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Modeling of Ca²⁺ flux in pancreatic -cells: role of the plasma membrane and intracellular stores

Department of Medicine, University of Chicago, Chicago, Illinois 60637

Submitted 6 May 2002 ; accepted in final form 7 March 2003

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: NA+/K+ ACTIVE... APPENDIX B: GENERAL GATING... APPENDIX C: FRACTION OF... REFERENCES

 
We have developed a detailed mathematical model of ionic flux in -cells that includes the most essential channels and pumps in the plasma membrane. This model is coupled to equations describing Ca²⁺, inositol 1,4,5-trisphosphate (IP₃), ATP, and Na⁺ homeostasis, including the uptake and release of Ca²⁺ by the endoplasmic reticulum (ER). In our model, metabolically derived ATP activates inward Ca²⁺ flux by regulation of ATP-sensitive K⁺ channels and depolarization of the plasma membrane. Results from the simulations support the hypothesis that intracellular Na⁺ and Ca²⁺ in the ER can be the main variables driving both fast (2–7 osc/min) and slow intracellular Ca²⁺ concentration oscillations (0.3–0.9 osc/min) and that the effect of IP₃ on Ca²⁺ leak from the ER contributes to the pattern of slow calcium oscillations. Simulations also show that filling the ER Ca²⁺ stores leads to faster electrical bursting and Ca²⁺ oscillations. Specific Ca²⁺ oscillations in isolated -cell lines can also be simulated.

Ca²⁺ oscillations; endoplasmic reticulum; pancreatic islets; theoretical model

There is extensive literature describing -cell electrical activity and its relationship to [Ca²⁺]_i in intact islets of Langerhans, isolated islet cells, and insulinoma cell lines. Most of the work has been carried out using mouse islets, with some studies using islets from rat, hamster, human, and other species.

Mouse pancreatic -cells exhibit complex and cyclic spike-burst activity in response to a rise in extracellular glucose concentration. The bursts consist of a depolarized phase of Ca²⁺-carrying action potentials alternating with a silent phase of repolarization, resulting in oscillations in intracellular Ca²⁺, which can drive pulses of insulin secretion (28, 37).

The only stimulus required for a complex cyclic spike-burst activity and corresponding [Ca²⁺]_i oscillations in islets and -cell clusters is elevation of glucose to levels above 5 and less than 20 mM. Intermediate glucose concentrations induce two main types of oscillations in mouse pancreatic islets: fast, where the period ranges from 10 to 30 s, and slow, with periods of several minutes (37, 54, 83). Single mouse -cells can also respond to glucose stimulation with regular oscillations (37).

We have previously studied slow and fast [Ca²⁺]_i oscillations in islets in response to a variety of conditions (70, 73; unpublished observations). We have also previously reported that a stable, transgenically derived murine insulinoma cell line (TC3-neo) responds to glucose with slow, large amplitude [Ca²⁺]_i oscillations but only in the presence of 10–20 mM tetraethylammonium (TEA), a blocker of K⁺ channels (74). We have utilized this cell line to characterize glucose-stimulated oscillatory activity (74).

However, the precise interpretation of previous results is limited due to the numerous channels and pumps in -cells that work concurrently, and identification of physiologically slow variables that drive oscillations remains unclear. To clarify these complex experimental results, we used a mathematical modeling approach. Our goals, then, are twofold: to develop a model for -cell ion homeostasis, including the bursts and [Ca²⁺]_i oscillations that can simulate cellular behavior, and to explain on this basis the experimental data for single cells and islets.

Several mathematical approaches in the literature have provided quantitative estimates of glucose-induced insulin secretion with corresponding descriptions of glucose transport, metabolism, and ion regulation (most recent are Refs. 6, 7, 13, 14, 26, 30, 62, 65, and 78). Many phenomena were successfully explained using these models (see DISCUSSION). However, most models described one specific phenomenon and therefore included a very restricted set of channels and pumps, so that it is difficult to apply these models to another aim. In addition, newly described experimental results and signaling molecules should be considered.

For this reason, we developed a new model to consider an extended variety of ionic channels and pumps as well as endoplasmic reticulum (ER) calcium sequestration mechanisms that have been identified in -cells. We simulated whole cell electrical activity and [Ca²⁺]_i, free calcium in the ER ([Ca²⁺]_ER), intracellular Na⁺ ([Na⁺]_i), cytosolic ATP ([ATP]_i), and inositol 1,4,5-trisphosphate ([IP₃]_i) concentrations by use of parameters derived mainly for mouse -cells. However, in this article, we do not make special consideration of metabolic processes or insulin secretion.

To our knowledge, this article represents the first attempt to include in one model the key ion channels and pumps on both the PM and the ER membrane. Several detailed and comprehensive models have been developed for cardiac muscle and other cell types (67, 71). Here, we employ this approach to investigate the possible role of K⁺ channels, Na⁺, and IP₃ in regulating -cell Ca²⁺ oscillations.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: NA+/K+ ACTIVE... APPENDIX B: GENERAL GATING... APPENDIX C: FRACTION OF... REFERENCES

 
Islet Isolation and [Ca²⁺]_i Measurement

Isolation of rodent islets. Islets of Langerhans were isolated from the pancreata of 8- to 10-wk-old C57BL/6J mice by collagenase digestion and discontinuous Ficoll gradient methods described previously (85).

Measurement of [Ca²⁺]_i. Islets were loaded with fura 2 for 25 min at 37°C in growth medium (RPMI 1640–10% fetal calf serum, penicillin, and streptomycin) supplemented with 5 mM acetoxymethyl ester of fura 2 (Molecular Probes). [Ca²⁺]_i was estimated as described elsewhere (74). Dual-wavelength digitized video fluorescent microscopy with fura 2 in single islets was performed using an intensified charge-coupled device (Hammatsu C2400) and Metafluor imaging software (Universal Imaging).

Model Development

Our model of the -cell combines a parallel conductance membrane model and an inside fluid compartmental model (Fig. 1). The compartmental model describes the time rate of changes in [Ca²⁺]_i, [Ca²⁺]_ER, [Na⁺]_i, and [ATP]_i, and [IP₃]_i. The extracellular space is assumed to have a relatively large volume, so that ionic concentrations of Ca²⁺, Na⁺, and K⁺ there are assumed to be constant.

View larger version (16K): [in this window] [in a new window]   Fig. 1. Schematic representation of currents and ion fluxes through the plasma membrane (PM) and endoplasmic reticulum (ER) membrane that have been included in the whole -cell model. Top: PM currents: IVCa, voltage-dependent Ca2+ current; ICa,pump, calcium pump current; INa,Ca, Na+/Ca2+ exchange current; ICRAN, Ca2+ release-activated nonselective cation current; INa, inward Na+ currents; INa,K, Na/K pump current; IKDr, delayed rectifying K+ current; IKCa, voltage-independent small-conductance Ca2+-activated K+ current; IKATP, ATP-sensitive K+ (KATP) current; ksg, coefficient of the sequestration rate of intracellular free Ca+ concentration ([Ca2+]i) by the secretory granules; Jer,p, uptake of Ca2+ by the ER; Jout, Ca2+ leak current from the ER; IP3, inositol 1,4,5-trisphosphate.

The corresponding equations for particular channels or pumps that we found in the literature were used even though they were in some cases obtained from cells other than pancreatic -cells. However, we have considered the origin of the equations employed and their correspondence to biological processes. Model coefficients from the literature were adjusted to simulate the corresponding experimental data as indicated. The basic set of evaluated coefficients is shown in Table 1, denoted as "standard."

Plasma membrane currents. The differential equation describing time-dependent changes in the plasma membrane potential (V) is the current balance equation (41)

(1)

where C_m is the total membrane capacitance. Model currents include a voltage-dependent Ca²⁺ current (I_VCa), a PM calcium pump current (I_Ca,pump), Na⁺/Ca²⁺ exchange current (I_Na,Ca), Ca²⁺ release-activated nonselective cation current (I_CRAN), inward Na⁺ currents (I_Na), a Na⁺/K⁺ pump current (I_Na,K), a delayed rectifier K⁺ current (I_KDr), voltage-independent small conductance Ca²⁺-activated K⁺ current (I_KCa), and the K_ATP current (I_KATP) (Fig. 1).

Ionic flux across the membrane, J_xi, is related to the ionic current by the expression

(2)

where F and z_x, are Faraday's constant and ionic charge, respectively.

General current equation. The ionic current across channels may be given by the empirical equation (41, p.18)

(3)

where

(4)

(5)

and where g_Xi is the whole cell conductance, V_X is the equilibrium potential, R is the universal gas constant, T is absolute temperature, and X_o and X_i are the extra- and intracellular concentrations, respectively.

Whole cell conductance may depend on ion concentrations and voltage. These dependencies will be specified for each conductance as they are discussed.

I_VCa. The whole cell Ca²⁺ current in mouse -cells flows principally through voltage-activated Ca²⁺ channels (29, 80). This current increases when the membrane is depolarized. Equation 3 was used for this current. The current/voltage relationship of this current [p_VCa(V)] can be described by a Boltzmann-type activation curve (7, 9). Then

(6)

where

(7)

and where g_mVCa is the maximum whole cell conductance, V_Cah is the half-activation potential, and K_Cah is the slope at half-maximal potential.

At physiological Ca²⁺ concentrations, the measured V_Cah varies from -3.8 mV (80) and -16.7 mV (45) to -19 mV (9), and K_Cah extends from 6.6 mV (45) and 8.4 mV (80) to 13 mV (9). In our model, these coefficients were fitted to be inside these regions (Table 1).

According to Göpel et al. (33) the mean integrated Ca²⁺ current observed during a 100-ms depolarization to -20 mV in the -cell in situ was 7.7 pC at [Ca²⁺]_o = 2.6 mM. This result can be obtained from Eq. 6 of our model, with g_mVCa = 995 pS and [Ca²⁺]_i = 0.05 µM. We set g_mVCa = 770 pS as the standard condition for our model (Table 1). This value is close to that estimated above. The current/voltage relationship calculated using Eq. 6 is in good correlation with data for -cells in situ (33).

PM Ca²⁺ pumps (I_Ca,pump). PM Ca²⁺ pumps are now well recognized as a primary system for the specific expulsion of Ca²⁺ from -cells (87). However, their kinetic properties are not well studied in these cells. For this reason, the general properties of PM Ca²⁺ pumps were used for the calculations.

It is now generally established that the Ca²⁺/ATP stoichiometry of the PM Ca²⁺ pump is 1 (10, 66) and that the reconstituted PM pumps are capable of establishing a membrane potential while operating with H⁺/Ca²⁺ = 1 (40). PM and ER Ca²⁺ pumps do not have the sensitivity to ATP at the millimolar ATP concentrations that exist in physiological conditions (10). The K_m is <0.5 µM Ca²⁺ for activated PM pumps (10, 66). On the basis of these data, the Ca²⁺ current through PM Ca²⁺ pumps was expressed in our model as (see 62)

(8)

where P_mCap is a maximum current and K_Cap is the value for the half-activation calcium concentration. K_Cap was set to 0.1 µM, which is within the range of experimental observations for other preparations. P_mCap (Table 1) is correspondingly above the evaluated lower limit (830 fA) that we obtained (see RESULTS).

Na⁺/Ca²⁺ exchanger (I_Na,Ca). The lack of specific inhibitors hinders the experimental assessment of the role of the Na⁺/Ca²⁺ exchanger. However, a detailed investigation of the -cell electrogenic 3 Na⁺/Ca²⁺ exchanger was recently reported (25), and the exchanger current (I_Na,Ca) was taken as follows from this work

(9)

where

(10)

and where g_NaCa is the maximum whole cell conductance, and K_NaCa is affinity constant.

One can calculate from Gall et al. (25) that g_NaCa = 271 pS (as C_m·_Na/Ca from their Eq. 2), and we used this value for g_NaCa in our model.

Similarly, those authors found K_NaCa to be 1.5 µM (25). However, an insignificant I_Na,Ca and a correspondingly very small influence of the Na⁺/Ca²⁺ exchanger on the model solutions were obtained at K_NaCa = 1.5 µM. This result does not correspond to the data involving the effect of this exchanger on glucose-induced electrical activity in intact pancreatic islets (25). This apparent contradiction is probably due to the use of low ATP concentration in the perforated patch whole cell experiments in Ref. 25; therefore, K_NaCa (1.5 µM) should only be considered as an upper limit. Data from different cell types show that ATP induces a dramatic increase in the intracellular Ca²⁺ affinity for the Na⁺/Ca²⁺ exchanger (20). Correspondingly, a lower value for K_NaCa (0.75 µM) was employed in our model to fit glucose-induced -cell oscillation patterns at an increased ATP content.

I_CRAN. I_CRAN is a nonselective cation current whose conductance is regulated by the ER Ca²⁺ content. In mouse -cells, it could be activated either indirectly by ER Ca²⁺ store depletion or directly by maitotoxin (84, 88, 89). However, the mechanism for coupling ER Ca²⁺ store depletion with CRAN remains an unresolved question.

In our model, only Na⁺ can penetrate into the cell via this channel, reflecting experimental results under physiological concentrations of cations (52, 84, 88). The current/voltage relationship of this current [p_CRAN(V)] was roughly linear, with a reversal potential close to 0 mV (52, 84, 88). In previous models, it was suggested that the regulation of I_CRAN depends on [Ca²⁺]_ER (8, 14, 62). We take this dependence following Ref. 62. Equation 3 can be written for this Na⁺ inward current

(11)

where

(12)

(13)

and g_mCRAN is the maximum whole cell conductance, f_CRAN is the voltage-independent part of the current, K_Car is the half-activation [Ca²⁺]_ER level, and V_CRAN is the reversal potential.

K_Car has been modified in our model, and it is now 200 rather than 40 µM (as accepted in Ref. 62), because the [Ca²⁺]_ER is usually found to be in the hundred micromolar range (see Ref. 86).

I_Na. The general equation for I_Na is derived from Eq. 3. The relationship between conditioning voltage and relative current amplitude for tetrodotoxin-blockable component of the I_Na [p_Na(V)] was characterized (33) as the Boltzmann equation. Then

(14)

where

(15)

The maximum value of whole cell conductance (g_mNa) was evaluated from the data (33), where it was found that I_Na = -392 pA at V = -140 mV. This I_Na value can be obtained from Eq. 14, with g_mNa as 1,200 pS when [Na⁺]_i is close to 0 and [Na⁺]_o = 140 mM. (We used [Na⁺]_i = 0.1 mM for this calculation). This value was used for standard conditions (Table 1).

Na⁺-K⁺ active transport (I_Na,K). Electrogenic Na⁺-K⁺-ATPase extrudes three Na⁺ ions in exchange for two K⁺ ions for each molecule of ATP hydrolyzed, generating a net outward flow of cations through the -cell PM (68). However, its kinetic properties have not been studied in -cells. For this reason, we employed the general model (12) as was done previously for -cells (65) (see APPENDIX A).

I_KDr. The principal type of voltage-dependent K⁺ current in -cells is the I_KDr. Kv2.1 is a likely candidate for the principal delayed rectifier isoform expressed in -cells (56, 70).

Equation 3 was used; however, the formulation (26, 78) is employed for whole cell conductance.

(16)

where g'_mKDr is the maximal whole cell conductance, and n is the general gating variable, which is determined in APPENDIX B.

The value of g'_mKDr (Eq. 16) is calculated to be 3,000 pS for standard conditions. This is comparable to the value used in Refs. 8, 26, and 65.

Family of KCa channels (I_KCa). The voltage-independent small-conductance Ca²⁺-activated K⁺ channels (SK channels) are also expressed in -cells (unpublished observations). It is quite plausible that these channels play an important role in [Ca²⁺]_i oscillations (32).

Equation 3 was used for this channel. Ca²⁺ activates I_KCa currents [f(Ca²⁺)] in a sigmoidal fashion (49, 53). However, there are no data on the exact dependence in -cells. The Hill equation was used to model this relationship, where the Hill coefficient equals 3 (14) or 5 (62). We also use a Hill equation, but with a Hill coefficient equal to 4

(17)

where

(18)

and K_KCa is affinity constant.

K_KCa varies in the 0.05–0.9 µM range (Ref. 41, p. 144). However, SK3 channels may be the predominant SK family member in -cells (unpublished observations), and because K_KCa = 0.1 µM was found for SK3 channels by Carignani et al. (11), this value was used in our model.

It is unlikely that the large-conductance Ca²⁺-activated K⁺ channel (BK channel) encoded by the slo gene plays an important role in the generation of oscillatory activity in -cells (50), although several previous models of -cell bursting have included this channel (78). We therefore did not include it in our considerations.

K_ATP channels (I_KATP). Equation 3 was used for this channel. However, we adopt here a kinetic model for the value of whole cell conductance (see Refs. 43, 58, 65). Then

(19)

where g'_mKATP is the maximal value of the whole cell conductance, and O_KATP is the fraction of channels open, which is considered in APPENDIX C.

The value g'_mKATP was evaluated to be 24 nS for the standard coefficient, which is comparable to the value used in Refs. 59 and 65. Then it can be calculated from Eqs. 19 and C1 (APPENDIX C) that the value of the whole cell conductance (g'_mKATP O_KATP) is 0.94 nS at [ATP]_i = 0. This value falls in the range between 0.2 and 3.0 nS that was measured for the maximum K_ATP conductance reported for single mouse -cells (48).

Fluid Compartmental Model

The material balance equation for calcium depends on three processes: entry, extrusion, and buffering (Fig. 1).

Calcium enters the -cells primarily through voltage-activated Ca²⁺ channels by diffusion along an inwardly directed electrochemical gradient (I_VCa). The maintenance of this ionic gradient depends on a Ca²⁺-extruding mechanism in the PM and Ca²⁺ sequestration by intracellular organelles. At the PM, three processes are involved in transporting Ca²⁺ out of the cell: a Ca²⁺ pump, a Na⁺/Ca²⁺ exchanger, and removal of Ca²⁺ sequestrated in insulin granules by exocytosis. In addition, both the ER and mitochondria can accumulate Ca²⁺ via pumps. Even though Ca²⁺ is critical for mitochondrial function, for the present we do not include mitochondrial Ca²⁺ stores, since it appears that both the volume of mitochondria and its Ca²⁺ concentration are small relative to the ER (2, 18).

Sarco(endo)plasmic reticulum Ca²⁺-ATPase pump. Uptake of Ca²⁺ into the ER is mediated in -cells by sarco(endo)plasmic reticulum Ca²⁺-ATPases (SERCAs) with an unusually low value for the half-activation calcium concentration (≤0.4 µM) (27). The Ca²⁺/ATP stoichiometry of the SERCA is 2 (10, 66). The equation for SERCA was written in the same form as the Ca²⁺ PM pump (Eq. 8; see also Ref. 62)

(20)

where J_er,p is the flux into the ER through SERCA pumps per cytosol volume, K_Carp is the half-maximal pump activity, and P_CaER is the maximum rate of pumping. The lower limit for this parameter was evaluated (see RESULTS) as P_CaER = 0.026 µM/ms for the data from Ref. 25. The evaluated parameter for standard conditions (Table 1) is correspondingly above this lower limit. K_Carp was taken as 0.5 µM, a value close to experimental data (27).

IP₃ metabolism. In the pancreatic -cell, there is clear evidence that IP₃ is an important cellular messenger inducing Ca²⁺ mobilization from intracellular stores by binding to specific receptors located on intracellular Ca²⁺ stores and is the main calcium release channel from the ER in -cell tissues (38). However, we could not find any mathematical models of -cell regulation that used IP₃ as an independent variable.

Different mechanisms seem to be available in the -cells for stimulating IP₃ production; however, only the modulation of inositol lipid-specific phospholipase activity by changes in Ca²⁺ is well studied (3). For this reason, the [Ca²⁺]_i dependence was only modeled as the Hill function with a Hill coefficient of 2 (3).

The subsequent kinetics of IP₃ in -cells is unknown. Therefore, we assume simply that IP₃ is converted to inactive metabolites at a rate proportional to its concentration. Then synthesis and degradation of IP₃ are described by the equation

(21)

where k_IP is the rate constant of IP₃ production, k_dIP is the rate constant of IP₃ degradation, and K_IPCa is the half-activation of the IP₃ production by calcium concentration.

The time constants k_IP and k_dIP (Table 1) were picked to correctly reproduce the experimentally observed time course of the average [Ca²⁺]_i during the slow [Ca²⁺]_i oscillations (see RESULTS). K_IPCa is taken as 0.4 µM (36). The fitted parameter k_dIP (0.04 s^-1, Table 1; half-life is 17.3 s) is inside the region of a half-life from 1 to 30 s that has been found for different cells with a radius of 5 µm and above (79), similar to that of -cells.

Ca_ER²⁺ mobilization. A Ca²⁺ leak current (J_out) from the ER per whole cell was taken according to Mears et al. (62)

(22)

where P_leak is the Ca²⁺ leak permeability out of the ER, P_IP3 is the maximum permeability through open IP₃-activated channels, and O is the fraction of open IP₃-activated Ca²⁺ channels.

The type III IP₃ receptor (IP₃R) is the main calcium release channel in the -cell ER (38). Apparently, in normal physiological conditions, the O does not depend on the change of ATP concentration (57). The initial dependence of the IP₃R maximum open probability vs. cytosolic Ca²⁺ could be well fitted to a Michaelis-Menten-type saturating equation with an affinity of 0.077 µM for [Ca²⁺]_i (60).

However, there are conflicting data concerning IP₃ action. According to Hagar and Ehrlich (39), the Hill equation with a Hill coefficient close to 2 was found for the effect of IP₃ on O in vitro. On the other hand, the Hill number for low degrees of saturation of the IP₃R was estimated at 3.5 in vivo for rat basophilic leukemia cells (63). On the basis of these data, the Hill coefficient is set equal to 3. Then

(23)

where K_RCa is the affinity constant of the [Ca²⁺]_i to IP₃R, and K_IP3 is the constant for IP₃ binding to a channel. K_IP3 is 3.2 µM (39).

Ca²⁺ and Na⁺ dynamics. The cytosolic and ER calcium concentrations and Na⁺ cytosolic concentration are determined by the ion fluxes across the plasma and ER membranes. However, calcium concentrations are strongly buffered in cells (77). This can be modeled using special coefficients for the fraction of free Ca²⁺ in the cytoplasm and ER (14, 62, 71). The Ca²⁺ leak from cells by -granule exocytosis was modeled previously (14) as a rate proportional to [Ca²⁺]_i. We employed a similar approach.

On the basis of the foregoing consideration, the equations for Ca²⁺ and Na⁺ concentrations can be written as

(24)

(25)

(26)

where f_i and f_er are the fractions of free Ca²⁺ in cytoplasm and ER, V_er and V_i are the effective volumes of the ER and cytosolic compartments, and k_sg is a coefficient of the sequestration rate of Ca²⁺_i by the secretory granules.

It has been proposed that the Ca²⁺-binding capacity inside the ER is lower than in the cytoplasm (69). This can increase the fraction of free Ca²⁺ in ER; therefore, we used a value of f_er threefold higher than f_i (Table 2).

ATP homeostasis. ATP concentration in the cytosol increases somewhat with an increase in glucose supply (2). However, the rate of ATP production and utilization is a complex function of the concentrations of ATP_i, ADP_i, Ca²⁺, glucose, and numerous other factors. For simplicity, we have chosen a first-order reaction to express the rate of ATP_i production from ADP_i.

Clearly, there is some basal level of ATP_i consumption in a cell at low glucose and [Ca²⁺]_i. There is also ATP consumption by the PM and ER Ca²⁺ pumps and by the electrogenic Na⁺-K⁺-ATPase in our model. However, there is considerable evidence that other ATP consumption processes are accelerated by an increase in [Ca²⁺]_i in -cells (2, 23, 76). For this reason, an additional term for ATP consumption was introduced, which depends on an increase in [Ca²⁺]_i. Then

(27)

where

(28)

k_ADP is the rate constant of ATP_i production, k_ATP,Ca is the rate constant of ATP_i consumption that accelerates as [Ca²⁺]_i increases, and k_ATP is the permanent rate constant of ATP_i consumption.

The general concentration of intracellular nucleotides ([A]_o) was assumed to be constant. The measured concentration of adenine nucleotides in pancreatic islets varies within the limits of 2–10 mM (23). We have taken [A]_o to be inside this range (4 mM).

It is important to find a range of parameters for ATP dynamics. According to the calculations in Ref. 23, the rate of ATP_i production in -cells may run to 0.5 µmol ATP·s^-1·g dry wt^-1 at low glucose concentrations and 1 µmol ATP·s^-1·g dry wt^-1 at high glucose concentrations. Matschinsky et al. (61) found the corresponding level of adenine nucleotides to be 20 µmol/g dry wt. Then the mean rate of ATP_i production per second can be estimated roughly as a part of the general quantity of intracellular nucleotides per unit of volume, that is, as 0.025 or 0.05 [A_o]/s (or 0.1 and 0.2 mM/s) at low and high glucose concentrations, respectively.

The constants k_ATP,Ca and k_ATP (Eq. 27 and Table 1) were fitted to simulate both the observed pattern of [Ca²⁺]_i oscillations and the evaluated rate of ATP_i production in -cells above. The simulation of low glucose concentration (k_ADP = 0.03 s^-1) leads in our model to a steady-state [ADP]_i of 3.068 mM (Table 3) and to a corresponding rate of ATP_i production (k_ADP [ADP]_i) of 0.092 mM/s. The simulation of slow [Ca²⁺]_i oscillations at high glucose levels (k_ADP = 0.37 s^-1; Fig. 3) yields a mean [ADP]_i of 0.7 mM and to the rate of ATP_i production as 0.26 mM/s. This correlates well with the values evaluated above.

View larger version (18K): [in this window] [in a new window]   Fig. 3. Glucose-induced slow electrical bursting and [Ca2+]i oscillations were simulated at a step increase of the rate constant of ATPi production from a low to an intermediate value (from kADP = 0.03 s-1 to 0.37 s-1) at t = 0 at initial concentration as in Table 3; all other parameter settings are standard (see Tables 1 and 2). A: [Ca2+]i; B: membrane potential. C: Ca2+ in ER ([Ca2+]ER). D: cytosolic ATP ([ATP]i). E: cytosolic IP3 ([IP3]i). F: intracellular Na+ concentration ([Na+]i). For simulation of Tg action, the maximal rate of sarco(endo)plasmic reticulum (SERCA; PCaER) was decreased from 0.105 to 0.013 µM/ms at arrow 3 in B.

Computational aspects. The spread of current between electrically coupled -cells most probably contributes to the synchronization of the electrical activity and [Ca²⁺]_i oscillations throughout an islet (33, 37). For this reason, we consider islets as an assemblage of cells with similar properties and perform the computer simulation only for some mean individual cell.

The complete system consists of seven state variables: differential equations describing the time rate of change in PM potential (V), Ca²⁺, Na⁺, ATP, and IP₃ concentrations in cytoplasm, calcium concentration in ER, and the differential equations characterizing the voltage-dependent gating variable (n) for the delayed rectifier K⁺ channels (Eqs. 1, 21, 24–27, B1). The units used in the model are time in milliseconds (ms), voltage in millivolts (mV), current in femtoamperes (fA), concentration in micromoles per liter (µM), conductance in picosiemens (pS), capacitance in femtofarads (fF), and temperature in degrees Kelvin (°K).

The model and cell parameters are given in Tables 1, 2, 3. These tables and the appendixes contain all the information necessary to carry out the simulations presented in this paper. However, the rate constant for ATP_i production (k_ADP) is represented in every figure legend.

Simulations were performed by forward integration of the coupled system of differential equations. We used two integrated software environments: "Content" for the IBM-compatible personal computer using an implicit fourth-order Runge-Kutta or Rodau IIA method (15) and "Virtual Cell" accessible via the Internet (55).

This model is available for direct simulation on the website "Virtual Cell" (www.nrcam.uchc.edu) in "MathModel Database" on the "math workspace" in the library "Fridlyand" with the name "Chicago.1".

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: NA+/K+ ACTIVE... APPENDIX B: GENERAL GATING... APPENDIX C: FRACTION OF... REFERENCES

 
Validation of the Model

It was possible to evaluate the values of the rates of Ca²⁺ pumps from the simulation of the data from Ref. 25, where the [Ca²⁺]_i transients were induced by a 2-s depolarization. Gall et al. (25) found that the time constant for the decrease in [Ca²⁺]_i is 1.8 s when Ca²⁺-ATP pumps on the PM and ER were activated and 4.6 s when Ca²⁺-ATP pumps on the ER were inhibited by thapsigargin (Tg). This was determined under conditions where the Na⁺/Ca²⁺ exchanger was inactive, i.e., g_NaCa = 0, granule exocytosis was lacking (k_sg = 0), and IP₃R was closed (P_IP3 = 0). Using Eqs. 9 and 21, we found that these data can be well simulated at P_CaER = 0.026 µM/ms and P_mCap = 830 fA. However, these values should be considered as a lower limit, whereas the experiments in Ref. 25 were made in conditions of low ATP concentration, and it is probable that Ca²⁺ pumps were in an inactivated form. Correspondingly, higher values were fitted in our model (see Table 1).

A number of parameters were taken from the literature, such as channel and pump kinetics (see MATERIALS AND METHODS). However, because these values were sometimes derived from experiments with different cell lines and specific experimental conditions, these parameters were constrained to be within a range consistent with our own system. Other parameters were fitted to simulate the pattern of fast and slow bursting of electrical activity and [Ca²⁺]_i oscillations in -cells. The parameters that yield these oscillations are the standard values given in Table 1. A computer simulation at a low ATP production rate and the coefficients from Tables 1 and 2 were used to obtain the steady-state data corresponding to low glucose concentrations (Table 3).

Islet Simulations

Slow Ca²⁺ oscillations. Slow Ca²⁺ oscillations occur spontaneously during glucose stimulation (Fig. 2). They are characterized by a descending phase in two components. A rapid decrease in [Ca²⁺]_i is followed by a slower phase. Similar data were obtained by many investigators for slow oscillations (27, 37, 64). It was also shown that a rapid decrease in [Ca²⁺]_i coincided with the closure of voltage-dependent Ca²⁺ channels (27).

View larger version (13K): [in this window] [in a new window]   Fig. 2. Slow Ca2+ oscillations and effect of thapsigargin (Tg) in mouse islets. Mouse islets were loaded with fura 2-AM, and changes in [Ca2+]i [expressed as the 340/380 nm ratio (F340/380)] were recorded during 14 mM glucose and subsequent addition of 1 µM Tg (shown).

A typical computer simulation of slow oscillations with the patterns of membrane potential and corresponding concentrations is illustrated in Fig. 3, left. It was obtained by introducing a step increase in the rate constant of ATP_i production from the resting conditions (Table 3) to an intermediate value. A detailed pattern of the concentration changes and corresponding currents is shown in Fig. 5A for a single oscillation.

View larger version (28K): [in this window] [in a new window]   Fig. 5. Simulation of single oscillations. Model solutions were obtained from the oscillation pattern after simulation of glucose addition as in Figs. 3 and 4. Concentrations, PM potential, and some currents are represented for 1 characteristic oscillation. Broken lines in Fig. 5.9 correspond to a membrane potential-independent part of ICRAN conductance (fCRAN, Eq. 12). Units are shown in series, labeled in A with the exception of Fig. 5.3 for [Ca2+]ER, where units are shown for every case. A: single characteristic slow oscillation from Fig. 3 (left). Broken line in A.1 corresponds to a single [Ca2+]i oscillation, when only [IP3]i was fixed at its mean level (3 µM) with other parameters, as at the beginning of Fig. 3. B: single characteristic fast oscillation from beginning of Fig. 4. C: simulation of single characteristic oscillation determined only by [Ca2+]ER changes at kADP = 0.45 s-1. Other slow variables were frozen at their mean level for fast oscillations in B: [IP3]i = 1 µM, [Na+]i = 6 mM, [ATP]i = 3.5 mM. D: simulation of single characteristic oscillation determined only by [Na+]i changes at kADP = 0.37 s-1. Other slow variables were frozen at their mean level for slow oscillation in A: [IP3]i = 3 µM, [Ca2+]ER = 50 µM, [ATP]i = 3.3 mM.

View larger version (17K): [in this window] [in a new window]   Fig. 4. Biphasic response to glucose in islets, simulated by increase of kADP from 0.03 to 0.45 s-1 at low rate of IP3 synthesis (kIP from Eq. 21 is 0.1 instead 0.3 µM/s in Fig. 3) at t = 0. All other parameter settings are as in Fig. 3. For simulation of glucose addition at arrow 1, kADP was increased from 0.45 to 0.7 s-1 and at arrow 2 from 0.7 to 2 s-1.

Biphasic responses and fast Ca²⁺ oscillations. Fast oscillations usually show a more complex picture. In this case, the electrical response of pancreatic islets to step increases in glucose concentration is often biphasic, consisting of a prolonged depolarization with action potentials (Phase 1) followed by fast bursts of membrane potential and Ca²⁺ oscillations. The initial tonic electrical activity is accompanied by a sustained elevation of [Ca²⁺]_i (8, 72, 89).

It was suggested by model simulations and experimental results that the Phase 1 response may result from the combined influences of depolarizing K_ATP channel closure and I_CRAN deactivation. The Ca²⁺ fills the ER during Phase 1, deactivating I_CRAN and repolarizing the PM, allowing steady-state bursting to commence (8, 62). We are also able to simulate these results on the basis of our model when Ca²⁺ fills the ER to considerably elevated concentrations. This increase of [Ca²⁺]_ER can occur if Ca²⁺ output from the ER is restricted, for example, by closing IP₃R channels. This can be achieved in our model, for example, by decreasing the [IP₃]_i as a consequence of decreased IP₃ production. The typical Phase 1 response followed by spike-burst behavior of the membrane potential and the fast oscillation patterns of ions and [ATP]_i are illustrated in Fig. 4. It was simulated at a decreased IP₃ synthesis rate constant and an intermediate value of the rate constant of ATP_i production.

The same results can be simply obtained by a partial closing of the IP₃R channels in place of the decreased IP₃ production (not shown). Our model also shows that the duration of Phase 1 decreases with increased initial [Ca²⁺]_ER at t = 0 (not shown; see Ref. 62).

Simulating increased ATP_i production leads to increased oscillation frequency, progressing to continuous spiking with a further increase in the ATP_i production rate constant (Fig. 4). This is in reasonably good agreement with the experimental data (13, 37).

Note that fast [Ca²⁺]_i oscillations simulated by our model (Fig. 4) have an appearance very similar to those observed in pancreatic -cells (8, 72). They also resemble the fast oscillations that were obtained in preceding models (8, 14, 62). This is not surprising, because our simulation of fast oscillations is close to those employed in these models.

Model Sensitivity to Slow Variables

Four parameters can change slowly in our model ([ATP]_i, [Ca²⁺]_ER, [IP₃]_i, and [Na⁺]_i), and their variations can lead to a slow membrane repolarization during a depolarized phase and to a depolarization in a silent phase. To determine the role of different processes and parameters, we performed an analysis of model sensitivity to slow variables to fix some parameters at their mean level for some kind of oscillations. It was found that the slow oscillations take place when only [Na⁺]_i ([ATP]_i, [Ca²⁺]_ER, and [IP₃]_i were constant) can change (Fig. 5D), and the characteristic fast oscillations occur when only [Ca²⁺]_ER ([ATP]_i, [Na⁺]_i, and [IP₃]_i were constant) can vary (Fig. 5C). Consideration of these two cases simplifies the general analysis.

The behavior of [Ca²⁺]_i, V, [Ca²⁺]_ER, [IP₃]_i, and [Na⁺]_i and the most essential current components is shown in Fig. 5, A and B, for single oscillations of slow and fast types. Because of the large fluctuations of I_CRAN, the voltage-independent part of this current (f_CRAN, Eq. 12) is also represented to facilitate the analysis (broken lines in Fig. 5.9). Corresponding concentrations and currents are also represented for cases when only [Ca²⁺]_ER (Fig. 5C) or [Na⁺]_i (Fig. 5D) can change. The contribution of other currents was either insignificant (I_Na) or does not change considerably between cases (I_Ca,pump, I_KDr, I_KCa, I_KATP).

ATP sensitivity. Activation of the Na⁺/K⁺ pump with increased [Na⁺]_i concentration during the depolarized phase accelerates ATP consumption in our model. Total Ca²⁺ pump rates and Ca²⁺-dependent ATP_i consumption also increase during the depolarized phase with [Ca²⁺]_i rising (see Eq. 27). The acceleration of ATP_i consumption leads to a decreased [ATP]_i/[ADP]_i ratio (Figs. 3 and 4). It is therefore possible that cyclic changes in the [ATP]_i/[ADP]_i ratio control the cyclic closure and reopening of K_ATP channels. This has been proposed as a mechanism underlying the oscillatory behavior of -cells (1, 65, 82).

We evaluated this possibility and found that fixing [ATP]_i at a mean level yields no noticeable effect on the oscillatory pattern shown in Fig. 5, A and B. This can be explained by insignificant changes of the fraction of K_ATP channels open (APPENDIX C) during the ATP_i variations around its mean value (not shown). This suggests that [ATP]_i is not the variable that drives slow oscillations in our model. On the other hand, considerable changes in mean [ATP]_i and correspondingly in I_KATP can lead to a dramatic change in oscillations (see Figs. 3 and 4).

Single-oscillation analyses. It can be seen that spike activity is generated by the alternate activation of delayed rectifying K⁺ channels and voltage-gated Ca²⁺ channels in every case. It is analogous to other models and has been previously analyzed (14, 78). The main difference between the models includes the nature of processes that lead to slow depolarization of PM potential during the depolarized phase and to its repolarizing during a silent phase.

When [Ca²⁺]_ER is the only slow parameter ([ATP]_i, [Na⁺]_i, and [IP₃]_i were constants; Fig. 5C), the following mechanism is responsible for oscillations: rising Ca_i²⁺ during the depolarized phase leads to increased Ca²⁺ pumping from the cytoplasm to the ER and to Ca_ER²⁺ accumulation, with corresponding closure of CRAN channels and decreased I_CRAN, that can be observed more clearly from the change of the voltage-independent part of I_CRAN (f_CRAN, broken line in Fig. 5C.9). This, in turn, leads to PM repolarization. These processes have opposite directions in the silent phase.

Figure 5D shows that, when [Na⁺]_i is the only slow variable ([ATP]_i, [Ca²⁺]_ER, and [IP₃]_i were constant), I_Na,K is the main current that manages the slow PM potential changes leading to oscillations. The mechanism involves increased [Ca²⁺]_i during a depolarized period, activating the outward Ca²⁺ flux through Na⁺/Ca²⁺ exchangers and a corresponding inward Na⁺ flux. The resultant increase in [Na⁺]_i leads to the slow increase of outward current through electrogenic Na⁺/K⁺ pumps (I_Na,K; Fig. 5D.7) with corresponding PM repolarization. Falling [Na⁺]_i during a silent phase leads to decreased I_Na,K and then in turn to PM depolarization.

In the simulation of slow oscillations shown in Fig. 5A, [Na⁺]_i and the current I_Na,K change as in Fig. 5D, with no change of the voltage-independent part of the current I_CRAN (f_CRAN). Correspondingly, the mechanism of slow oscillations is the mechanism described above, with [Na⁺]_i as the only slow variable.

In the simulation of fast oscillations shown in Fig. 5B, [Ca²⁺]_ER and f_CRAN change as in Fig. 5C, with little change of [Na⁺]_i and the corresponding current I_Na,K. In this case, the currents I_Na,K and I_CRAN act together. However, I_CRAN plays the major part, as the concentrations and currents are similar in Fig. 5, B and C. This is in agreement with the conclusion that [Ca²⁺]_ER drives the fast oscillations but does not need to oscillate to have slow oscillations, whereas [Na⁺]_i gives slow oscillations but does not need to oscillate to have fast oscillations.

However, in the case of fast oscillations, CRAN channels are working under conditions when the ER is filled with Ca²⁺, and the [Ca²⁺]_ER change occurs close to the half-activation level of [Ca²⁺]_ER for CRAN channels (200 µM; see Eq. 12 and Table 1). This leads to an increased rate of both PM potential depolarization and repolarization, when [Ca²⁺]_ER correspondingly increases and decreases following [Ca²⁺]_i changes.

Shape of slow [Ca²⁺]_i oscillations and role of IP₃. It can be seen in our simulations that, at the beginning of the silent period of slow oscillations, Ca_ER²⁺ is gradually released into the cytosol (Figs. 3 and 5A.3) and the shape of the [Ca²⁺]_i curve is close to what was found by us (Fig. 2) and others (27, 83) in experiments. Such a slow phase is absent at the simulation of fast oscillations (Figs. 3 and 5B.1).

When [IP₃]_i was fixed at its mean level at the conditions that were used to simulate slow oscillations, there was still the ability to produce some oscillations. However, the slow [Ca²⁺]_i decrease disappears at the beginning of the silent period. This resulting single [Ca²⁺]_i oscillation is shown as a broken line in Fig. 5A.1.

The relationship between [Ca²⁺]_i and [Ca²⁺]_ER needs additional clarification. In our model, a dynamic equilibrium is established between [Ca²⁺]_i and [Ca²⁺]_ER for several seconds when there is no spiking activity; that is, [Ca²⁺]_i and [Ca²⁺]_ER are closely connected. The decrease in [IP₃]_i leads to a corresponding, slow IP₃R channel closure that decreases the rate of Ca²⁺ flux from the ER. A slow [Ca²⁺]_ER drop leads to a slow decrease in [Ca²⁺]_i at their dynamic equilibrium. When the [IP₃]_i is fixed, this mechanism does not operate, and the slow [Ca²⁺]_i drop disappears.

No remarkable changes in the fast oscillations pattern (shown in Fig. 5B) were observed if [IP₃]_i was fixed at its mean level with otherwise identical conditions (not shown). The lack of an effect can be understood in light of the small changes in [IP₃]_i that occur during fast oscillations (see Figs. 4 and 5B.4).

This suggests that changes in [IP₃]_i are not the main driving force for the fast and slow oscillations in our model. However, the change of mean [IP₃]_i can have a considerable effect, including a transformation of slow oscillations to fast at decreasing [IP₃]_i (compare Figs. 3 and 4). Furthermore, [IP₃]_i changes can have a considerable effect on the pattern of slow oscillations.

Role of the ER. In islets pretreated with Tg to block the SERCA pump, the frequency was increased, the amplitude of slow [Ca²⁺]_i oscillations was larger than in control islets, and the descending phase of each [Ca²⁺]_i oscillation was fast with no slow descending second phase (Fig. 2; see also Refs. 27, 64). Our model permits a simulation of these experiments (Fig. 3). In this case, the corresponding amplitude of slow [Ca²⁺]_i oscillations was larger, the period of oscillation was shorter, and the descending phase of each [Ca²⁺]_i oscillation was fast with no slow second phase (Fig. 3), all of which are in reasonably good agreement with the experimental data.

In this case, Ca²⁺ in the ER does not drive the oscillations. However, the remaining [Na⁺]_i-dependent mechanisms are able to create the relevant oscillations, and corresponding fluxes resemble the ones that were observed if the oscillations were driven only by Na_i variations, as in Fig. 5D.

A different pattern was found when SERCA pumps were blocked in fast oscillation experiments. Worley et al. (89) observed that, when Tg was added to the medium, the bursting was transformed to continuous spiking, and fast [Ca²⁺]_i oscillations were converted to a tonically elevated Ca²⁺ level. Bertram et al. (8) found that an addition of Tg was accompanied by a gradual increase in the plateau fraction and in electrical burst frequency.

However, in our general model, blocking of SERCA leads to the pattern of slow oscillations, as shown in Fig. 3, right. For this reason, other possibilities were investigated. When oscillations were simulated only by changes in [Ca²⁺]_ER (by fixing [Na⁺]_i, [IP₃]_i, and [ATP]_i to constant values, as in Fig. 5C), the block of the SERCA pumps increased the frequency of fast [Ca²⁺]_i oscillations, because I_CRAN is activated in consequence of [Ca²⁺]_ER decrease. A further increase of SERCA inhibition can then lead to a permanent PM depolarization and to a continuous rise in [Ca²⁺]_i (Fig. 6). This transformation is consistent with the aforementioned experiments for fast oscillations. It is also in agreement with the results of the modeling of Chay (14) and Bertram et al. (8), who had not included [Na⁺]_i changes in their models and had taken into account only Ca²⁺ and K⁺ currents and [Ca²⁺]_ER influence on I_CRAN.

View larger version (15K): [in this window] [in a new window]   Fig. 6. Model simulation of Tg action on fast oscillations at kADP = 0.45 s-1 with the initial simulation as in Fig. 4 and frozen concentrations of [ATP]i, [Na+]i, and [IP3]i as in Fig. 5C. Maximal rate of SERCA (PCaER) was decreased from 0.105 to 0.0785 µM/ms at arrow 1 in B and from 0.0785 to 0.0524 µM/ms at arrow 2.

In other words, the model suggests that, under conditions when fast oscillations occur, something is blocking the [Na⁺]_i-dependent oscillation mechanisms, even though Tg action resulted in a depletion of the Ca_ER store. One possible mechanism is a partial inhibition of the Na⁺/K⁺ pump. For example, we found that only [Ca²⁺]_ER-dependent fast oscillations were possible in our model when the maximum activity of the Na⁺-K⁺-ATPase was diminished to 25% of basal activity, even though all other coefficients and initial concentrations were as in Fig. 4. In this case, a simulation of Tg action at fast oscillations resembles Fig. 6. However, the physiological differences between fast and slow oscillations need further examination.

Role of [Na⁺]_i. Slow glucose-induced [Na⁺]_i oscillations were found in individual -cells under conditions known to induce oscillations of [Ca²⁺]_i (34, 35). Partial suppression of the Na⁺/K⁺ pump by ouabain resulted in an increased amplitude of slow Na⁺ and Ca²⁺ oscillations and decreased their frequency. Slow [Na⁺]_i and [Ca²⁺]_i oscillations were converted to tonically elevated [Na⁺]_i and [Ca²⁺]_i levels following increased ouabain concentration (34, 35).

Our simulation leads to analogous results (Fig. 7). This result can be explained on the basis of our model: the partial suppression of the Na⁺/K⁺ pump leads to decreased I_Na,K, which is responsible for slow PM potential variations. For this reason, to obtain the same value of I_Na,K variation that leads to a corresponding PM potential change, it was necessary to have larger [Na⁺]_i changes. This extends the time necessary for corresponding [Na⁺]_i accumulation or dissipation. The increase in Na⁺/K⁺ pump inhibition leads to collapse of this [Na⁺]_i-connected mechanism and to the disappearance of slow oscillations.

View larger version (16K): [in this window] [in a new window]   Fig. 7. Simulated applications of ouabain to a cell that exhibits slow oscillations with the initial simulation as in Fig. 3. Addition of ouabain is modeled by decreasing PNaK from 600 to 200 fA (arrow 1) in B and from 200 to 150 fA (arrow 2).

To check the role of [Na⁺]_i in driving slow oscillations, we also froze [Na⁺]_i at its mean level characterized for slow oscillations (7 mM) and made a simulation at the initial conditions by using the coefficients characterized for slow oscillations (as in Fig. 3). No oscillations were found (not shown).

Our model simulations demonstrate the necessity for [Na⁺]_i oscillations to drive the slow Ca²⁺ oscillations and suggest that [Na⁺]_i is a candidate pacemaker for slow oscillations.

-Cell Lines

We have previously studied the properties of a clonal -cell line termed TC3-neo (74). In contrast to normal mouse islets, this cell line does not usually respond to a step increase in glucose concentration with regular oscillatory increases in [Ca²⁺]_i. Instead, a slow rise in [Ca²⁺]_i occurs with occasional intermittent spikes. However, the exposure of TC3-neo cells to 20 mM TEA, a blocker of K⁺ channels (24), permitted the generation by glucose of large, regular, slow oscillations in [Ca²⁺]_i. In the absence of glucose, TEA was without effect (74). However, the molecular differences between TC3-neo cells and primary -cells have not been completely studied.

The magnitude of the Ca²⁺ inward current in -cells within intact pancreatic islets is considerably more than in individual -cells maintained in tissue culture [up to twice (33)]. This difference can be expected to have dramatic effects on the generation of -cell electrical activity and leads to failure of dispersed cells to generate the characteristic oscillatory electrical activity (33, 81). It is reasonable to assume that TC3-neo cells also have a decreased Ca²⁺ inward current that can be simulated by a reduced conductance of Ca²⁺ channels in our model.

Under these conditions, i.e., reduced conductance of Ca²⁺ channels, simulation of glucose addition does not lead to oscillations. However, a partial block of delayed-rectifier K⁺ channels, small-conductance Ca²⁺-activated K⁺ channels, and K_ATP channels, which simulate TEA action, leads to large, regular oscillations in our model (Fig. 8). This occurs because the decreased inward Ca²⁺ current leads to PM repolarization. To restore the possibility of oscillations, a corresponding decrease of K⁺ outward currents, as by TEA action, is required to allow PM depolarization.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Model simulation of a -cell line experiment at intermediate glucose level was made at a step increase of rate constant of ATP production from low to intermediate value (from kADP = 0.03 to 0.25 s-1) at t = 0; all other parameter setting are standard (see Tables 1, 2, 3), with the exception of gmVCa for voltage-activated Ca2+ current that was decreased from 770 to 300 pS. For simulation of tetraethylammonium (TEA) action at arrow 1, maximal conductance for K+ channels was decreased: for KATP from 24,000 to 12,000 pS, for a delayed rectifier K+ current from 3,000 to 800 pS, and for the voltage-independent small conductance Ca2+-activated K+ current from 130 to 60 pS.

The exposure of TC3-neo cells to 1 µM nitrendipine, which blocks L-type Ca²⁺ channels, can reversibly suppress oscillatory activity (Fig. 1C in Ref. 74). In our model, a similar decrease in Ca²⁺ channel activity also leads to a suppression of oscillatory activity (not shown).

In the absence of glucose, exposure of TC3-neo cells to 100 µM tolbutamide to block I_KATP induced [Ca²⁺]_i oscillations only in the presence of TEA (Fig. 2A in Ref. 74). The same results were obtained with our model (Fig. 9) when an additional block of K_ATP channels was used to simulate tolbutamide effects. In this case, the simulation of tolbutamide addition on I_KATP was equivalent to the simulation of glucose additions in Fig. 8.

View larger version (10K): [in this window] [in a new window]   Fig. 9. Model simulation of a -cell line experiment at a low glucose level (kADP = 0.03 s-1 for all time) and the additional action of TEA and tolbutamide. To simulate TEA action, coefficients were taken as in Fig. 8B at arrow 1. To simulate tolbutamide action, maximal conductance for KATP was decreased from from 12,000 to 4,000 pS at arrow 2.

Similar experimental results were obtained by Eberhardson and Grapengiesser (21) for single -cells isolated from mouse islets, in which the [Ca²⁺]_i oscillations could be obtained at low glucose concentration only when both TEA and tolbutamide were added.

	DISCUSSION

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: NA+/K+ ACTIVE... APPENDIX B: GENERAL GATING... AP