Adenine nucleotide regulation in pancreatic {beta}-cells: modeling of ATP/ADP-Ca2+ interactions

Leonid E. Fridlyand, Li Ma, and Louis H. Philipson

Department of Medicine, University of Chicago, Chicago, Illinois

Submitted 16 December 2004 ; accepted in final form 20 June 2005


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Glucose metabolism stimulates insulin secretion in pancreatic {beta}-cells. A consequence of metabolism is an increase in the ratio of ATP to ADP ([ATP]/[ADP]) that contributes to depolarization of the plasma membrane via inhibition of ATP-sensitive K+ (KATP) channels. The subsequent activation of calcium channels and increased intracellular calcium leads to insulin exocytosis. Here we evaluate new data and review the literature on nucleotide pool regulation to determine the utility and predictive value of a new mathematical model of ion and metabolic flux regulation in {beta}-cells. The model relates glucose consumption, nucleotide pool concentration, respiration, Ca2+ flux, and KATP channel activity. The results support the hypothesis that {beta}-cells maintain a relatively high [ATP]/[ADP] value even in low glucose and that dramatically decreased free ADP with only modestly increased ATP follows from glucose metabolism. We suggest that the mechanism in {beta}-cells that leads to this result can simply involve keeping the total adenine nucleotide concentration unchanged during a glucose elevation if a high [ATP]/[ADP] ratio exits even at low glucose levels. Furthermore, modeling shows that independent glucose-induced oscillations of intracellular calcium can lead to slow oscillations in nucleotide concentrations, further predicting an influence of calcium flux on other metabolic oscillations. The results demonstrate the utility of comprehensive mathematical modeling in understanding the ramifications of potential defects in {beta}-cell function in diabetes.

insulin; islets; adenosine triphosphate-sensitive potassium channels; mathematical model; oscillations


GLUCOSE METABOLISM LEADS TO INSULIN SECRETION in pancreatic {beta}-cells. A consequence of metabolism is an increase in the ratio of ATP to ADP that contributes to depolarization of the plasma membrane (PM) via inhibition of ATP-sensitive K+ (KATP) channels. The subsequent activation of calcium channels and increased intracellular calcium leads to insulin exocytosis (Fig. 1; reviewed in Refs. 3, 17, 53, 63). Glucose rapidly equilibrates across the plasma membrane and is phosphorylated by glucokinase, which determines metabolic flux through glycolysis and ATP production in mitochondria. As a result, the concentrations of ATP and ADP reflect the increased concentration of glucose. The {beta}-cell PM contains KATP channels. In the absence of glucose, enough KATP channels are open to dictate a hyperpolarized resting membrane potential (–60 mV) determined by outward K+ flux. In the presence of elevated glucose, the increase in intracellular [ATP]/[ADP] closes the KATP channels, which in turn results in depolarization of the PM. Once the membrane potential is more positive than about –40 mV, Ca2+ channels open, allowing the influx of extracellular Ca2+. The resulting rise in cytoplasmic Ca2+ concentration then triggers exocytosis of secretory granules containing insulin, and insulin secretion can then be further augmented (66).



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Fig. 1. Schematic diagram of currents and ion fluxes through the plasma membrane (PM) and endoplasmic reticulum (ER) membrane, and mechanisms of adenine nucleotide regulation that have been included in the whole {beta}-cell model. Top: PM: voltage-dependent Ca2+ current (IVCa), a calcium pump current (ICa,pump), Na+/Ca2+ exchange current (INa,Ca), Ca2+ release-activated nonselective cation current (ICRAN), inward Na+ currents (INa), a sodium-potassium pump current (INa,K), a delayed rectifying K+ current (IKDr), the voltage-independent small-conductance Ca2+-activated K+ current (IKCa), and ATP-sensitive K+ current (IKATP); ksg, a coefficient of the sequestration rate of Ca2+ by the secretory granules; Jer,p, uptake of Ca2+ by ER; Jout, Ca2+ leak current from ER. IP3, inositol 1,4,5-trisphosphate, ATP, the free cytosolic form of ATP; ADPf, free cytosolic ADP and MgADP; ADPb, bound cytosolic ADP and MgADP. Solid lines, flux of metabolic substrates; dashed lines, inhibitory effects of ATP and stimulatory influence of free ADP on KATP channel conductance.

 
Changes in [ATP]/[ADP] connect metabolic flux with PM depolarization and Ca2+ flux. However, the mechanisms of adenine nucleotide regulation and interrelationships with other metabolic and ionic fluxes in pancreatic {beta}-cells are incompletely understood. Adenine nucleotide concentrations in pancreatic islets have been determined under a variety of conditions by enzymatic techniques and high-pressure liquid chromatography in cell extracts (25). However, these measurements report their total intracellular level, whereas nucleotides are distributed among various intracellular compartments (e.g., cytosol, mitochondria, secretory granules) with different concentrations and kinetics (19, 41). The concentrations of adenine nucleotides (ATP in particular) in the individual compartments can, in principle, be measured by targeted expression of the ATP-dependent luminescent protein firefly luciferase (2, 50).

Exactly how KATP channels are regulated in vivo by adenine nucleotides is still unresolved. When pancreatic {beta}-cells are exposed to increasing concentrations of glucose the activity of KATP channels decreases (3, 63). However, the ATP concentration required to cause half-maximal inhibition of channel activity is ~10 µM in in vitro experiments. Because intracellular [ATP] ([ATP]i) in {beta}-cells is normally in the millimolar range, essentially no channel activity would occur (63). The other problem is that, in most investigations, the effect of various substrates, including those that markedly enhance insulin secretion (such as glucose) on ATP concentration is relatively small (25, 62). Our studies also lead to this conclusion. Therefore, one might not expect ATP to be a critical physiological regulator of KATP channel activity (63).

On the other hand, intracellular free MgADP stimulates KATP channel activity, and it has been suggested that ADP, or the ATP/ADP ratio, is responsible for channel regulation in vivo (3, 14, 44, 63). There have been few estimates of free MgADP in {beta}-cells, but increasing concentrations of glucose are associated with a decline in the concentration of free ADP in the range that can inhibit KATP channel activity (29, 62). However, the specific regulatory mechanisms for the nucleotides regulating KATP are still unclear.

Changes in [ATP]/[ADP] are tightly coupled to oscillations in intracellular free Ca2+ ([Ca2+]i), oxygen, and glucose consumption in pancreatic {beta}-cells (2, 42). However, intermediate glucose concentrations induce two main types of [Ca2+]i oscillations in pancreatic {beta}-cells: fast, where the period ranges from 10 to 30 s; and slow, with periods of several minutes (30, 32). Fast [Ca2+]i oscillations can follow very small changes in [ATP]i and other components, and they have been difficult to measure and interpret (26, 42). Slow oscillations most likely constitute a physiological oscillatory pattern in {beta}-cells that may constitute the framework for pulsatile insulin release observed in vivo (30, 32). For this reason, we focus here only on the slow oscillations in pancreatic {beta}-cells.

Mathematical analysis of complex systems provides a quantitative framework within which the control of individual processes, cellular fluxes, and metabolite levels can be discussed. Several mechanisms and corresponding mathematical models have been proposed to connect changes in [Ca2+]i, with regulation in cytoplasmic [ATP]/[ADP] and metabolic oscillations. However, the proposed models fall short of a comprehensive explanation of existing data (see Ref. 26 and DISCUSSION). We have recently developed a computational model of the {beta}-cell, where a driving force for slow [Ca2+]i and [ATP]i oscillations is the periodic change in cytoplasmic Na+ concentration (26). Here, we have attempted to uncover the links between the changes in [Ca2+]i, [ATP]/[ADP], KATP channel conductivity, respiration, and glucose consumption by use of a refined model. The results support the idea that {beta}-cells maintain a relatively high [ATP]/[ADP] value even in low glucose and that dramatically decreased free ADP with only modestly increased ATP follows from glucose metabolism. The model was employed to test hypotheses for a pacemaker underlying high-glucose-induced oscillations in intracellular calcium. We found that these can lead to oscillations in nucleotide concentration, supporting a feedback of calcium flux on other metabolic oscillations.


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 ABSTRACT
 METHODS AND MODEL
 RESULTS
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Experimental Procedures

Recombinant baculovirus construction. The recombinant baculoviruses shuttle vector (pFB-IRES-GFP) was derived from pFastBac1 (Life Technologies). Plasmid DNA was digested with SnaBI and NotI to remove the baculovirus Polh promoter sequences and subcloned with a 1977-nucleotide NsiI-NotI fragment from pIRES2-EGFP (Clontech). The cytosolic firefly luciferase from plasmid pGL3 control (Promega) was cloned 2421-nucleotide NheI-BamHI fragment into the pFB-IRES-GFP shuttle vector (L. Ma and L. H. Philipson, unpublished data). Recombinant baculovirus was prepared, amplified, and titrated as described previously (47).

Transfection and cell culture. The islet and dispersed islet cells were isolated from pancreata of 8- to 10-wk-old C57BL/6J mice (Jackson Laboratory, Bar Harbor, ME) using collagenase digestion followed by discontinuous Ficoll gradient centrifugation. The islet cells were dissociated using 0.25 mg/ml trypsin. The cells were then plated on glass coverslips, incubated, and transfected with baculovirus as described (47). Green fluorescent protein (GFP) expression was determined using fluorescence microscopy.

Measurement of intracellular ATP. After transduction with the recombinant baculovirus, the positive infected cells were visualized by GFP detection. The islet cells were cultured at 2 mM glucose with DMEM for 2 h and, then incubated in Krebs-Ringer bicarbonate (KRB) buffer [125 mM NaCl, 5 mM KCl, 1 mM NaH3PO4, 1 mM MgSO4, 1 mM CaCl2, 500 µM luciferin (Molecular Probes), 20 mM HEPES and 2mM glucose, pH 7.4] for 5 min at 37°C. Cell luminescence was measured in a luminometer. Results are expressed as means ± SE unless otherwise stated.

Model Development

Glucose consumption. We propose in the present model that glucose phosphorylation by glucokinase is the only limiting step in glucose consumption in pancreatic {beta}-cells under physiological conditions (51, 54, 68). Glucose is phosphorylated in a sigmoidal fashion, so the Hill equation was used to model this process. The MgATP dependence of this reaction could be well fit to a Michaelis-Menten-type saturation equation (16). Therefore, we employed an empirically derived rate expression for glucokinase from (16)

(1)
where Jglc is the glucokinase reaction rate, [Glc] is extracellular D-glucose concentration, Pglc is the maximum rate of glucose consumption; Km ATP is the Michaelis-Menten constant, KG is the half-maximal glucose concentration, and hgl is the Hill coefficient.

The mean measured KG (at physiological glucose levels and MgATP concentration taken from human {beta}-cells) varies from: 6 mM (72) or 8.17 mM (55) to 8.33 mM (16), and hgl extends from 1.57 (55) or 1.73 (72) to 1.8 (16). Km ATP varies from 0.31 mM (16) or 0.58 mM (55) to 0.63 mM (72). In our model, the coefficient values were fitted to lie inside these bounds (Table 1).


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Table 1. Standard parameter values added or changed in expanded model

 
Reduced metabolic compounds and oxidative phosphorylation. To reduce the complexity of terms, we assume that the glucokinase reaction determines the glycolytic flux, and we employ a term for the total pool of intermediate metabolites available for oxidative phosphorylation. Their lumped synthesis rate is determined by the rate of glucose phosphorylation. ATP production determines the consumption rate of these intermediate metabolites. The total pool of these intermediate metabolites can be described with the kinetic equation

(2)
where [Re]i is the concentration of the intermediate metabolites, KRe is the stoichiometric coefficient for Re production from glucose, and JOP is the oxidative phosphorylation rate, measured as the rate of ATP production. The concentration of Re is conveniently expressed in ATP units; i.e., we assume that one ATP molecule is produced from one molecule of this intermediate metabolite. Then KRe is determined by the quantity of ATP molecules that is produced from one glucose molecule, which equals 31 according to present estimations (64).

Oxidative phosphorylation processes use two kinds of metabolic substrates for ATP synthesis: the reduced equivalents such as NAD(P)H (or FADH2) and free cytosolic MgADP. The dependence of oxidative phosphorylation (JOP) on free MgADP may be calculated using the Hill equation (35, 49). Then, an empirical equation can be written, assuming the simplest linear dependence of reaction rate on [Re]i:

(3)
where [MgADPf]i is concentration of free cytosolic MgADP, POP is the maximum rate of ATP production, KOP is the activation rate constant, and hop is the Hill coefficient.

Recent experimental data suggest that mitochondrial Ca2+ stimulates oxidative phosphorylation (7). However, the data on the possible magnitude of this stimulation are contradictory, because Ca2+ can also have energy-dissipative effects, decreasing oxidative phosphorylation (15, 51, 52). The calculated ATP production rate increased by only 18% following an increase in [Ca2+]i (from 0.02 to 0.6 µM) in a recent model of cardiac mitochondrial energy metabolism (15). For this reason, we do not take into account effects of Ca2+ on the rate of ATP production in our model.

Expressed in terms of [MgADPf]i, the apparent Km of 20 µM was obtained in rat liver mitochondria (8). A reasonable value for the Hill coefficient can be assumed to be in the range of 1.4 or higher, to establish a slightly sigmoidal activation characteristic (35, 49). In our model, these coefficients were fitted as KOP = 20 µM, and hop = 2. The maximum rate POP was fitted to simulate the observed pattern of [Ca2+]i oscillations (Table 1).

ATP and ADP homeostasis. Islets derive >95% of their energy supply from mitochondrial oxidative phosphorylation. The contribution from glycolysis is only ~2% (25). For this reason, the rate of oxidative phosphorylation (Eq. 3) can be used as the ATP production rate. In ATP hydrolysis (as well as in the creatine kinase reaction), the relevant reactants are Mg2+ complexes of the nucleotides. Because the overwhelming proportion of cellular ATP exists as such a complex [>90% in liver (13)], the error in approximating cytosolic MgATP by total cytosolic ATP is not significant, and we consider the cytoplasmic ATP concentration to be the measure of MgATP in this article. The majority of ATP is in the free form in the cytoplasm. However, in contrast to ATP, only a small fraction of total cellular ADP is free (25, 70).

To account for the proposed critical role of ATPases in ATP consumption (see Refs. 20, 40, 67), we previously incorporated equations for ATP consumption by the PM and endoplasmic reticulum (ER) Ca2+ pumps and by the Na+-K+-ATPase (26). Our model also includes a Ca2+-dependent ATP consumption term to account for utilization of ATP during insulin secretion (26). Then, on the basis of Eq. 3 for oxidative phosphorylation and Eq. 27 from Ref. 26, we can write the balance equation for [ATP]i:

(4)
In this expression, INa,K is a sodium-potassium pump current, ICa,pump is a PM calcium pump current, Jer,p is the Ca2+ flux into the ER through Ca2+ pumps per cytosol volume, kATP,Ca is the rate constant of ATP consumption that accelerates as [Ca2+]i increases, and kATP is the rate constant of sustained ATP consumption. F is Faraday's constant and Vi is the cytosol volume. The coefficients were fitted as previously described (26). However, for better simulation of the observed pattern of [Ca2+]i oscillations, we increased kATP,Ca from 0.00005 to 0.00008 µM–1·ms–1, increasing the sensitivity of ATP consumption from [Ca2+]i.

We added the balance equations for free ADP ([ADPf]i) and bound ADP ([ADPb]i) to the previous model, where the terms of free ADP production correspond to the terms of ATP consumption in Eq. 4, and the interaction between [ADPf]i and [ADPb]i was described by linear flux exchange terms:

(5)
and

(6)
where kADPf is the rate constant of ADPb production from ADPf, and kADPf is the rate constant of ADPf production from ADPb.

The total concentration of ATP and ADP is kept constant during experimental stimulation. On the basis of the experimental results by Ghosh et al. (29), we assume that the concentration of free MgADP in pancreatic {beta}-cells is 1/20 that of total cytosolic ADP. To calculate [ADPf]I, we set the free MgADP to 55% of total free ADP as estimated for rat hepatocytes (13). Then [ADPf]i = 1.82x[MgATPf]i, and it can be calculated from Eq. 6 that kADPb/kADPf {approx} 0.1 in steady state.

Several circumstances and reactions can determine the time to establish steady-state concentrations of adenine nucleotide pools in {beta}-cells. In particular, the creatine kinase reaction can be important. However, according to calculations by Ronner at al. (62), the chemical equilibrium in free ADP concentration resulting from the creatine kinase reaction is established within 0.2 s in {beta}HC9 insulin-secreting cells. Although no information could be found regarding the transition time between free and bound ADP and MgADP, this process is not catalytic and is at least diffusion limited. It is unlikely to be fast relative to the creatine kinase reaction. We used kADPb = 0.02 s–1 and kADPf = 0.2 s–1 as a reasonable estimation, where kADPb/kADPf = 0.1.

KATP channels. Free ATP inhibits, whereas free MgADP activates, KATP channels (24). However, as discussed above in the introductory section, the regulation of KATP channels in vivo is not clearly understood, and the current data are inadequate to create a detailed mathematical model of KATP channel regulation. Therefore, we previously used (26) the draft kinetic model (34) as modified (51, 52), where free ATP inhibits, whereas MgADP activates, KATP channels. However, we unmasked this equation to clearly show the dependence of channel opening on specific forms of nucleotides. After rearrangement, the equation for the whole cell conductance of KATP channels (Eq. 31 from Ref. 51), can be represented as

(7)
where OKATP is the fraction of channels open, [ATPf]i is the concentration of free ATP in cytoplasm, Po = 0.08, Pd = 0.89, and Kdd, Ktd, and Ktt are the respective coefficients representing the affinity of components to the channels.

After taking into account the initial coefficient values from Hopkins at al. (34) and the existence of different forms of adenine nucleotides (Table 2 from Ref. 51), we were able to recalculate the coefficients from the work by Magnus and Keizer (51) as: Kdd = 17 µM, Ktd = 26 µM, and Ktt = 20 µM (26). However, in this article, Ktt, representing ATP affinity, was increased from 20 to 50 µM. This was done to incorporate new data showing that phosphorylated inositol compounds can decrease the sensitivity of KATP channels to ATP (3, 4, 24).


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Table 2. PM potential and intracellular ion concentrations at steady state at simulation of low glucose concentration (4.6 mM)

 
The simulated dependence of the open probability on the overall free concentrations of both nucleotides in the physiological range of concentrations is shown in Fig. 2. At physiological ATP concentrations, OKATP is small and does not depend sharply on [ATP]i. This is consistent with the data on the weak dependence of OKATP on ATP (see introductory remarks) and with the evidence suggesting a large excess of KATP channels in {beta}-cells. Therefore, only a small part of the whole KATP channel conductance takes part in the regulation of PM potential at any physiological ATP level (12).



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Fig. 2. Simulated equilibrium fraction of open KATP channels (OKATP) calculated using Eq. 7 with respect to concentration of unbound cytosolic ADP ([ADPf]i) for various concentrations cytosolic ATP ([ATP]i). 1) [ATP]i = 3,000 µM; 2) [ATP]i = 4,000 µM.

 
The simulated data in Fig. 2 correspond to experimental data showing that decreased free ADP concentration in its physiological range can decrease the open probability of KATP channels at constant ATP concentrations (34, 39). In Eq. 7 we propose that a fall in [ADPf]i can be the dominant regulator of the open probability of KATP with an increase in glucose levels as suggested by biochemical studies (24). Recent data (3, 4, 24) show that KATP channel regulation is more complex than was proposed by Hopkins at al. (34). For this reason, Eq. 7 is only a useful approximation of the experimental data.

Computational aspects. The complete system consists of seven state variables that were introduced in the previous model (26) and three new variables ([Re]i, [ADPf]i, and [ADPb]i). In total, 10 differential equations describe their behavior, including Eq. 2 for [Re]i, Eq. 4 for [ATP]i, Eq. 5 for [ADPf]i, and Eq. 6 for [ADPb]i. Equation 7 was used for OKATP. The units and coefficients used in the model are, for the most part, similar to those used in Ref. 26. New and adjusted coefficients are shown in Table 1. The total concentration of intracellular nucleotides ([ATP]i + [ADPf]i + [ADPb]i) is kept constant during simulations, and it is taken as 4 mM (26). For computational purposes, we considered islets as an assemblage of the component {beta}-cells with similar properties and performed computer simulations only for some mean individual cell (26). Simulations were performed as noted previously, using the same software environment (26). A steady state is achieved with time during simulations if no sustained oscillations emerge. 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 name "Chicago.2".


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Free ATP Measurement

Using baculovirus transduction to express firefly luciferase, we estimated cytosolic free ATP in primary islet {beta}-cells by monitoring ATP-dependent luciferase activity in living cells using photon detection (Fig. 3). We found that an increase in glucose concentration from 2 to 14 mM caused only a 19.8 ± 7.9% increase in free ATP levels.



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Fig. 3. Relative intracellular ATP measurements in mouse pancreatic {beta}-cells. Photon output is shown during incubation with 2 mM glucose and after transition to 14 mM glucose. Results are average of 3 independent experiments. Data represent means ± SE.

 
Computer Simulation

Computer simulation at low glucose concentration ([Glc] = 4.6 mM) with the coefficients from Ref. 26 and Table 1 leads to a steady-state [Ca2+]i of ~0.1 µM and [ATP]i/[ADPtot]i ratio close to 3 (Fig. 4, left, and Table 2), where [ADPtot]i is the concentration of intracellular ADP ([ADPtot]i = [ADPf]i + [ADPb]i). The corresponding rate of ATPi production (Jop = 0.218 mM·s–1) was similar to that we reported previously for the simulation of low glucose concentrations (26).



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Fig. 4. Glucose-induced slow electrical bursting and [Ca2+]i oscillations were simulated at a step increase of the glucose concentration [Glc] from 4.6 to 8 mM at arrow in (1) at initial concentration as in Table 2; all other parameter setting are standard (Tables 1 and 2). PNa,K was taken as 400 fA and gmVCa as 670 pS. 1) [Ca2+]i; 2) Membrane potential; 3) [Na+]i; 4) [Re]i; 5) [ATP]i; 6) [ADPf]i; 7) oxidative phosphorylation rate (JOP, Eq. 3); 8) glucokinase reaction rate (Jglc, Eq. 1).

 
Simulations show slow [Ca2+] oscillations when [Glc] is above a threshold level of ~7 mM. A typical computer simulation with the patterns of membrane potential, calculated concentrations, and flux is shown in Fig. 4. As can be seen, a step increase of [Glc] accelerates glucose uptake, increasing [Re]i and oxidative phosphorylation rate. This slightly increases relative [ATP]i (~20%) and sharply decreases [ADPf]i severalfold. This decreased KATP current leads to depolarization of PM and, via opening of voltage-dependent Ca2+ channels, to a rise in [Ca2+]i.

The simulations of slow oscillations correlate well with our previous model (see Figs. 3 and 5 from Ref. 26), reliably simulating the basic characteristics of slow Ca2+ oscillations in pancreatic {beta}-cells. It was also possible to simulate the fast oscillations in this system similarly as in Ref. 26 (by decreasing kIP from 0.3 to 0.1 s–1; not shown).



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Fig. 5. Steady-state simulation of [ATP]i (1) and [ADPf]i (2) vs. [Glc] as calculated using the full {beta}-cell model. All other parameter settings are as in Fig. 4 (dotted lines). For simulation of {beta}-cell lines, voltage-dependent Ca2+ channel conductance was decreased (gmVCa = 450 pS; solid lines).

 
The simulated phase relations in metabolism during slow oscillations are also illustrated in Fig. 4. [Ca2+]i elevation during the active phase of oscillations increases ATP consumption in Ca2+ pumps (PM and ER) and in other reactions (Eq. 4) that increase ADP production. This causes a small decrease in [ATP]i and a corresponding sharp increase in [ADPf]i. Opposite processes occur with a [Ca2+]i decrease during oscillations.

The small oscillations in the glucokinase rate (Eq. 1) were secondary to the oscillations in [ATP]i, since the external [Glc] is fixed. For this reason, the rate of glucose consumption determined by glucokinase follows the [ATP]i changes (Fig. 4, 8), out of phase with [Ca2+]i oscillations.

Variations in the oxidative phosphorylation rate are determined primarily by the [MgADPf]i changes (Eq. 3), and this rate increases with [Ca2+]i increase. Concentrations of Re (Fig. 4, 4) begin to decrease near the midpoint of the [Ca2+]i increase due to consumption of intermediate metabolites for ATP production. As a result, the oxidative phosphorylation rate is in phase with [Ca2+]i oscillations (Fig. 4, 7). However, the rate of oxidative phosphorylation determines the oxygen consumption rate, if oxygen concentration is not limiting. For this reason, oxygen consumption rate (the inverse of O2 concentration inside an islet) should be in phase with [Ca2+]i oscillations.

The simulated steady-state [ATP]i and [ADPf]i shown in Fig. 5 (dotted lines) were generated for different glucose levels. However, slow sustained oscillations of metabolic parameters emerge beginning at 7 mM glucose, after which no steady-state solution is achieved. On the other hand, several measurements of free ADP changes were made in rat pancreatic islets (29, 67) and in {beta}-cell lines (62), where Ca2+ oscillations do not occur or are not as uniform as in mouse islets. Lack of Ca2+ oscillations in {beta}-cell lines can be explained by decreased expression of voltage-dependent Ca2+ channels (26). Incorporating this suggestion, steady-state solutions were also obtained using a decreased Ca2+ channel conductance (gmVCa). This expands the [Glc] interval where adenine nucleotide changes can be simulated in steady-state conditions (Fig. 5, solid lines).


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Regulation of ATP and ADP Levels

In early studies, ATP levels were determined in total cell homogenates (25). However, ATP is concentrated in subcellular domains such as mitochondria or vesicles (19, 41). Luciferase expression can be targeted to specific compartments to measure free ATP (2, 50). Here, we employed recombinant baculovirus to express firefly luciferase in pancreatic {beta}-cells, enabling estimation of cytoplasmic free [ATP] by photon detection.

In one study with luciferase in single human {beta}-cells, the average increase in relative intracellular free ATP, expressed in relative light output, was found to be 9% following a step increase in glucose from 3 to 15 mmol/l (2). A similar increase was also observed in mouse islet {beta}-cells following glucose challenge (2). In living INS-1 insulinoma cells, ATP-dependent luminescence was increased by a mean of 18% following a glucose step from 2.8 to 12.8 mM (50) and 6% following a glucose step from 3 to 16 mM (1). Similarly, a modest increase in free cytoplasmic ATP was found in intact rat islets following glucose challenge (1).

Our data are consistent with these measurements (Fig. 2) and lead to the conclusion that intracellular free ATP concentration increases only modestly with increased glucose concentration in pancreatic {beta}-cells or islets. All these data are consistent with early evidence, obtained using homogenized islets (20, 25).

Measurements of free ADP by Ghosh et al. (29) indicate that, in {beta}-cell-rich rat pancreatic islet cores (with a background of 4 mmol/l amino acid), an increase of glucose from 4 to 8 mmol/l led to a decrease of free MgADP from ~44 to ~31 µM (pooled data from Table 5 of Ref. 29) with no significant change in ATP concentration. Ronner et al. (62) found, for clonal {beta}HC9 insulin-secreting cells, that increased [Glc] was associated with an exponential decline of the concentration of free ADP from ~50 µM at 0 mM glucose to ~5 µM at 30 mM glucose, whereas the concentration of ATP remained nearly constant. Detimary et al. (19) also concluded that glucose induces larger changes in [ATP]/[ADP] in the cytoplasmic pool than in the whole cell and that these changes are largely due to a fall in ADP concentration.

Recently, Sweet et al. (67), evaluated [ATP]i/([ADPf]i [Pi]) in response to glucose, using measurement of cytochrome c redox state and oxygen consumption in perifused isolated rat islets. They found that this ratio increases up to 10-fold following a glucose step increase from 3 to 20 mM. However, the [Pi] change was not studied. In other experiments the [Pi] change was insignificant following glucose increase (29). The [Pi] decreased twofold following a glucose step increase from 3 to 30 mM (Fig. 5A from Ref. 21). Even with a twofold decrease in [Pi], the tenfold increase of [ATP]i/([ADPf]i [Pi]) in response to glucose means that [ATP]i/[ADPf]i increases fivefold.

Our simulations also show only a small relative increase in cytosolic ATP in response to glucose stimulation with a significant fall in relative cytoplasmic [ADPf]i and an increase in [ATP]i/[ADPtot]i (Figs. 4 and 5) that agrees closely with these published data. A small increase in the ATP level following glucose stimulation could reflect the greater consumption of ATP by pancreatic {beta}-cells with increasing glucose concentrations (25). Our model reflects this behavior where an increased ATP consumption occurs with glucose-induced [Ca2+]i increase (Eq. 4) along with increased ATP production. However, the initial abrupt decrease of [ADPf]i and increase in [ATP]i/[ADPtot]i following glucose stimulation in Figs. 4, 6, and 5 requires clarification.

We assumed that the total adenine nucleotide concentration is constant during short-term experiments. In this case, our model leads to a large relative decrease in [ADPtot]i (and corresponding [ADPf]i and [MgADPf]i) compared with the small relative [ATP]i increase with increased glucose, if an initial [ATP]i/[ADPtot]i ratio is considerably more than 1, even at a low glucose level. It is best explained in terms of a simple numerical example [following a brief consideration by Sweet et al. (68)]: for example, if [ATP]i/[ADPtot]i = 3 and [ATP]i + [ADPtot]i = 4 mM for low glucose level in our model (Table 2), then [ATP]i = 3 mM and [ADPtot] = 1 mM. If [ATP]i/[ADPtot]i = 9 for increased glucose, and total adenine nucleotide concentration is kept constant, then [ATP]i = 3.6 mM and [ADPtot]i = 0.4 mM. This means that [ATP]i increases by only 20%, whereas [ADPtot]i (and [MgADPf]i) decreases 2.5-fold. This is a consequence of the initial high [ATP] that cannot be increased significantly if total adenine nucleotide concentration is kept constant, whereas the relative [ADP] may undergo a pronounced decrease. In a contrasting example, if the [ATP] and [ADP] are equal ([ATP]i/[ADPtot]i = 1), then a 20% increase in [ATP]i corresponds to only a 20% decrease in [ADPtot]i.

Our calculations support the suggestion that was previously proposed (see, e.g., Refs. 3, 14, 44, 63, 68), that decreased free ADP can indeed drive closure of KATP channels at increased glucose concentrations, if [ATP]i in {beta}-cells is kept nearly constant (however, we are not addressing the exact KATP nucleotide-binding constants here). This decrease in [ADP] is a specific property of {beta}-cell stimulus-secretion coupling possibly shared with other cell types that have a fuel-sensing function. In contrast to {beta}-cells, no increase in the [ATP]/[ADP] value was found in purified rat {alpha}-cells following an increase in [Glc] (18). Also, muscle work during aerobic exercise leads to increased [ADP] (49).

In addition, our analysis stimulated a search for mechanisms underlying the large decrease in free [ADP] when the relative [ATP] increases only slightly, which, apparently, has not been previously considered in the literature. We suggest that {beta}-cells can achieve this aim simply by keeping the total adenine nucleotide concentration unchanged during a glucose elevation and having a high [ATP]i/[ADPtot]i ratio even at low glucose levels. However, given the lack of appropriate methods to determine absolute values of both cytosolic [ATP]i and [MgADPf]i or their ratio in real time in specific {beta}-cell compartments, this question invites further investigation.

Mechanisms of Metabolic Oscillations

We also sought to identify pacemaker candidates in the interrelationships between Ca2+ and [ATP]/[ADP] oscillations. Using our model, we evaluated three hypotheses relating {beta}-cell calcium and metabolic oscillations. The first hypothesis to consider is that slow [Ca2+]i oscillations are the driving force for metabolic oscillations in pancreatic {beta}-cells. Indeed, there are several theoretical studies and mathematical models proposing that cytoplasmic Ca2+ oscillations can be created independently (9, 10, 28, 30, 31, 60, 65, 73) and that thereafter they can stimulate metabolic oscillations (10, 30, 61). For example, the oscillations of Ca2+ in ER could be the pacemaker of cytoplasmic Ca2+ oscillations (9, 28, 60, 73). We also suggested in our model that independent [Ca2+]i oscillations can drive slow [ATP]/[ADP] changes and corresponding metabolic oscillations in {beta}-cells (Fig. 4; see also Ref. 26). In this case, slow [Ca2+]i oscillations can evoke IKATP oscillations through oscillations in [ATP]/[ADP] as was proposed (61).

We have also found that essentially any model of independent slow Ca2+ oscillations (e.g., Refs. 9, 28, 31, 65, 73), when coupled with Eqs. 16, will lead to corresponding ATP, ADP, and metabolic oscillations. For this reason, our explanation of the processes that underlie metabolic oscillation does not limit the use of this model to the generation of slow Ca2+ oscillations. The results from our modeling can be considered as a general characteristic of metabolic oscillations when Ca2+ oscillations are a driving force and increased [Ca2+]i during oscillations leads to increased ATP consumption.

Our model is in good agreement with recently published studies favoring this first hypothesis. For example, simultaneous measurements of oxygen and glucose consumption, the processes tightly coupled with adenine nucleotide regulation and [Ca2+]i during glucose-stimulated oscillations, showed that glucose consumption rate was out of phase with slow [Ca2+] oscillations (37). Oxygen consumption rate and [Ca2+]i changes were approximately in phase (37). This means that increased [Ca2+]i during oscillations is accompanied by a decreased glucose consumption rate and by an increased respiration rate. The mechanism of this phenomenon is as yet unknown (42). However, these data are in accord with our model simulation (RESULTS, Fig. 4) and could be explained by a decrease in ATP and an increase in free ADP concentrations with [Ca2+]i increase during the appropriate phase of slow oscillations. However, [Ca2+]i and changes in [ATP]i/[ADP]i have not yet been measured simultaneously during oscillations.

In intact INS-1 insulinoma cells, citrate and ATP oscillations are in phase with each other (48). Citrate changes can be evaluated in our model through a variation of the total pool of intermediate metabolites (Re) that oscillates in phase with ATP (Fig. 4). This can be explained by an increased consumption of the reduced molecules for oxidative phosphorylation in phase with [Ca2+]i increases during oscillations (see RESULTS).

Glucose-induced NAD(P)H and [Ca2+]i slow oscillations were measured simultaneously in mouse pancreatic islets (46). It was found that these oscillations were nearly in phase, although NAD(P)H oscillations preceded those of calcium by ~0.1 of a period. In our model, NAD(P)H concentration is included as a component of the total pool Re, as is citrate. Similarly, as is the case for citrate, such NAD(P)H oscillations ([Re]i changes in the model) are nearly in phase with [Ca2+]i slightly preceding [Ca2+]i changes (Fig. 4), also in reasonable correspondence with the data.

The concept that slow metabolic oscillations could be driven by cytoplasmic Ca2+ oscillations has previously been dismissed on the basis of the proposal that such a scenario would contradict observations that increases in metabolism lead to Ca2+ influx and insulin secretion (69). Indeed, the [ATP]/[ADP] ratio (11, 20), NADH levels (59), and respiration (25, 37) all increase before an increase in [Ca2+]i following glucose stimulation. At first glance, these data are in contradiction with our suggestion.

However, as was pointed out by Kennedy et al. (42), creation of oscillations by changes in metabolism does not prove that these metabolic variations are the real pacemaker of oscillations. We have shown in Fig. 4 (left) that our simulated glucose challenge leads at first to a slow increase in [ATP]i, [Re]i, and oxidative phosphorylation rate and to a decrease in free ADP and cytoplasmic Na+ concentration ([Na+]i) before a creation of Ca2+ oscillations; i.e., this behavior corresponds to the experimental data. However, these slow processes are necessary to depolarize the PM to the threshold for calcium influx when slow Ca2+ oscillations emerge. Then, the driving force for slow [Ca2+]i oscillations is the periodic changes of some mediator (e.g., [Na+]i in our model). Our calculations show clearly that, during oscillations, the opposite changes in concentrations of some metabolites could occur compared with initial response to external stimulation. For example, [ATP] decreases and free [ADP] increases with [Ca2+]i increase during oscillations (Fig. 4); i.e., this is opposite to the changes during the initial glucose-induced [Ca2+]i increase. On the other hand, [Re]i increase is accompanied by [Ca2+]i increase both during oscillations and after simulation of glucose increase (Fig. 4, 4). This shows clearly that data obtained by changing metabolism should be used with care for interpretation of the oscillation processes.

In conclusion, our mathematical modeling shows that slow [Ca2+]i oscillations can be the driving force for metabolic oscillations in pancreatic {beta}-cells. We can also point out that, although the mechanisms underlying Ca2+ oscillations, which are independent of [ATP]/[ADP] value changes, were considered in several mathematical models (9, 28, 65, 73), these did not include a detailed analysis connecting metabolic changes with Ca2+ oscillations.

The second hypothesis suggests that some metabolic pathways serve as a pacemaker and inherently oscillate to give rise to oscillations in [ATP]/[ADP], [Ca2+]I, and respiration (5, 38, 58, 69). For example, oscillations in glycolysis could lead to [ATP]/[ADP] oscillations that influence KATP channel conductance serving as a pacemaker of slow Ca2+ oscillations (6, 69, 71). This possibility was recently analyzed in detail using mathematical modeling (6, 71).

The third hypothesis assumes an interaction among the [ATP]i/[ADPf]i, [Ca2+]I, and KATP channels as the mechanism underlying oscillatory behavior of {beta}-cells. Here, [Ca2+]i increase during the active phase leads to [ATP]i/[ADPtot]i decrease via [Ca2+]i-induced decreases in ATP production (43, 51, 52) or by an increase in ATP consumption (2, 20, 56). In turn, changes in [ATP]/[ADP] cause decreased KATP channel conductance, leading to PM repolarization and Ca2+ channel closing. [Ca2+]i decrease during a resting phase of oscillations acts in the opposite direction, stimulating a new cycle. This mechanism can create sustained Ca2+ oscillations (2, 20, 51, 52, 56).

However, in the most computationally developed model, by Magnus and Kaizer (51, 52), that used this hypothesis, it was suggested "that the uptake of Ca2+ by {beta}-cell mitochondria suppressed the rate of production of ATP via oxidative phosphorylation" (51). Recent experimental data contradict this conclusion and favor the opposite. For example, in a recent review it was pointed out that "the primary role of mitochondrial Ca2+ is the stimulation of oxidative phosphorylation" (7). Decreased oxidative phosphorylation following [Ca2+]i increase in the model of Magnus and Kaiser (51, 52) leads directly to the conclusion that oxygen consumption should decrease with increased [Ca2+]i during the active phase of oscillations if we incorporate the experimental evidence that oxygen consumption accounts for the oxidative phosphorylation rate in vivo. However, this conclusion seems contrary to the experimental evidence by Jung et al. (37) considered above, that oxygen consumption increases with increased [Ca2+]i during slow oscillations. Decreased ATP production with increased Ca2+ was also used in other model (6), casting some doubt on the result of these simulations.

The second and third hypotheses above propose that regulation of KATP channel conductance by [ATP]/[ADP] changes play a decisive role in driving slow Ca2+ oscillations. However, KATP channel blockers such as tolbutamide can create slow bursting and Ca2+ oscillations at low glucose concentrations in pancreatic islets (33, 45), in single {beta}-cells, or in {beta}-cell clusters isolated from mouse islets (23, 36), and in {beta}TC3-neo cells (60). The existence of slow Ca2+ oscillations was found in SUR1–/– knockout mouse lacking functional KATP channels (22). These results using KATP channel blockers and knockout mouse models argue against an important role of KATP channel conductivity in slow Ca2+oscillations. This contradicts the second and third hypotheses, where Ca2+ oscillations can be created only if oscillations in [ATP]/[ADP] lead to oscillations in conductance of KATP channels. This mechanism could not work if KATP channels are blocked. Furthermore, stimulation of slow Ca2+oscillations by KATP channel blockers at low glucose level is opposite to the proposal that oscillations in glycolysis could lead to [ATP]/[ADP] oscillations that influence KATP channel conductance (second hypothesis), since, apparently, these oscillations could not exist at low glucose levels (6, 71). Because the second and third hypotheses cannot explain these critical experiments with KATP channel blockers, we believe that they are incorrect, and they will not be further considered here.

However, it should be pointed out that mainly fast oscillations were considered in many models (28, 51, 52, 56). According to Kanno et al. (40), KATP channel modulation could take part in the creation of {beta}-cell fast [Ca2+]i oscillations, which we do not consider in this article (see introductory remarks). This means that appropriate simulation of fast metabolic and Ca2+ oscillations on the basis of the second and third hypotheses is possible; however, it is not taken into consideration here.

In our model, slow Ca2+ oscillations can be still simulated with considerable decrease in KATP conductance that is accompanied by the corresponding shift of glucose level from which a simulation of oscillations takes place to a region of lower glucose concentration. For example, a 10-fold decrease of KATP conductance (from 24 to 2.4 nS) still permits us to simulate the slow Ca2+ oscillations at 8 mM glucose; however, the concentration of glucose from which these oscillations could be created was shifted from 7 to 4.2 mM (not shown). These simulations correlate well with experimental data on KATP channel blocker action described above. This effect is possible in our model because KATP channels serve as a trigger of Ca2+ oscillations at their closing, not as a pacemaker. In this case, closing of KATP channels by specific blockers at low glucose levels should create oscillations similar to those caused by increased glucose [as, for example, we have demonstrated in experiments (60)].

In conclusion, by employing mouse {beta}-cells expressing firefly luciferase, we have found that a glucose challenge caused only an ~20% increase in free ATP levels, confirming previous measurements and highlighting the potential importance of free ADP in glucose signaling. The integrated {beta}-cell mathematical model that we have developed reproduces key experimental relationships among ATP, ADP, and cytoplasmic Ca2+ changes. Variations in steady-state concentrations after glucose challenge are also in good agreement with published data. We used the model to test the proposal that slow Ca2+ oscillations are the driving force of metabolic oscillations in pancreatic {beta}-cells. We found that considerable experimental data on oscillations in glucose consumption, metabolites, mitochondrial respiration, and ATP concentrations were also reflected in the simulations from the mathematical model. This analysis supports the hypothesis that ATP, and particularly free ADP, can be the critical regulators of glucose-stimulated calcium flux. Because most ATP production (≤95%) occurs in mitochondria, this view supports the recent suggestion that subtle variation in mitochondrial function could underlie {beta}-cell defects in type 2 diabetes (57). The glucose-dependent changes in [ATP]/[ADP] levels could also underlie the development of oxidative stress in pancreatic {beta}-cells (27).


    GRANTS
 TOP
 ABSTRACT
 METHODS AND MODEL
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
This work was partially supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-48494, DK-20595, and DK-44840.


    ACKNOWLEDGMENTS
 
We thank Yong Wang and Yingli Quan for technical support.


    FOOTNOTES
 

Address for reprint requests and other correspondence: L. H. Philipson, Dept. of Medicine, MC-1027, The Univ. of Chicago, 5841 S. Maryland Ave., Chicago, IL 60637 (e-mail: l-philipson{at}uchicago.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


    REFERENCES
 TOP
 ABSTRACT
 METHODS AND MODEL
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 

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