Glucose induces transient changes in islet [Ca²⁺]_i
In glucose-free solution, islet [Ca²⁺]_i was 72.3 ± 5.5 nM (n = 19). In 63% of the islets examined, increasing glucose to 10 mM produced an initial decrease in [Ca²⁺]_i to 66.5 ± 5.1 nM. The mean latency of this phase was 145 ± 36 s (n = 12; Fig. 1, A and B). A similar transient reduction in [Ca²⁺]_i has been reported for mouse (Valdeolmillos et al., 1989, 1993; Gilon et al., 1994; Worley et al., 1994; Bergsten et al., 1994; Bergsten, 1995; Liu et al., 1998; Henquin et al., 1998; Barbosa et al., 1998), rat (Antunes et al., 2000), pig (Bertuzzi et al., 1996), and human islets (Martin and Soria, 1996), and has been attributed to increased Ca²⁺ sequestration into the endoplasmic reticulum as the ratio of ATP/ADP increases (Yada et al., 1992; Hellman and Gylfe, 1984; Roe et al., 1994).

View larger version (40K): [in this window] [in a new window]   FIGURE 1  [Ca2+]i responses of individual mouse islets to changing glucose from 0 to 10 mM. (A) Representative islet exhibiting mainly fast phase 2 oscillations. (B) Islet showing a mixture of fast and slow [Ca2+]i oscillations. (C) Islet showing primarily slow [Ca2+]i oscillations. Initial islet [Ca2+]i changes often included initial decreases, a plateau phase, and then oscillations (see text). All time bars represent 2 min.

View larger version (27K): [in this window] [in a new window]   FIGURE 3  Slow [Ca2+]i oscillations with prominent interburst decaying phases. (Top) Islet where calcium slowly rose for the entire duration of the active phase, and slowly decayed during the interburst phase. (Bottom) Another islet where faster oscillations occurred on the slower oscillations as "bursts of bursts" (see text). All time bars represent 2 min.

View larger version (25K): [in this window] [in a new window]   FIGURE 4  Simultaneous recordings of [Ca2+]i oscillations (top) and electrical activity (bottom) in two slow islets. (A) Islet where faster electrical bursts occurred on slower bursts and triggered concomitant fast [Ca2+]i oscillations. (B) An islet having very slow electrical activity and the [Ca2+]i oscillations, indicating that electrical activity can account for the slow as well as medium [Ca2+]i oscillations.

One consistent feature of the data, which is apparent from Fig. 1, is that the kinetics of the [Ca²⁺]_i oscillations are independent of the duration of the initial [Ca²⁺]_i plateau seen when glucose is raised. We further observed that the duration of the initial transient was always greater than the duration of succeeding plateaus, with the latter approaching the duration of the transients in the slowest cases. Based on past experience with the phantom-bursting model (Bertram et al., 2000), we hypothesized that the kinetics of both the transients and the slow oscillations reflect the timescale of the slower negative feedback variable, S₂, (here, c_ER, Fig. 2, C and F; see Methods and Goforth et al., 2002) and thus would always be on the order of a few minutes. The succeeding oscillations, in contrast, would represent the "phantom" interactions between S₂ and S₁ (here, c, which is not shown but is reflected in the fast component of c_avg), and could thus occur on a timescale anywhere between those of S₁ and S₂, or from minutes to seconds. To test this hypothesis, we simulated the [Ca²⁺] and electrical responses of islets to a step change in glucose concentration. As shown in Fig. 2, the model successfully reproduces an initial plateau in c_avg after glucose exposure, as well as the electrical and [Ca²⁺]_i oscillations that follow. In Fig. 2, A–C, oscillations of membrane potential, free cytosolic [Ca²⁺] (c_avg), and ER [Ca²⁺] (c_ER) are shown. The oscillations occur on the "medium" timescale (2 min^-1) because of the relatively large contribution from S₁. In Fig. 2, D-F, "slow" oscillations (0.7 min^-1) are seen because the oscillation frequency is dominated by the timescale of c_ER. In both cases, the slow loading of the ER due to Ca influx determines the initial c_avg plateau, which lasts 2 min.

Thus, by adjusting parameters, medium or slow oscillations could be produced in two model islets having identical plateau phases, in agreement with our experimental observation that the period of the [Ca²⁺]_i oscillations is not correlated to the duration of the initial plateau. In addition, the model predicted that in slower islets, [Ca²⁺]_i would decay more slowly after each period of Ca²⁺ influx. A representative example of the slow tails during steady-state oscillations is shown in Fig. 3 (top). In the model, this slow decay is due to the slow, passive release of Ca²⁺ from the ER after each burst of action potential-mediated Ca²⁺ influx, as anticipated by Gilon et al. (1999), who noted that the slow tails have kinetics similar to the dissipation phase of [Ca²⁺]_ER. The model also suggests that, other things being equal, the slow tails will be more prominent in cases where [Ca²⁺]_i rises throughout the active phase of the oscillation, i.e., when the filling of the ER dominates over the slowing of Ca²⁺ entry (compare Figs. 3, top and 4 A with Fig. 4 B). Such slow tails of [Ca²⁺]_i were a conspicuous feature not only of islet [Ca²⁺]_i recordings (Figs. 3 and 4), but of single cells when artificially induced to oscillate on timescales comparable to those of islets (Fig. 9).

View larger version (29K): [in this window] [in a new window]   FIGURE 9  Converting fast single-cell electrical activity to an isletlike pattern with dynamic clamp resulted in corresponding [Ca2+]i oscillations that resembled those of islets. Starting at the arrow, 0.008 nS of maintained exogenous conductance were added in accordance with Eqs. 11–13. Calcium changes are shown at the top; membrane potential changes at the bottom.

Fast and slow [Ca²⁺]_i oscillations are due to fast and slow forms of islet electrical activity
Using the perforated patch technique to record from ß-cells on the surface of intact islets (Göpel et al., 1999), and a fast photomultiplier and fura-2 fluorescence to monitor simultaneous changes in islet [Ca²⁺]_i, we determined the extent to which the various patterns of [Ca²⁺]_i oscillations we observed were determined by the electrical activity of individual islets. We were especially interested in determining whether islets displaying slow [Ca²⁺]_i oscillations (0.2 min^-1 or less) had correspondingly slow electrical activity, as a variety of nonelectrical mechanisms have been proposed to account for the slow [Ca²⁺]_i oscillations in islets and single ß-cells (Hellman et al., 1992; Dryselius et al., 1994; Liu et al., 1998; Leech et al., 1994; Gylfe et al., 1991; Tornheim, 1997; Porterfield et al., 2000; Jung et al., 2000).

Fig. 4 shows data obtained from two representative islets where [Ca²⁺]_i and membrane potential were recorded simultaneously. As expected from previous recordings of mouse islet electrical activity, islet silent phase potential was between -55 and -65 mV, plateau potential was near -40 mV, and the peaks of the voltage spikes approached -20 mV (Henquin, 1987). Summary results for the islets studied are given in Table 2; comparison of islets with single cells will be discussed below. There was a high degree of synchrony between [Ca²⁺]_i and the membrane potential, with changes in membrane potential leading the changes in [Ca²⁺]_i (cf. Valdeolmillos et al., 1989; Santos et al., 1991; Barbosa et al., 1998). The islet shown in Fig. 4 A had 50-s duration slow waves that occurred at a frequency of 0.8 min^-1. The bursts of bursts of electrical activity mentioned above are apparent here, with the faster electrical events driving the faster [Ca²⁺]_i oscillations within the slower plateaus. The close synchrony between [Ca²⁺]_i and membrane potential is even more impressive in this case, considering that the electrical activity was recorded in a superficial ß-cell whereas the fura-2 signal represented the average fluorescence of many ß-cells within the islet syncytium. This suggests that the extensive gap junctional coupling known to exist within the islet is very effective in synchronizing the activity of the islet cells (Mears et al., 1995; Serre-Beinier et al., 2000). In contrast, the islet shown in Fig. 4 B displayed much slower [Ca²⁺]_i oscillations, which occurred at 0.2 min^-1. These slow [Ca²⁺]_i oscillations were also highly synchronized with islet bursting electrical activity (Valdeolmillos et al., 1989). This suggests that electrical activity is likely to be a critical mediator of all the islet [Ca²⁺]_i oscillations we observe, fast, medium, and slow (Dryselius et al., 1994; Valdeolmillos et al., 1989). Consistent with this, the application of Ca²⁺-free saline, diazoxide, or the L-type Ca²⁺ channel blocker nimodipine abolished all the [Ca²⁺]_i oscillations of islets (data not shown).

Glucose induces multiple transient changes in [Ca²⁺]_i in single mouse ß-cells as in islets, but single ß-cells typically lack robust slow [Ca²⁺]_i oscillations
As we found with intact islets, exposing single mouse ß-cells to 10 mM glucose resulted in three phases of activity, although there were both qualitative and quantitative differences between the [Ca²⁺]_i changes of isolated ß-cells and islets. The [Ca²⁺]_i properties of single ß-cells are summarized in Table 1. In glucose-free saline, single ß-cells had a mean basal [Ca²⁺]_i of 65.2 ± 2.7 nM (n = 37), as in islets (p > 0.05), and was similar to previous reports (Gylfe, 1988; Wang and McDaniel, 1990; Yada et al., 1992; Gilon et al., 1994; Krippeit-Drews et al., 2000). Upon exposure to 10 mM glucose, [Ca²⁺]_i decreased to 54.1 ± 3.5 nM (in 23/37 cells, or 62.2%), which was not significantly different from that of islets (Fig. 4; Table 1). This suggests that ER Ca²⁺ sequestration is likely to be similar in single ß-cells and islets. Decreases in [Ca²⁺]_i levels upon glucose stimulation have been reported previously in single rodent ß-cells by others (Gylfe, 1988; Wang and McDaniel, 1990; Yada et al., 1992).

After the initial decrease, [Ca²⁺]_i then increased to reach a plateau of 437.5 ± 20.1 nM (n = 37). The mean latency measured from the time of glucose addition was 301.2 ± 42.1 s. As with islets, the [Ca²⁺]_i plateau usually decreased over several minutes time. Only the peak Phase I [Ca²⁺]_i (437.5 ± 20.1 nM) and silent phase [Ca²⁺]_i (176.1 ± 12.3 nM) differed significantly between islets and single cells, suggesting that there is no gross difference in the Ca²⁺ handling capabilities of single cells and islets. Despite this, there were striking differences in oscillatory behavior. Whereas single ß-cells on occasion exhibited one or more [Ca²⁺]_i transients after the plateau phase (Fig. 6 A), it was more often the case that irregular [Ca²⁺]_i changes were seen (Fig. 6 C). The paucity of slow [Ca²⁺]_i oscillations in the isolated ß-cells stands in sharp contrast to previous reports of others (Gylfe, 1988; Hellman et al., 1992; Larsson et al., 1996; Smith et al., 1997; Jonkers et al., 1999; Leech et al., 1994). Also absent were oscillations corresponding to medium islet bursting.

View larger version (28K): [in this window] [in a new window]   FIGURE 6  [Ca2+]i responses of individual mouse ß-cells to changing glucose from 0 to 10 mM. (A) Representative ß-cell exhibiting mainly fast and stochastic [Ca2+]i changes, including some near oscillations. (B) ß-cell showing a mixture of fast and vestigial slow [Ca2+]i changes. (C) ß-cell showing very irregular [Ca2+]i transients after glucose addition. Initial ß-cell [Ca2+]i changes often included initial decreases, a plateau phase (see text). All time bars represent 2 min.

Given that the [Ca²⁺]_i signaling patterns that were associated with these three electrical patterns might differentially support insulin secretion, we measured the electrical activity of single ß-cells from each class using perforated patch, while simultaneously monitoring [Ca²⁺]_i with fura-2. As in our previous study of single ß-cell electrical activity (Kinard et al., 1999), we used the weighted percent time that cells were depolarized, or delta peak, to differentiate among the three classes (see Methods). Our aim was to determine whether the three electrical classes I–III in turn produced distinct [Ca²⁺]_i patterns.

As shown in Fig. 7, ß-cells of the three classes indeed had characteristic [Ca²⁺]_i patterns in 10 mM glucose. Thus, Class I cells, which were continuously spiking from a threshold of near -40 mV, and had a mean delta peak value of 66.7 ± 10.6, were associated with a sustained [Ca²⁺]_i level that was either nonoscillatory or exhibited very small oscillations that were in phase with the sharp electrical spikes. The peak [Ca²⁺]_i measured in these cells was 328.3 ± 23.2 nM. Interestingly, [Ca²⁺]_i and membrane spiking activity appeared to be most clearly synchronized in Class I ß-cells when spike frequency either abruptly increased or decreased (Fig. 7 A), resulting in either increased or decreased [Ca²⁺]_i. Thus, in these cells the frequency of electrical spiking appeared to be a major determinant of [Ca²⁺]_i.

View larger version (30K): [in this window] [in a new window]   FIGURE 7  The electrical activity and [Ca2+]i changes of single ß-cells exposed to 11.1 mM glucose. (A) Class I continuously spiking ß-cell having sustained [Ca2+]i and small oscillations. (B) Class II ß-cell having fast bursts of electrical activity and corresponding oscillations in [Ca2+]i. (C) Class III plateau cell with close synchrony between calcium and electrical activity and negligible electrical voltage spikes.

Thus, in single ß-cells, individual voltage spikes tended to produce only small increases in [Ca²⁺]_i, whereas trains of spikes maintained [Ca²⁺]_i at a steady elevated level. In contrast, larger and more pulsatile changes in [Ca²⁺]_i were associated with more defined episodes of electrical bursting. The [Ca²⁺]_i oscillations associated with Class II and III type electrical patterns, although more burstlike, were also much faster (i.e., 12–35 min^-1) than even the fastest [Ca²⁺]_i oscillations of intact islets, but were close to the rate of spiking seen during bursts in islets (Fig. 4 A; Bertram et al., 2000; Kinard et al., 1999). Overall, it appeared that electrical activity was an important determinant of [Ca²⁺]_i signaling in single ß-cells as well as in islets. Whereas the highly ordered electrical activity of islets produced very regular [Ca²⁺]_i oscillations, the more stochastic electrical activity of isolated ß-cells correspondingly produced faster and less regular [Ca²⁺]_i oscillations. Given that single cells in our hands are almost exclusively fast, with oscillations lasting <5 s, it is particularly striking that our islets are so slow.

We next tested whether the subspace model could simulate the behavior of isolated ß-cells as well as islets. Beginning with a medium-bursting islet, we reduced the conductance of the KATP channels to make the bursting fast. We also added simulated stochastic channel noise, as noise was very apparent in the single-cell recordings (Sherman et al., 1988). The result was a pattern much like our Class II cells (Fig. 8 B). Note that this fast-bursting pattern is primarily a consequence of the changed conductance; the noise changes the pattern quantitatively, not qualitatively, so as to enhance agreement with the experiments. When in addition to the above changes we increased the conductance of the L-type Ca²⁺ channels, spike amplitude also increased, resulting in a pattern much like our Class I cells (Fig. 8 A). In contrast, decreasing the Ca²⁺ conductance reduced spike amplitude, resulting in a pattern like our Class III cells (Fig. 8 C). In addition to simulating reasonably well the three observed patterns, the features of the fast oscillations also resembled the experimental observations in three respects. Going from Class I to Class II to Class III, oscillation frequency decreases, spike amplitude decreases, activity fraction increases, and delta peak decreases (compare Table 2). The pattern of [Ca²⁺]_i also adjusts much as seen in the experiments, with longer electrical events producing more pronounced fluctuations in [Ca²⁺]_i.

These simulations suggest that the marked differences in behavior between islets and isolated cells may be due to modest variation of cellular parameters, such as conductances. To determine whether single ß-cells were intrinsically capable of producing isletlike [Ca²⁺]_i oscillations, we took advantage of our earlier observation that single ß-cells can be induced to exhibit isletlike electrical bursting if an appropriate artificial conductance is introduced with dynamic clamp (Bertram et al., 2000; Kinard et al., 1999; Sharp et al., 1993).

As shown in Fig. 9 (top), a single ß-cell showing characteristic fast, irregular electrical activity in 10 mM glucose produced correspondingly irregular changes in [Ca²⁺]_i under control conditions. However, the application of 8 pS of an artificial, voltage-dependent Ca²⁺ conductance (Bertram et al., 2000; Kinard et al., 1999) resulted in slow bursts lasting 20–30 s (bottom) and more pronounced, isletlike [Ca²⁺]_i oscillations (compare with Fig. 4 A). Similar results were seen after application of dynamic-clamp conductances to 15/20 isolated ß-cells. In the representative cell shown in Fig. 9 (top), the application of dynamic clamp resulted in a drop in [Ca²⁺]_i from around 400 nM to <200 nM once bursting commenced. We attribute this effect to the very strong influence of the interburst period in lowering [Ca²⁺]_i. Thus, the termination of spiking consistently decreased [Ca²⁺]_i presumably by suppressing Ca²⁺ influx at a more hyperpolarized potential. Results from our dynamic-clamp studies thus show that single ß-cells are capable of producing isletlike changes in [Ca²⁺]_i once islet electrical bursting is enabled. The implication is that isolated ß-cells can produce more regular and more pronounced [Ca²⁺]_i oscillations, but that this does not normally occur in the absence of cell-cell coupling by gap junctions.

Single cells may be fast because of heterogeneity or noise
We have shown that the fast [Ca²⁺]_i patterns of single cells result from the fast electrical pattern, but the question remains: Why is the electrical activity fast? Two main hypotheses have been considered: one, that isolated ß-cells differ in their properties from cells in islets (Smolen et al., 1993), and the other, that isolated cells have the same properties as cells in islets, but are more susceptible to stochastic channel fluctuations, which are averaged out when cells are electrically coupled (Sherman et al., 1988). These are in fact not mutually exclusive possibilities, but let us consider the two hypotheses separately.

One possibility is that there are small, but systematic, quantitative differences in the mixture of ionic conductances of single ß-cells compared to islets. For example, Göpel et al. (1999) proposed that single ß-cells have less g_Ca and hence less K_slow than cells in islets. Alternatively, single cells may possess the right mix of conductances on average when coupled, but not individually when isolated. It is clear that in the former case, the cells can be "corrected" (i.e., made to behave like islets) by using dynamic clamp to augment an inadequate conductance or subtract a conductance present in excess. However, more subtle forms of correction are also possible. We applied the same type of exogenous conductance used in the experiments depicted in Fig. 9 to both our Class I (Fig. 8 A) and Class II (Fig. 8 B) model cells. Both of these cell types were fast because of reduced outward g_KATP, yet adding an inward current readily converted them to isletlike bursters (not shown). Furthermore, this was true despite the fact that the Class I cells also had excess g_Ca whereas the Class II cells had insufficient g_Ca compared to our standard slow islet model cells (Figs. 2 and 5). We conclude that the success of experimental dynamic-clamp conversion in cases such as Fig. 9 is compatible with either systematic parameter deviation or random parameter heterogeneity being the source of fast single ß-cell behavior.

We went on to confirm that we could also produce cells with Class II patterns by the addition of channel noise, even when the cell parameters were appropriate for isletlike (medium) bursting. As shown in Fig. 10, such a cell could also be converted to a regular islet burster having pronounced medium period [Ca²⁺]_i oscillations by the addition of voltage-dependent conductance with the same characteristics (see Methods) and nearly the same magnitude as that used in the experimental case. The response of the model in particular captures two key elements of the experiments. Immediately after initiation of dynamic clamp, the cell goes into continuous firing followed by slower bursting (compare Figs. 8 and 10); this was invariably observed in all successful conversions in both experiments and simulations. Also, the conversion to slower bursting results in the appearance of slow tails in the [Ca²⁺]_avg time course, though not as dramatically as in the experiments. The most significant discrepancy between the experiment and the model is that interburst cytosolic [Ca²⁺] falls in the former but rises in the latter. Overall, the simulation of Fig. 10 supports the hypothesis that experimental dynamic-clamp conversion succeeds because it overcomes channel noise.

We will return to the issue of noise versus heterogeneity in the Discussion, but first make some final comments about Fig. 10. The model suggests that the transient firing period is a miniature version of the first phase transient seen when bursting is initiated by a step increase in glucose (Fig. 2): First the ER fills, then bursting commences when [Ca²⁺]_ER begins to oscillate. Before initiation of dynamic clamp, [Ca²⁺]_ER was noisy but flat. As in our previous simulations, which first predicted dynamic-clamp conversion (see Fig. 8 in Bertram et al., 2000), the effect of the added inward conductance is to prevent S₁ (here, [Ca²⁺]_i) from driving fast bursts and allow S₂ (here, [Ca²⁺]_ER) to vary, initiating the slower rhythm.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
The goal of the present study was to systematically compare [Ca²⁺]_i signaling and electrical activity in mouse islets and isolated ß-cells under similar experimental conditions to help refine and constrain current models of islet dynamics, as well as newer models that we and others are currently developing. Our long-range goal is to construct a model that can reproduce the entire spectrum of behaviors observed in single ß-cells and in islets and to account for the modulation of these behaviors under varying conditions. A hindrance to this objective has been the fact that few studies have appeared in which both single ß-cells and islets from the same preparation have been studied under identical conditions by the same investigators. We began this investigation with the hypothesis that the single ß-cell, which is the basic building block of the islet, intrinsically has most or all of the machinery needed to reproduce islet bursting and its concomitant [Ca²⁺]_i oscillations.

Although both single ß-cells and islets were found to be glucose responsive, there were very clear differences in terms of their [Ca²⁺]_i signaling and their electrical activity. Thus, whereas intact islets and isolated ß-cells showed similar initial decreases in [Ca²⁺]_i followed by plateaus when challenged with glucose, isolated cells typically exhibited faster, less pronounced, and much less regular [Ca²⁺]_i oscillations than islets. This difference reflects the very different electrophysiological properties of islets and isolated ß-cells, since the latter in our hands only spike repetitively or show fast (<5 s) bursting in 11 mM glucose, rather than medium or slow electrical bursting (Kinard et al., 1999). The three electrical patterns were indeed associated with distinct changes in [Ca²⁺]_i, ranging from elevated steady levels (Class I cells), to fast oscillations lasting either a few seconds (Class II cells) or up to 10 s (Class III cells). Thus, despite marked quantitative differences between the stochastic electrical activity of isolated ß-cells, and the more regular and more oscillatory electrical activity of islets, in both preparations we found that [Ca²⁺]_i closely tracked the electrical activity. Most notably, when single cells were induced by exogenous current injection to behave electrically like islets, the [Ca²⁺]_i pattern followed suit (Fig. 9).

We found that single voltage spikes were not very effective in raising [Ca²⁺]_i above baseline, whereas bursts of spikes were. This is because the timescale of [Ca²⁺]_i accumulation is slow compared to the duration of a spike (Rorsman et al., 1992). Continuous spiking nonetheless resulted in a high average level of [Ca²⁺]_i because the interspike potential in single ß-cells was very depolarized, comparable to the plateau potential seen in bursting islets. As oscillations in ß-cells [Ca²⁺]_i are believed to be responsible for triggering oscillations in insulin secretion (Henquin, 1987, 1992; Rosario et al., 1986; Kinard and Satin, 1996), these differences in [Ca²⁺]_i oscillations would likely be reflected in different secretion patterns, though we did not measure secretion in this study.

Why don't single ß-cells exhibit islet-like bursting?
Unlike other workers, we rarely observed either pronounced slow [Ca²⁺]_i or membrane potential oscillations in our single ß-cells (Hellman et al., 1992; Jonkers et al., 1999). A number of explanations have been put forth to explain why single ß-cells do not exhibit the slower bursting that is characteristic of intact islets. Most recently, Göpel and colleagues have proposed that the amplitudes of voltage-gated L-type Ca²⁺ currents, and concomitantly, a novel Ca²⁺-activated K⁺ current called K_slow, are smaller in isolated ß-cells compared to islets, resulting in abnormal bursting (Göpel et al., 1999). However, the K_slow current we observed in isolated ß-cells is in fact comparable in size to islet K_slow (Goforth et al., 2002). Thus, it is unlikely that differences in K_slow amplitude alone are sufficient to explain the different bursting patterns of islets and isolated ß-cells.

We consistently found that single ß-cells were more depolarized than islets, namely near -40 mV, and that their electrical activity resembled the fast spiking seen at the tops of the plateaus of bursting islets (also see Fig. 3, A and B, in Lebrun and Atwater, 1985). Intriguingly, the resting membrane potential of isolated ß-cells is also near -40 mV. This observation led us to hyperpolarize single ß-cells with direct injected current or with excess artificial K_ATP conductance with dynamic clamp; these maneuvers did not result in medium bursting (Kinard et al., 1999). Nonetheless, we cannot rule out the possibility that single ß-cells have insufficient K⁺ conductance or systematically differ in some other current compared to islets. Interestingly, we were able to convert fast bursters and spikers to medium bursters by adding an inward, voltage-dependent conductance (Fig. 10), as suggested by previous mathematical modeling (Bertram et al., 2000). The ability of single cells to be converted in this manner strongly suggests that single ß-cells possess the slow negative feedback process required for isletlik