MODELING IN PHYSIOLOGY
Model of beta -cell mitochondrial calcium handling and electrical activity. I. Cytoplasmic variables

Gerhard Magnus1 and Joel Keizer1,2

1 Institute of Theoretical Dynamics and 2 Section on Neurobiology, Physiology, and Behavior, University of California, Davis, California 95616

    ABSTRACT
Top
Abstract
Introduction
Discussion
Appendix A
Appendix B
References

We continue our development of a kinetic model of bursting electrical activity in the pancreatic beta -cell ( J. Keizer and G. Magnus. Biophys. J. 56: 229-242, 1989), including the influence of Ca2+ handling by the mitochondria. Our minimal model of mitochondrial Ca2+ handling [G. Magnus and J. Keizer. Am. J. Physiol. 273 (Cell Physiol. 42): C717-C733, 1997] is expanded to include the D-glucose dependence of the rate of production of mitochondrial reducing equivalents. The Ca2+ dependence of the mitochondrial dehydrogenases, which is also included in the model, plays only a small role in the simulations, since the dehydrogenases appear to be maximally activated when D-glucose concentrations are sufficient to produce bursting. A previous model of ionic currents in the plasma membrane is updated using a recent experimental characterization of the dependence of the conductance of the ATP-sensitive K+ (KATP) current on adenine nucleotides. The resulting whole cell model is complex, involving 12 dynamic variables that couple Ca2+ handling in the cytoplasm and the mitochondria with electrical activity in the plasma and inner mitochondrial membranes. Simulations with the whole cell model give rise to bursting electrical activity similar to that seen in pancreatic islets and clusters of pancreatic beta -cells. The full D-glucose dose response of electrical activity is obtained if the cytosolic rate of ATP hydrolysis is a sigmoidal function of glucose. The simulations give the correct shape, period, and phase of the associated oscillations in cytosolic Ca2+, predict that the conductance of the KATP current oscillates out of phase with electrical activity [as recently observed in ob/ob mice (O. Larsson, H. Kindmark, R. Bränstrom, B. Fredholm, and P.-O. Berggren. Proc. Natl. Acad. Sci. USA 93: 5161-5165, 1996)], and make other novel predictions. In this model, bursting results because Ca2+ uptake into mitochondria during the active phase reduces the mitochondrial inner membrane potential, reducing the rate of production of ATP, which in turn activates the KATP current and repolarizes the plasma membrane.

pancreatic beta -cell; electrical activity; adenosine triphosphate-sensitive potassium channel

    INTRODUCTION
Top
Abstract
Introduction
Discussion
Appendix A
Appendix B
References

WHEN SUBJECTED to >5-8 mM D-glucose, pancreatic beta -cells from a wide range of species exhibit a complicated pattern of electrical activity (5, 13, 49). In an intermediate range of D-glucose concentrations, bursts of action potential spikes (the "active" phase) are observed separated by a "silent" phase, during which the membrane repolarizes. At even higher glucose concentrations, continuous, uninterrupted action potentials are seen. This type of electrical activity has been observed in clusters of dissociated beta -cells using patch electrodes (27) and in both microdissected islets and intact islets in the pancreas using microelectrodes (49). This electrical activity has two important physiological correlates: increased cytosolic Ca2+ concentration ([Ca2+]i) (50) and increased rate of insulin secretion during the active phase (5). It is generally accepted that the rise in [Ca2+]i plays a major role in insulin secretion and that the action potential spikes during a burst are responsible for the rise in [Ca2+]i.

Because the glucose signal for insulin secretion operates via metabolism rather than through a plasma membrane-bound receptor, the details of how glucose stimulates electrical activity have been difficult to resolve. Nonetheless, the discovery and subsequent characterization of an ATP-sensitive K+ current (IKATP) in the beta -cell (3, 8) have suggested that both ATP and ADP may be responsible for transduction of the D-glucose signal. Increases in ATP and decreases in ADP concentrations associated with D-glucose metabolism have been proposed to depolarize the plasma membrane of the beta -cell by inactivating IKATP (4). Experiments and models (25) suggest that this depolarization suffices to activate the outward delayed rectifier K+ current and inward Ca2+ currents that are responsible for the action potential spikes. Although there appears to be broad consensus about this role for IKATP in the glucose signal, general agreement about which cellular processes control the repolarization of the burst is lacking. This is an important, unresolved issue because of the correlation of the duration of the active phase with the rise in [Ca2+]i and insulin secretion.

Here we continue our exploration (25) of one hypothesis that could explain the repolarization: that the uptake of Ca2+ by beta -cell mitochondria suppresses the rate of production of ATP via oxidative phosphorylation, which subsequently activates IKATP and repolarizes the burst. Previously, we argued the plausibility of this hypothesis using a kinetic model of electrical activity in the beta -cell combined with an extremely simplified model of the influence of Ca2+ on the production of ATP (25). Here we take a similar approach, but now we use a much more complete model that is based on six key mechanisms involved in mitochondrial Ca2+ handling (32). To this model we have added several more refinements. First, we have included the D-glucose dependence of the production of NADH based on the control of glycolysis by glucokinase. Second, we have included the Ca2+ stimulation of respiration due to two key mitochondrial dehydrogenases, pyruvate dehydrogenase (PDH) and glycerol phosphate dehydrogenase (GPDH). Third, we have updated our model of regulation of IKATP using the data of Hopkins et al. (22).

Recently, the endoplasmic reticulum (ER) has been implicated in agonist-induced electrical activity in beta -cells (6, 24, 60). However, glucose itself appears to induce only a transient increase in ER Ca2+ uptake in beta -cells. Furthermore, although Ca2+ uptake into the ER occurs via sarcoplasmic reticulum Ca2+-ATPase-type pumps, ATP is not rate determining for the pumps under physiological conditions (24). For these reasons, we have chosen not to include ER Ca2+ handling in the model developed here, focusing instead on the interactions between plasma membrane and mitochondrial Ca2+ handling.

Simulations with the model, described in subsequent sections, support this hypothesis. As we show, its validity is dependent on two other conditions: 1) that maximal stimulation of the mitochondrial dehydrogenases occurs rapidly and near the D-glucose threshold for electrical activity and 2) that the rate of ATP hydrolysis in the cytosol is an increasing sigmoidal function of the D-glucose concentration. The model makes a variety of other predictions that should be amenable to experimental tests. Included among these is that oscillations in the conductance of the IKATP should accompany electrical activity and [Ca2+]i oscillations during bursting. Experimental evidence for this type of behavior in clusters of beta -cells has recently appeared (27). This and other predictions of the model for cytoplasmic activity are also described. The companion article in this series (33) is devoted to the behavior of mitochondrial variables during bursting.

A complete summary of the equations used in the model is given in Ref. 31 and can be found on our website (http://www.itd.ucdavis.edu/odegallery/).

    MITOCHONDRIAL REDUCING EQUIVALENTS FROM D-GLUCOSE

Here we use the stoichiometry involved in glucose metabolism to express the rates of production of reducing equivalents for mitochondrial respiration. The main pathways of beta -cell metabolism involved in mitochondrial respiration are illustrated in Fig. 1. Five fluxes, all of which have been measured using isotopic labeling experiments (7), are indicated explicitly and expressed in terms of D-glucose concentration units: the D-glucose utilization rate (Delta Jgly,total where gly represents glycolysis), the lactate dehydrogenase rate (Delta Jgly,anaerobic), the PDH rate (Delta JPDH), and an average rate for the tricarboxylic acid (TCA) cycle (Delta JTCA). (Here the "Delta " notation is used to represent the contribution of the flux due exclusively to D-glucose metabolism.) Note that the TCA cycle produces NADH, which enters the mitochondrial respiratory chain at complex I, and FADH2, which enters respiration at complex II. NADH for complex I is also made by PDH, whereas the glycerol phosphate shunt transfers cytosolic reducing equivalents to the mitochondrial respiratory enzymes at complex II.


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Fig. 1.   Fluxes of D-glucose metabolism (Delta Jgly) as related to generation of reducing equivalents presented to complexes I and II of respiratory chain. Numbers of oxidized (oxd) and reduced (red) species are relative to a single D-glucose molecule broken down along each corresponding pathway at indicated rate. Delta JTCA, rate for tricarboxylic acid cycle; Delta JPDH, rate for pyruvate dehydrogenase.

On the basis of the stoichiometry in Fig. 1 and the assumption of a quasi-steady state for D-glucose metabolism, we can write expressions for the rate of production of NADH and FADH2 in terms of the D-glucose utilization rate. Using the fact that glucokinase is the proximal metabolic "glucose sensor" (16, 40), we then express the rate of production of reducing equivalents in terms of the rate of glucokinase phosphorylation of D-glucose. Our quasi-steady-state assumption precludes the possibility of spontaneous oscillations in glycolysis that have been proposed to play a role in the beta -cell (29). Nonetheless, we are unaware of evidence for glycolytic oscillations in islets on the time scale of a burst.

The quasi-steady-state assumption and the stoichiometry in Fig. 1 imply that the rate of production of NADH at complex I from D-glucose can be written
&Dgr;<IT>J</IT><SUB>red,I</SUB> = 2&Dgr;<IT>J</IT><SUB>PDH</SUB> + 6&Dgr;<IT>J</IT><SUB>TCA</SUB> (1)
Similarly the rate of production of FADH2 and FMNH2 for complex II due to mitochondrial succinate dehydrogenase (SDH) and cytosolic GPDH is
&Dgr;<IT>J</IT><SUB>red,II</SUB> = &Dgr;<IT>J</IT><SUB>SDH</SUB> + &Dgr;<IT>J</IT><SUB>shunt</SUB> (2)
where Delta Jshunt is the rate of the glycerol phosphate shunt. We then neglect any efflux of pyruvate from islets and write the SDH rate in terms of D-glucose concentration units (Delta JSDH = 2Delta JTCA). Making the approximations that 1) all beta -cell D-glucose is metabolized to either pyruvate or lactate (Delta Jgly,total = Delta Jgly,anaerobic + Delta JPDH) and 2) the NADH produced in the cytosol at steady state either reduces pyruvate or enters the glycerol phosphate shunt (Delta Jshunt = 2Delta Jgly,total - 2Delta Jgly,anaerobic) allows Eq. 2 to be rewritten as
&Dgr;<IT>J</IT><SUB>red,II</SUB> = 2&Dgr;<IT>J</IT><SUB>TCA</SUB> + 2&Dgr;<IT>J</IT><SUB>PDH</SUB> (3)

To simplify Eqs. 1 and 3, we rewrite the TCA cycle flux in terms of the PDH flux using their experimental ratio in rat islets. Measurements of 14CO2 output show that the ratio Delta JTCA/Delta JPDH remains constant at substimulatory and maximum concentrations of labeled D-glucose (7)
&Dgr;<IT>J</IT><SUB>TCA</SUB> = 0.42&Dgr;<IT>J</IT><SUB>PDH</SUB> (4)
Thus writing Delta JTCA in terms of Delta JPDH in Eqs. 1 and 3 gives
&Dgr;<IT>J</IT><SUB>red,I</SUB> = 4.52&Dgr;<IT>J</IT><SUB>PDH</SUB> (5)
&Dgr;<IT>J</IT><SUB>red,II</SUB> = 2.84&Dgr;<IT>J</IT><SUB>PDH</SUB> (6)

We can then eliminate Delta JPDH in these expressions in favor of Delta Jgly,total, the D-glucose utilization rate. Indeed, labeling experiments show that Delta JPDH is ~0.28 of Delta Jgly,total at 2.8 mM D-glucose and increases to ~0.48 at 16.7 mM (7). These results are consistent with independent measurements of the TCA cycle and enolase reaction fluxes (7, 51, 52). We express the increase in the PDH rate using a function f ([Glc]) of the concentration of D-Glucose in the external medium ([Glc]) that increases from 0.28 to 0.48 when [Glc] increases from 2.8 to 16.7 mM. Thus
&Dgr;<IT>J</IT><SUB>PDH</SUB> = <IT>f</IT>([Glc]) &Dgr;<IT>J</IT><SUB>gly,total</SUB> (7)
or using Eqs. 5 and 6
&Dgr;<IT>J</IT><SUB>red,I</SUB> = 4.52 <IT>f</IT>([Glc]) &Dgr;<IT>J</IT><SUB>gly,total</SUB> (8)
&Dgr;<IT>J</IT><SUB>red,II</SUB> = 2.84 <IT>f</IT>([Glc]) &Dgr;<IT>J</IT><SUB>gly,total</SUB> (9)
The form of f ([Glc]) is given in CALCIUM DEPENDENCE OF NADH PRODUCTION, where we propose that it is due to the activation of key dehydrogenase enzymes by Ca2+.

Using Eqs. 8 and 9, we can now express the rates of production of reducing equivalents in terms of the concentration of D-glucose applied to an islet. This requires two observations: 1) that D-glucose transport across the plasma membrane is not rate limiting for glycolysis (35, 38), so that D-glucose concentrations inside the beta -cell and in the external medium are essentially the same, and 2) that the in vitro kinetic properties of glucokinase are nearly identical to those of D-glucose utilization in islets (16). Thus we can equate Delta Jgly,total to the empirical expression for the rate of glucokinase (1, 16)
&Dgr;<IT>J</IT><SUB>gly,total</SUB> = <FR><NU>&bgr;<SUB>max</SUB>⋅(1 + &bgr;<SUB>1</SUB>[Glc])⋅[Glc][ATP]<SUB>i</SUB></NU><DE><AR><R><C>1 + &bgr;<SUB>3</SUB>[ATP]<SUB>i</SUB> + (1 + &bgr;<SUB>4</SUB>[ATP]<SUB>i</SUB>)</C></R><R><C> ⋅&bgr;<SUB>5</SUB>[Glc] + (1 + &bgr;<SUB>6</SUB>[ATP]<SUB>i</SUB>)⋅&bgr;<SUB>7</SUB>[Glc]<SUP>2</SUP></C></R></AR></DE></FR> (10)
where [ATP]i is intracellular ATP concentration. The parameter beta max in Eq. 10 is the maximum rate, which we base on experimental values for Delta Jgly,total (~130 pmol · h-1 · islet-1) (61). Its value in Table 3 also includes a factor of 1/0.09 to convert cytosolic units of millimolar per minute to mitochondrial units of nanomoles per minute per milligram of protein used for all metabolic fluxes (see APPENDIX A for unit conversion factors). Note that the ATP dependence of the D-glucose utilization rate is extremely weak and raises Delta Jgly,total by a maximum of ~3% for saturating [Glc] of 25 mM and [ATP]i in the physiological range (1.5-2.0 mM). Substituting the expression for Delta Jgly,total from Eq. 10 into Eqs. 8 and 6 gives the D-glucose dependence of the rate of production of reducing equivalents for mitochondrial respiratory complexes I and II.

    CALCIUM DEPENDENCE OF NADH PRODUCTION

Although glucose is known to raise ([Ca2+]i) in the beta -cell, a direct connection between such elevations and the redox state of mitochondrial NAD in islets has not been demonstrated. However, the addition of D-glucose to intact islets has been shown to activate PDH (39), whereas the similar stimulation of beta -cell clusters increases both [Ca2+]i and pyridine nucleotide autofluorescence, the latter derived predominantly from mitochondrial NADH (15). The assumption of Ca2+ uptake by mitochondria as an intermediate step in the amplification of NADH production is supported by numerous experiments using organelle preparations from heart and liver cells, where the sensitivity of dehydrogenase activation to [Ca2+]i is related to the external concentrations of Na+, spermine, and other effectors of Ca2+ transport across the inner membrane (35, 38). Stimulated increases in mitochondrial Ca2+ concentration ([Ca2+]m) have been measured for mitochondria of the insulin-secreting cell line INS-1 in situ (47), and other evidence suggests that mitochondrial sequestration of Ca2+ uptake is a reasonable consequence of parallel cytosolic increases (see discussion in Ref. 32).

The PDH complex of the mitochondrial matrix catalyzes the net reaction
pyruvate + CoA + NAD<SUP>+</SUP> = acetyl-CoA
 + CO<SUB>2</SUB> + NADH + H<SUP>+</SUP> (11)
The enzyme PDH, which catalyzes the initial decarboxylation step, has an active form (PDHa), which becomes completely inactivated when phosphorylated. Interconversion between these two forms is controlled by a kinase and phosphatase, i.e.
PDH<SUB>a</SUB> + MgATP<SUP>2−</SUP>
 <AR><R><C>PDH<SUB>a</SUB> kinase</C></R><R><C>→</C></R><R><C>←</C></R><R><C>PDH-P phosphatase</C></R></AR> PDH-P + MgADP<SUP>−</SUP> (12)
where the relative rates of PDHa kinase and PDH phosphate (PDH-P) phosphatase determine the fraction of activated PDH ( fPDHa) and, hence, set the maximum rate for the decarboxylation of pyruvate at steady state (44).

The products acetyl-CoA and NADH of the PDH reaction activate PDHa kinase and competitively inhibit PDHa. However, these effects are strongest in state 4 mitochondrial preparations and may be neglected for phosphorylating mitochondria in situ (21). We assume, in addition, that inhibition of PDHa kinase by pyruvate is negligible in the beta -cell, as was shown to be the case for heart mitochondria respiring at 50% of their maximal state 3 rates. In the latter experiments, 100-500 µM pyruvate increased the flux of acetyl-CoA formation from 55 to 72%. Because pyruvate reaches levels as high as 1.69 pmol/islet approx  525 µM (2), it is reasonable to assume that pyruvate exerts the strongest effect of all the PDH reaction metabolites on the rate of Eq. 11 in the beta -cell (21). This is expressed through the D-glucose dependence of Delta Jgly,total in the model (Eqs. 8 and 9).

The dephosphorylation of PDH-P proceeds with approximately first-order kinetics and is stimulated by both Mg2+ and Ca2+ (56). Because both of these cations also affect mitochondrial Ca2+ transport, their independent actions in determining the PDH-P phosphatase rate must be established from experiments in which the inner membrane permeability is not a factor. Isolated PDH-P phosphatase is completely inhibited in the absence of Mg2+ (14), whereas in toluene-permeabilized fat cell mitochondria, a sigmoidal Mg2+ dependence persists for 1 nM to 100 µM Ca2+ (56). These results suggest that Mg2+, rather than Ca2+, is the primary effector.

If it is assumed that the properties of PDH-P phosphatase in situ are roughly similar to those measured in extracts, the enzyme's K0.5 for Mg2+ in the absence of Ca2+ is ~2-3 mM, with a Hill constant of 1.5-2.5 (44, 56). The concentration of free Mg2+ in the matrix is ~0.35 mM (11) and, therefore, subsaturating. A reasonable approximation for the reaction flux ( Jphos) is then first order in the PDH-P concentration, with the rate constant determined by the rapid equilibrium binding of Mg2+. If Jphos,max is the maximum reaction velocity when fPDHa = 0, then
<IT>J</IT><SUB>phos</SUB> = <FR><NU><IT>J</IT><SUB>phos,max</SUB></NU><DE>1 + <FENCE><FR><NU><IT>K</IT><SUB>Mg<SUP>2+</SUP></SUB></NU><DE>[Mg<SUP>2+</SUP>]<SUB>m</SUB></DE></FR></FENCE><SUP>2</SUP></DE></FR>⋅(1 − <IT>f</IT><SUB>PDH<SUB>a</SUB></SUB>) (13)
where [Mg2+]m is mitochondrial Mg2+ concentration. Because Ca2+ is believed to enhance binding of PDH-P phosphatase to the PDH phosphorylation site, a Ca2+-dependent increase in Jphos can be expressed through an elevation of the affinity for Mg2+ (14, 56). If KMg2+,max corresponds to the absence of Ca2+ and KCa2+ is the concentration constant producing half the maximum Ca2+-dependent increase in the affinity of PDH-P phosphatase for Mg2+, then
<IT>K</IT><SUB>Mg<SUP>2+</SUP></SUB> = <FR><NU><IT>K</IT><SUB>Mg<SUP>2+</SUP>,max</SUB></NU><DE>1 + <FR><NU>[Ca<SUP>2+</SUP>]<SUB>m</SUB></NU><DE><IT>K</IT><SUB>Ca<SUP>2+</SUP></SUB></DE></FR></DE></FR> (14)
Defining
<IT>u</IT><SUB>1</SUB> = <FENCE><FR><NU><IT>K</IT><SUB>Mg<SUP>2+</SUP>,max</SUB></NU><DE>[Mg<SUP>2+</SUP>]<SUB>m</SUB></DE></FR></FENCE><SUP>2</SUP> (15)
and combining Eq. 15 with Eqs. 13 and 14 gives
<IT>J</IT><SUB>phos</SUB> = <FR><NU><IT>J</IT><SUB>phos,max</SUB></NU><DE>1 + <FR><NU><IT>u</IT><SUB>1</SUB></NU><DE><FENCE>1 + <FR><NU>[Ca<SUP>2+</SUP>]<SUB>m</SUB></NU><DE><IT>K</IT><SUB>Ca<SUP>2+</SUP></SUB></DE></FR></FENCE><SUP>2</SUP></DE></FR></DE></FR>⋅(1 − <IT>f</IT><SUB>PDH<SUB>a</SUB></SUB>) (16)

Like Ca2+, spermine does not modulate the PDH-P phosphatase rate at saturating levels of Mg2+, suggesting that its effects on the enzyme are also indirect (12, 56). Spermine, which is present at high concentrations in beta -cells, also acts independently of its role in the regulation of inner membrane Ca2+ transport (31, 32), lowering the range of KMg2+,max (Eq. 15) to 1-2 mM in mitochondrial extracts (56). Because [Mg2+]m approx  0.35 mM (11), it is reasonable to set u1 = 15 in Eq. 16. Also in Eq. 16, a hypothetical value for KCa2+ of 0.04 pmol/mg protein approx  0.05 µM (see APPENDIX A) produces a half-maximal free Ca2+ concentration of 0.15 µM for Ca2+ activation of PDH-P phosphatase. This simulated result is about an order of magnitude above that observed for uncoupled mitochondria and extracts (37). However, those experiments exclude spermine, which tends to raise the affinity of PDH-P phosphatase for Mg2+ and indirectly increase its stimulation by Ca2+.

PDHa kinase, unlike PDH-P phosphatase, is tightly bound to the PDH complex and not affected by Mg2+ or Ca2+ (56). Although PDHa kinase is inhibited by ADP acting competitively with ATP (10), this factor has not been included in the regulation of the enzyme, since the mitochondrial ATP-to-mitochondrial ADP concentration ratio ([ATP]m/[ADP]m) has been shown to increase only negligibly in excited islets stimulated by D-glucose (53). The time course of PDHa phosphorylation at fixed agonist concentrations displays roughly first-order kinetics similar to those of the PDH-P phosphatase reaction (56). A reasonable approximation of the PDHa kinase rate is then
<IT>J</IT><SUB>kin</SUB> = <IT>J</IT><SUB>kin,max</SUB>⋅<IT>f</IT><SUB>PDH<SUB>a</SUB></SUB> (17)
where Jkin,max is the maximal rate of the kinase. Because the rate of activation of PDH appears to be rapid, we have treated the equilibration of the active and inactive forms of PDH in Eq. 12 as instantaneous. Thus we equate Jkin and Jphos to obtain
<IT>f</IT><SUB>PDH<SUB>a</SUB></SUB> = <FR><NU>1</NU><DE>1 + <IT>u</IT><SUB>2</SUB><FENCE>1 + <IT>u</IT><SUB>1</SUB> <FENCE>1 + <FR><NU>[Ca<SUP>2+</SUP>]<SUB>m</SUB></NU><DE><IT>K</IT><SUB>Ca<SUP>2+</SUP></SUB></DE></FR></FENCE><SUP>−2</SUP></FENCE></DE></FR> (18)

Equation 18 does a good job of fitting experimental data from heart mitochondria if the parameter u2 = Jkin,max/Jphos,max = 1.1 (Fig. 2). The data points from heart mitochondria are typical, where the substrate-dependent maximum for fPDHa ranges from ~0.45 to 0.7 (36, 42), and the larger saturation values may be simulated by decreasing the parameter u2. The value of K0.5 for matrix Ca2+ activation of PDH, ~0.1 µM in Fig. 2, is also in good agreement with experiments (36, 42).


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Fig. 2.   Simulated fraction of activated pyruvate dehydrogenase ( fPDHa) with respect to mitochondrial Ca2+ concentration ([Ca2+]m) using Eq. 18, with parameters for fPDHa (u1 and u2) = 1.5 and 1, respectively, and Ca2+-dependent affinity of PDH-P phosphate for Mg2+ (KCa2+) = 0.05 µM. Experimental points (42) were determined by assay for PDH activity in samples withdrawn from preparations of rat heart mitochondria at pH 7.4 and 25°C in a sucrose-K+ medium containing 20 mM succinate, 2.5 µM rotenone, 10 mM NaCl, and 1 mM ATP. Indo 1 fluorescence measurements were used to calculate [Ca2+]m.

In islets, fPDHa rises from ~16 to 50% as the D-glucose concentration increases from 2 to 12 mM (39). Such results parallel the increase in the Delta JPDH/Delta Jgly,total ratio from 0.28 to 0.48 by glucose metabolism, as determined by the labeled D-glucose experiments discussed in MITOCHONDRIAL REDUCING EQUIVALENTS FROM D-GLUCOSE. Thus it is plausible to assume that the D-glucose-dependent factor in Eqs. 8 and 9 is due to the activation of PDH. We make this explicit in our model by writing
<IT>f</IT>([Glc]) = <IT>f</IT><SUB>PDH<SUB>a</SUB></SUB> (19)

The production rate for NADH and its equivalents now depends on the level of added D-glucose only through the glucokinase rate law, Delta Jgly,total (Eq. 10). Because Delta JPDH includes fluxes through the TCA cycle and the glycerol phosphate shunt that have been related to acetyl-CoA production stoichiometrically (Fig. 1), the dependence of these rates on fPDHa([Ca2+]m) rather than on f ([Glc]) implies accelerations of Delta JTCA and Delta Jshunt similar to that described explicitly for Delta JPDH. Such an approximation is not unreasonable, because submicromolar matrix Ca2+ is known to stimulate the alpha -ketoglutarate and NAD-isocitrate dehydrogenases of the TCA cycle by increasing the affinity of these enzymes for subsaturating levels of their respective substrates alpha -ketoglutarate and threo-DS-isocitrate (36-38, 53). Also, although it faces the extramitochondrial side of the inner membrane, the GPDH that determines Delta Jshunt undergoes a similar increase in its substrate affinity at the rising [Ca2+]i typical of electrically excited beta -cells (30, 58).

    MITOCHONDRIAL KINETIC EQUATIONS

To help simplify the mitochondrial variables in our model, we represent the reducing equivalents that flow into mitochondrial metabolism in terms of an effective concentration of NADH. Thus we define [NADH]m* as the effective concentration of NADH resulting from the total production and oxidation of reducing equivalents at complex I and complex II. The effective rate of production ( Jred) is written in terms of a basal, D-glucose-independent term ( Jred,basal) and the two D-glucose-dependent terms derived in MITOCHONDRIAL REDUCING EQUIVALENTS FROM D-GLUCOSE, i.e.
<IT>J</IT><SUB>red</SUB> = <IT>J</IT><SUB>red,basal</SUB> + &Dgr;<IT>J</IT><SUB>red,I</SUB> + 0.66&Dgr;<IT>J</IT><SUB>red,II</SUB> (20)
where we have used the experimental observation that reducing equivalents in complex II have roughly two-thirds the effect of NADH at complex I. Combining this expression with Eqs. 8, 9, and 19, we obtain
<IT>J</IT><SUB>red</SUB> = <IT>J</IT><SUB>red,basal</SUB> + 7.36<IT>f</IT><SUB>PDH<SUB>a</SUB></SUB>&Dgr;<IT>J</IT><SUB>gly,total</SUB> (21)
with fPDHa and Delta Jgly,total given by Eqs. 18 and 10, respectively. The basal rate of NADH production is that in the absence of D-glucose and is estimated from experiments in Table 1 (31).

                              
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Table 1.   Mitochondrial Ca2+ handling parameters altered from minimal model

Thus the balance equation for [NADH]m* becomes
 <FR><NU>d[NADH]<SUB>m</SUB>*</NU><DE>d<IT>t</IT></DE></FR> = <IT>J</IT><SUB>red</SUB> − <IT>J</IT><SUB>o</SUB> (22)
where Jo is the effective rate of oxidation defined previously (Eq. 5 in Ref. 32), except [NADH]m is replaced by [NADH]m* and [NAD+]m by
[NAD<SUP>+</SUP>]<SUB>m</SUB>* = 8 nmol/mg protein − [NADH]<SUB>m</SUB>* (23)
Equation 23 allows us to calculate [NAD+]m* in terms of [NADH]m*. We have argued previously (32) that the pyridine nucleotides are approximately conserved and assume for simplicity that it is true for their effective concentrations.

For our whole cell model, a term representing substrate level phosphorylation ( Jp,TCA) must be added to the balance equation for matrix ADP in Eq. 23 of Ref. 32. On the basis of an ideal stoichiometry of coupled NADH oxidation and ADP phosphorylation, 1 GTP = 1 ATP is produced by way of the mitochondrial succinyl-CoA synthase and nucleoside diphosphate kinase reactions for every 3 NADH. Thus basal (nonglucose) metabolism contributes Jred,basal/3 to Jp,TCA. The glucose-dependent contribution to Jp,TCA can be obtained by multiplying Eq. 4 by 2 to account for the fact that Delta JTCA is expressed in D-glucose concentration units. Adding these terms together gives
<IT>J</IT><SUB>p,TCA</SUB> = <FR><NU><IT>J</IT><SUB>red,basal</SUB></NU><DE>3</DE></FR> + 0.84⋅&Dgr;<IT>J</IT><SUB>PDH</SUB>
= <FR><NU><IT>J</IT><SUB>red,basal</SUB></NU><DE>3</DE></FR> + 0.84⋅<IT>f</IT><SUB>PDH<SUB>a</SUB></SUB>⋅&Dgr;<IT>J</IT><SUB>gly,total</SUB> (24)
The balance equation for mitochondrial ATP + ADP may now be expressed as
 <FR><NU>d[ADP]<SUB>m</SUB></NU><DE>d<IT>t</IT></DE></FR> = <IT>J</IT><SUB>ANT</SUB> − <IT>J</IT><SUB>p,TCA</SUB> − <IT>J</IT><SUB>p,F<SUB>1</SUB></SUB> (25)
where JANT is the exchange rate of cytosolic ADP3- for matrix ATP4- mediated by the adenine nucleotide translocator, Jp,F1 is the flux of ATP production by oxidative phosphorylation, and
[ATP]<SUB>m</SUB> + [ADP]<SUB>m</SUB> = 12 nmol/mg protein (26)
is a conservation condition for mitochondrial adenine nucleotides (32).

The balance for the matrix free Ca2+ concentration has the form
<FR><NU>d[Ca<SUP>2+</SUP>]<SUB>m</SUB></NU><DE>d<IT>t</IT></DE></FR> = <IT>f</IT><SUB>m</SUB>(<IT>J</IT><SUB>uni</SUB> − <IT>J</IT><SUB>Na<SUP>+</SUP>/Ca<SUP>2+</SUP></SUB>) (27)
where fm is the fraction of unbound mitochondrial Ca2+ and Juni and JNa+/Ca2+ are the influx and efflux of Ca2+ across the inner membrane mediated by the Ca2+ uniporter and the Na+/Ca2+ exchanger, respectively. Some parameter values for both of these transport mechanisms differ from those used in the minimal mitochondrial model and reflect whole cell conditions (see Table 1). Thus we account for activation of the uniporter by spermine (26), a polyamine that is abundant in beta -cells (23), by lowering the equilibrium constant L for the allosteric binding of Ca2+ to the uniporter. This has the effect of diminishing the sigmoidal dependence on [Ca2+]i. The values of maximal transport rate (vmax) and the dissociation constant for the influx of Ca2+ ( Jmax,uni and Ktrans) also have been changed. They remain, however, within the ranges dictated by experiment (20, 31, 45).

Parameter settings for mitochondrial Ca2+ efflux by way of the Na+/Ca2+ antiport have also been changed to include an inward flow of positive charge that corresponds to an electrogenic exchange of 3 cytosolic Na+ for 1 matrix Ca2+. In our previous work (31) we explored the electrogenic and the alternative electroneutral mechanism (2 Na+:1 Ca2+), since the issue of the carrier's stoichiometry is still somewhat controversial (31, 32). It has also been suggested that an electroneutral Na+/Ca2+ exchanger may receive energy directly from electron transport as the matrix Ca2+ level rises, thereby functioning as an active mechanism (19). In any case, the assumption that the carrier-mediated efflux augments the uniporter-driven dissipation of respiratory energy during the futile cycling of Ca2+ across the inner membrane affects simulations of the full model by making them more robust with respect to the parameter ranges that generate bursting electrical activity (see DISCUSSION).

Adding the electrogenic Ca2+ efflux JNa+/Ca2+ to the ordinary differential equation for the inner membrane voltage gives
<IT>C</IT><SUB>mito</SUB> <FR><NU>d&Dgr;&PSgr;</NU><DE>d<IT>t</IT></DE></FR>
= −(−<IT>J</IT><SUB>H,res</SUB> + <IT>J</IT><SUB>H,F<SUB>1</SUB></SUB> + <IT>J</IT><SUB>ANT</SUB> + <IT>J</IT><SUB>H,leak</SUB> + 2<IT>J</IT><SUB>uni</SUB> + <IT>J</IT><SUB>Na<SUP>+</SUP>/Ca<SUP>2+</SUP></SUB>) (28)
where Cmito is the membrane capacitance (in the empirical units "nmol · mV-1 · mg protein-1"), JH,res is the respiration-driven H+ ejection, HH,F1 is the H+ uptake through the F1Fo-ATPase, Delta Psi is inner membrane voltage and JH,leak is the ohmic proton leakage. The functional forms and the parameters for all of these rates have been reported previously (32).

    PLASMA MEMBRANE KINETIC EQUATIONS

Although many simplified models of plasma membrane currents in beta -cells have been proposed (54), we have chosen here to simulate the primary currents in mouse beta -cells (55). The currents in the plasma membrane used in our model of the beta -cell are illustrated in Fig. 3, grouped by whether they contribute predominantly to the spike or burst oscillation. Because the main features of this model of electrical activity have been described in detail elsewhere (55), we concentrate here on refinements of the currents based on recent experimental work. As is customary, we treat the plasma membrane as consisting of a membrane capacitance (C in pF) in series with various currents (In in fA, where the subscript defines the current type). The plasma membrane potential (V ) then satisfies the usual differential equation
<IT>C</IT> <FR><NU>d<IT>V</IT></NU><DE>d<IT>t</IT></DE></FR> = −(<IT>I</IT><SUB>K<SUB>dr</SUB></SUB> + <IT>I</IT><SUB>K<SUB>ATP</SUB></SUB> + <IT>I</IT><SUB>Ca<SUB>f</SUB></SUB> + <IT>I</IT><SUB>Ca<SUB>s</SUB></SUB> + <IT>I</IT><SUB>NS</SUB>) (29)
where IKdr is the delayed rectifier K+ current, ICaf and ICas are the fast Ca2+-inactivated and slow voltage-inactivated Ca2+ currents, and INS is a nonselective cation current that is activated by D-glucose, which for simplicity we assume carries only Ca2+. The dependence of IKdr, ICaf, and ICas on membrane potential and gating variables is exactly as assumed in previous work. Their form is given in APPENDIX B along with the parameter values for the currents.


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Fig. 3.   Mechanism for bursting assumed by whole cell model. Top: plasma membrane currents associated with burst and spike oscillations; area corresponding to cytosol gives a simplified description of Ca2+ feedback driving adenine nucleotide concentration oscillations and ATP-sensitive K+ (KATP) channel gating. Uptake of Ca2+ by mitochondria positively affects oxidative phosphorylation by activating PDH and other dehydrogenases; futile cycling of Ca2+ across mitochondrial inner membrane periodically diminishes ATP production by lowering inner membrane voltage (Delta Psi ). Heavy lines and arrows, ion fluxes; thin lines and arrows, activation (oplus ) or inactivation (ominus ) of membrane transport and other key processes by increasing values of indicated effectors.

We have updated the description of IKATP on the basis of the experiments of Hopkins et al. (22), which delineated the dependence of this current on the concentration of ATP and ADP. They fitted their data to a detailed kinetic model, which we adopt here. In that model, binding of ATP and ADP is treated as instantaneous, and the resulting current has the form
<IT>I</IT><SUB>K<SUB>ATP</SUB></SUB> = <OVL><IT>g</IT></OVL><SUB>K<SUB>ATP</SUB></SUB>⋅O<SUB>K<SUB> ATP</SUB></SUB>⋅(<IT>V</IT> − <IT>V</IT><SUB>K</SUB>) = <IT>g</IT><SUB>K<SUB>ATP</SUB></SUB>⋅(<IT>V</IT> − <IT>V</IT><SUB>K</SUB>) (30)
where gKATP is the whole cell KATP conductance and <OVL><IT>g</IT></OVL><SUB>K<SUB>ATP</SUB></SUB> is its maximal value, VK is the K+ Nernstian reversal potential, and OKATP is the fraction of channels open. According to the results of Hopkins et al. (22), when the channel has 1) no nucleotide or a single MgADP- bound or 2) two MgADP- bound, the channel is open with relative conductances of 0.08 and 0.89, respectively. This leads to the following expression for the dependence of the open probability on nucleotide concentration
O<SUB>K<SUB>ATP</SUB></SUB> 
= <FR><NU>0.08<FENCE>1 + <FR><NU>2[MgADP<SUP>−</SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB>dd</SUB></DE></FR></FENCE> + 0.89<FENCE><FR><NU>[MgADP<SUP>−</SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB>dd</SUB></DE></FR></FENCE><SUP>2</SUP></NU><DE><FENCE>1 + <FR><NU>[MgADP<SUP>−</SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB>dd</SUB></DE></FR></FENCE><SUP>2</SUP><FENCE>1 + <FR><NU>[ADP<SUP>3−</SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB>td</SUB></DE></FR> + <FR><NU>[ATP<SUP>4−</SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB>tt</SUB></DE></FR></FENCE></DE></FR> (31)
The dissociation constants (Kdd, Ktd, and Ktt) describe the binding equilibrium of the various nucleotide forms. Figure 4 illustrates the dependence of the open probability on the overall free concentration of both nucleotides.


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Fig. 4.   Simulated equilibrium fraction of open KATP channels (OKATP) as scaled to its value for 1 µM ADP and no ATP and calculated using Eq. 31 with respect to concentration of unbound cytosolic ADP ([ADP]i) for various concentrations of cytosolic ATP ([ATP]i). Activation following from higher levels of ADP is shown as being gradually overwhelmed by inactivating effects of nucleotides; greater ATP concentrations are increasingly inhibitory.

Depending on concentration, a rise in ATP or a comparable fall in ADP can be the dominant regulator of the open probability. Note, however, that whereas all increases in ATP concentration tend to lower OKATP, this effect is most dramatic when ADP is close to physiological values (~100 µM). Moreover, at fixed [ATP]i, the open probability has a bell-shaped dependence on [ADP]i. As Hopkins et al. (22) have shown, these curves reproduce experimental data.

We use the same Goldman-Hodgkin-Katz form for the current through the nonselective ion channel as used in previous work (55), namely
<IT>I</IT><SUB>NS</SUB> = <IT>g</IT><SUB>NS</SUB> <FR><NU>[Ca<SUP>2+</SUP>]<SUB>o</SUB><IT>V</IT></NU><DE>1 − exp (2<IT>FV</IT>/<IT>RT</IT>)</DE></FR> (32)
where R is the gas constant, T is the Kelvin temperature, and F is Faraday's constant. For simplicity, we assume that all the current is carried by Ca2+, so that [Ca2+]o is the external Ca2+ concentration. This current represents a D-glucose-dependent inward current that has been found in mouse beta -cells (46). Although the mechanism of this dependence is not known, we have assumed, again for simplicity, that the whole cell conductance of this current (gNS) increases hyperbolically with the total [ATP]i. Thus
<IT>I</IT><SUB>NS</SUB> = <FR><NU><IT>g</IT><SUB>NS</SUB></NU><DE>1 + <FR><NU><IT>K</IT><SUB>NS</SUB></NU><DE>[ATP]<SUB>i</SUB></DE></FR></DE></FR>⋅<FR><NU>[Ca<SUP>2+</SUP>]<SUB>o</SUB><IT>V</IT></NU><DE>1 − exp (2<IT>FV</IT>/<IT>RT</IT>)</DE></FR> (33)
where parameter values are given in Table 3. Neither of the specific assumptions in Eq. 32 or 33 is crucial to the simulations. Indeed, as shown in our previous work (55), all that is required is an inward leak with sufficient current to maintain the silent phase.

    CYTOSOLIC KINETIC EQUATIONS

The mechanisms used in this model involve three cytosolic concentrations as variables: [Ca2+]i, [ADP]i, and [ATP]i. Although the model explicitly takes into account the production and transport of ATP from the mitochondria, we have not included a mechanistic description of the hydrolysis and other potential reactions of ADP and ATP in the cytosol. Instead, to eliminate [ATP]i as a varible, we use the simplifying assumption that [ADP]i + [ATP]i = 2 mM, which is on the order of the measured total adenine nucleotide concentration in mouse beta -cells (31, 34, 53). This assumption is compatible with the 1:1 exchange of cytosolic ADP3- for matrix ATP4- via the mitochondrial adenine nucleotide translocator, as modeled previously (32). In making this assumption, we ignore other processes, such as the adenylate kinase reaction that converts AMP and ATP to 2 ADP. This allows us to write the following balance equation for total [ADP]i
<FR><NU>d[ADP]<SUB>i</SUB></NU><DE>d<IT>t</IT></DE></FR> = &ggr;<SUB>1</SUB>(−<IT>J</IT><SUB>ANT</SUB> + <IT>J</IT><SUB>hyd</SUB> − <IT>J</IT><SUB>p,gly</SUB>) (34)
where gamma 1 converts mitochondrial rate units to millimolar per millisecond (see APPENDIX A), Jhyd is the rate of hydrolysis of cytosolic ATP, and Jp,gly = 2Delta Jgly,total is the net glycolytic rate of ADP phosphorylation. Although compared with oxidative phosphorylation in the beta -cell, glycolysis contributes only 5-10% of the total phosphorylation of ADP (28), we include the final term in Eq. 34 for completeness. We write Jhyd as a sum of two components: one that represents the basal ATP hydrolysis rate in the cytosol of the unexcited beta -cell and another that depends on the concentration of added D-glucose at steady state
<IT>J</IT><SUB>hyd</SUB> = <IT>k</IT><SUB>hyd</SUB>[ATP]<SUB>i</SUB> + &Dgr;<IT>J</IT><SUB>hyd,ss</SUB> (35)
where Delta Jhyd,ss is the steady-state hydrolysis rate of cytosolic ATP. An empirical expression for Delta Jhyd,ss is proposed in WHOLE CELL MODEL: STEADY-STATE BEHAVIOR; an alternative form of Eq. 35 in which a dynamic D-glucose-dependent ATP hydrolysis rate is allowed to relax to Delta Jhyd,ss is discussed in ELECTRICAL ACTIVITY: TRANSIENT BEHAVIOR.

To obtain the concentrations of the specific species of ATP and ADP required for calculating the rate of the adenine nucleotide translocator (JANT) and for the regulation of the KATP channel conductance, we use fixed fractional values of the total [ADP]i and [ATP]i determined from data for beta -cells, as described elsewhere (31, 32). The numerical values are summarized in Table 2.

                              
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Table 2.   Parameters determining adenine nucleotide concentrations

The balance equation for [Ca2+]i consists of six terms
<FR><NU>d[Ca<SUP>2+</SUP>]<SUB>i</SUB></NU><DE>d<IT>t</IT></DE></FR> = <IT>f</IT><SUB>i</SUB>{&agr;(<IT>I</IT><SUB>NS</SUB> + <IT>I</IT><SUB>Ca<SUB>f</SUB></SUB> + <IT>I</IT><SUB>Ca<SUB>s</SUB></SUB>)
 − &ggr;<SUB>2</SUB>(<IT>J</IT><SUB>uni</SUB> − <IT>J</IT><SUB>Na<SUP>+</SUP>/Ca<SUP>2+</SUP></SUB>) − <IT>k</IT><SUB>Ca</SUB>[Ca<SUP>2+</SUP>]<SUB>i</SUB>} (36)
In this expression, alpha  = 1,000/2FVcyt converts between the plasma membrane Ca2+ current and the rate of change of Ca2+ concentration, where Vcyt is the cytosolic volume of a beta -cell (treated as a sphere with a radius of 7 µm). The factor gamma 2 = 1.53 × 10-3 converts mitochondrial rate units to micromolar per millisecond (see APPENDIX A), and fi = 0.01 is the fraction of Ca2+ that is free in the cytosol. The currents INS, ICaf, and ICas are defined in PLASMA MEMBRANE KINETIC EQUATIONS, and Juni and JNa+/Ca2+ are the uniporter flux into and the Na+/Ca2+ exchanger flux out of the mitochondria. Detailed expressions for these fluxes are given in our previous work (31, 32). The final term in Eq. 36 represents removal of cytosolic Ca2+ into nonmitochondrial stores and the intercellular space. At physiological concentrations of glucose, all three groups of terms in Eq. 36 make significant contributions to changes in [Ca2+]i.

    WHOLE CELL MODEL: STEADY-STATE BEHAVIOR

The steady-state oscillations shown in Fig. 5 were generated by the whole beta -cell model for 8.3 mM D-glucose and a D-glucose-dependent cytosolic ATP hydrolysis rate of 14 nmol · min-1 · mg protein-1 (Delta Jhyd,ss in Eq. 35); the remaining parameter values are from the standard set listed in Tables 1-3 or as given previously (32). The phase relations illustrated by these simulations are consistent with the mechanism for bursting discussed in the introduction. During the active phase (Fig. 5E ), Ca2+ uptake through the voltage-gated channels of the depolarized plasma membrane increases [Ca2+]i (Fig. 5C ). This increases influx of Ca2+ to the mitochondria via the uniporter as well as efflux via the Na+/Ca2+ exchanger, creating an oscillation of the matrix free Ca2+ concentration (Fig. 5A ) that peaks at the end of the active phase. The electrogenic cycling of Ca2+ across the mitochondrial inner membrane transiently lowers the inner membrane voltage, the rate of oxidative phosphorylation, and the contribution of the adenine nucleotide translocator to the rate at which ATP appears in the cytosol (JANT + Jp,gly in Fig. 5D ). The result is an increase of [ADP]i (Fig. 5B ) that is in phase with [Ca2+]i, which is transduced by the mitochondria into an adenine nucleotide concentration change that is two orders of magnitude greater (Fig. 11 in Ref. 33). The small decrease in the overall rate of cytosolic ATP hydrolysis (Jhyd in Fig. 5D ) reflects the linear dependence of the basal component of that flux on [ATP]i. Once the increase of [ADP]i is sufficient to open enough KATP channels to repolarize the plasma membrane, cytosolic and mitochondrial Ca2+ levels fall. Then, with the onset of the silent phase, ATP production recovers and the cytosolic ADP concentration declines.


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Fig. 5.   beta -Cell oscillations concurrent with bursting, as simulated using whole cell model for glucose concentration = 8.3 mM and Jhyd = 14 nmol · min-1 · mg protein-1; all other parameter values are in Tables 1-3 or as reported previously (32). A: [Ca2+]m; B: [ADP]i; C: [Ca2+]i; D: total rate at which ATP appears in cytosol, by way of mitochondrial adenine nucleotide translocator and as generated in glycolysis (JANT + Jp,gly), and total cytosolic ATP hydrolysis rate (Jhyd); E: membrane potential (V ).

The simulated voltage and [Ca2+]i oscillations are similar to those obtained using earlier beta -cell models, with some attributes of the spikes peculiar to the detailed modeling of the plasma membrane Ca2+ channels (25, 55). In the active phase the Ca2+ currents in the plasma membrane have a peak value that is four to five times greater than the other fluxes in Eq. 36. At the plateau of the active phase and in the silent phase, on the other hand, the mitochondrial fluxes and the efflux term are comparable and dominate the [Ca2+]i balance equation. The concentrations of cytosolic ADP are consistent with estimates for mouse islets (31) and measurements from fractionated rat beta -cell preparations (53), although the amplitude of the oscillations (15 µM) may be too small to have been observed in vivo. The oscillations of [Ca2+]m have also not been observed experimentally, but the range of values in Fig. 5A is reasonable (32).

Bursting is a transitional phenomenon of the beta -cell plasma membrane that occurs only at intermediate levels of islet excitability. Low D-glucose concentrations depolarize the cell 2-10 mV above its resting potential of about -70 mV. If metabolic stimulation continues to increase, a transition to the bursting regimen occurs for 5-7 mM D-glucose, whereas concentrations above ~16 mM generate states of continuous spiking from a depolarized voltage plateau. Another experimental effect of increasing D-glucose is an increase of the plateau fraction or relative duration of the active phase. Concomitant increases in the burst period have also been recorded, although many of the reported changes are considerably smaller or negligible (31).

Figure 6 shows the relation between these characteristics of beta -cell electrical activity and a hypothetical D-glucose dependence for Delta Jhyd,ss, the second term in Eq. 35. Values of Delta Jhyd,ss that will generate bursts, indicated by vertical lines in Fig. 6, have been determined from the simulations. Values of Delta Jhyd,ss above these ranges correspond to relatively large cytosolic ADP concentrations and, hence, hyperpolarized steady states of the membrane voltage. The lower D-glucose-dependent hydrolysis rates result in more ATP and the generation of continuous spiking.


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Fig. 6.   Two-parameter bifurcation diagram of whole cell model steady states. Dashed lines separate parameter space defined by stimulation-dependent ATP hydrolysis rate in cytosol (Delta Jhyd,ss) and concentration of added D-glucose into regions corresponding to 3 types of islet electrical activity shown. Solid line is a plot of Eq. 37, one of many possible relations of Delta Jhyd,ss to D-glucose concentration ([D-glucose]) that is consistent with thresholds for bursting and continuous spiking observed experimentally. All parameter settings are in Tables 1-3 or as reported previously (32).

The dashed curves of Fig. 6 separate the parameter space into three regions corresponding to bursting, continuous spiking, and hyperpolarization. The top dashed curve separates bursting from stable steady states (to the left). The bottom curve, which separates the bursting and continuous spiking, is less clearly defined but represents a transitional region that may include chaotic solutions. The solid curve between the two dashed lines represents a relationship between the D-glucose-dependent ATP hydrolysis rate (Delta Jhyd,ss) and the D-glucose concentration, using the Hill relation defined by Delta Jhyd, max = 30.1 nmol · min-1 · mg protein-1, KGlc = 8.7 mM, and nhyd = 2.7 
&Dgr;<IT>J</IT><SUB>hyd,ss</SUB> = <FR><NU>&Dgr;<IT>J</IT><SUB>hyd,max</SUB></NU><DE>1 + <FENCE><FR><NU><IT>K</IT><SUB>Glc</SUB></NU><DE>[Glc]</DE></FR></FENCE><SUP><IT>n</IT><SUB>hyd</SUB></SUP></DE></FR> (37)

If it is assumed that the hydrolysis rate has this dependence on D-glucose, then the usual D-glucose dose response of electrical activity, with its characteristics of a threshold near 5.6 mM, bursting in the 5.6-14 mM regime, and continuous spiking at higher concentrations, is obtained. Similar results are obtained for parameters in the ranges Delta Jhyd,max = 30-31.5 nmol · min-1 · mg protein-1, KGlc = 8.7-9.0 mM, and nhyd = 2.65-2.85. As is obvious, however, from Fig. 6, a dose-response curve that is compatible with experiment cannot be obtained if the ATP hydrolysis rate does not increase with D-glucose concentration.

    ELECTRICAL ACTIVITY: TRANSIENT BEHAVIOR

D-Glucose concentrations above the threshold for beta -cell excitability along with the corresponding values of Delta Jhyd,ss generated by Eq. 37 will not give rise to bursting if the initial variables of the model are typical of resting conditions in islets. This is due to the dependence of Delta JPDH and NADH production on [Ca2+]m (see CALCIUM DEPENDENCE ON NADH PRODUCTION) and on the high buffering capacity of mitochondrial Ca2+ (32). The fraction of activated PDH (Eq. 18) follows the slowly rising concentration of free Ca2+ in the matrix and may take >= 1 min to saturate in the simulations. This creates a delay for D-glucose-dependent ATP production by oxidative phosphorylation to reach its maximum level. Because the cytosolic ATP hydrolysis rate defined by Eq. 37 changes instantaneously when glucose is increased, values of [ATP]i are always too low to close enough KATP channels for membrane depolarization. This behavior suggests that, for a range of values of Delta Jhyd,ss, the bursting state and the depolarized state may coexist, although because of the complexity of the model we have not explored this potential bistability further.

To remedy this situation, we replace Eq. 35 with
<IT>J</IT><SUB>hyd</SUB> = <IT>k</IT><SUB>hyd</SUB>[ATP]<SUB>i</SUB> + &Dgr;<IT>J</IT><SUB>hyd</SUB> (38)
where Delta Jhyd is defined by the relaxation equation
 <FR><NU>d&Dgr;<IT>J</IT><SUB>hyd</SUB></NU><DE>d<IT>t</IT></DE></FR> = <FR><NU>1</NU><DE>&tgr;<SUB>hyd</SUB></DE></FR> (&Dgr;<IT>J</IT><SUB>hyd,ss</SUB> − &Dgr;<IT>J</IT><SUB>hyd</SUB>) (39)
Simulated V and [Ca2+]i time series generated by this augmented model for a beta -cell excited by 8.3 mM D-glucose are shown in Fig. 7, where tau hyd has been set at 50 s. The results duplicate characteristics of the prolonged transient phase of beta -cell excitability typically observed before the onset of fixed-phase bursting (18, 41).


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Fig. 7.   Simulated [Ca2+]i (A ) and V time series (B ) with glucose concentration = 8.3 mM and tau hyd = 50 s; all other parameter settings are standard. Version of full beta -cell model used here replaces Eq. 35 with Eqs. 38 and 39, allowing Delta Jhyd, the D-glucose-dependent hydrolysis rate for cytosolic ATP, to relax to steady-state value defined by Eq. 37. Results partially duplicate biphasic behavior typical of islets metabolically stimulated from resting conditions.

    DISCUSSION
Top
Abstract
Introduction
Discussion
Appendix A
Appendix B
References

We have combined a kinetic model of mitochondrial Ca2+ handling with a model of D-glucose-induced electrical activity in the plasma membrane of the pancreatic beta -cell to investigate the role of mitochondria in bursting electrical activity (32, 55). Both models were constructed separately in a modular fashion; i.e., all the individual kinetic steps in each model were fit carefully to available experimental data before the complete models were constructed. The key features that couple mitochondria to electrical activity in the beta -cell are [Ca2+]i, [ATP]i and [ADP]i. To account for this coupling, we have added three new kinetic features to the combined model: Ca2+ stimulation of mitochondrial dehydrogenases, the D-glucose dependence of the rate of production of NADH, and an updated kinetic model of the ATP and ADP regulation of the KATP channel in the plasma membrane. On the other hand, we have not included Ca2+ handling by the ER, which appears to be most significant in islets when muscarinic agonists are applied (6, 24).

The combined model thus allows us to test the hypothesis that increased Ca2+ uptake into the mitochondria, which results from action potential spikes at the plasma membrane, is capable of regulating IKATP by interfering with ATP production, thereby repolarizing the active phase of electrical activity. Simulations with the combined model support this hypothesis. The bursting electrical activity shown in Fig. 5 is typical of simulations at intermediate D-glucose concentrations. As found experimentally, the active phase of electrical activity occurs simultaneously with a rapid rise in [Ca2+]i to a nearly constant level of ~0.3 µM (18, 50). In the simulations this is accompanied by a slow rise in [ADP]i that reaches its maximum just as the silent phase begins. This slow increase in [ADP]i is accompanied by a decrease in [ATP]i, which together conspire to increase the conductance of IKATP (Fig. 4). In this range of concentrations, the open fraction for the KATP channels is actually proportional to [ADP]i, and the slow increase of IKATP during the active phase ultimately suffices to repolarize the burst. We have tested whether the slow inactivation of the "slow" Ca2+ current contributes significantly to repolarization by simply removing the slow current from the calculation. This has a minor influence on the shape of the active phase voltage but does not alter the existence of the burst (31). Thus we conclude that in this model it is the activation of IKATP that is responsible for termination of the active phase.

The simulations reveal two other important features. First, Fig. 6 illustrates that to achieve a dose response of active phase duration due to increasing concentrations of D-glucose, it is necessary that the rate of ATP hydrolysis in the cytosol increases with D-glucose concentration. Because ATP is a nearly universal energy source for cytosolic processes, we have not attempted to model explicitly its dependence on D-glucose concentration. However, given the fact that metabolic stimulation increases the concentration of ATP, which is then hydrolyzed by a variety of proteins (e.g., ATP-dependent pumps), a parallel D-glucose-dependent increase in the rate of hydrolysis of ATP is in line with expectations.

The simulations also allow us to examine the functioning of the two major Ca2+ control mechanisms on oxidative phosphorylation during a burst. On the one hand, Ca2+ uptake into the mitochondria stimulates NADH production via the mitochondrial dehydrogenases, thus increasing the driving force for oxidation. On the other hand, the electrogenic uptake of Ca2+ via the uniporter interfers with oxidative phosphorylation. The simulations show that during bursting the latter effect dominates. Indeed, as Fig. 5 illustrates, [ADP]i increases and the efflux of ATP through the adenine nucleotide translocator decreases during the active phase. In the simulations this occurs because the mitochondrial dehydrogenases are maximally stimulated during a burst and, therefore, cannot stimulate oxidation further. As Fig. 2 illustrates, the active form of PDH is most sensitive to Ca2+ uptake at resting levels of [Ca2+]m (~0.1 µM) but saturates (both experimentally and in the model) at ~0.5 µM (42). Thus the elevated [Ca2+]m during a burst (Fig. 5) functions to inhibit, rather than to stimulate, oxidative phosphorylation.

It is important to note that no adjustments in the rate parameters are made to reach these conclusions. Furthermore, the component kinetic mechanisms have been fit wherever possible to experimental data for pancreatic beta -cells. This includes not only the currents in the plasma membrane (55) but also the density of mitochondria in the beta -cell (Table 3) and the high concentrations of spermine (23), which activate the mitochondrial uniporter. The controversial stoichiometry of the mitochondrial Na+/Ca2+ exchanger may also be assumed to be electroneutral rather than electrogenic (see MITOCHONDRIAL KINETIC EQUATIONS). Such a change does not alter the behavior of the model other than by narrowing the parameter range of Delta Jhyd,ss at each D-glucose concentration for which bursting occurs in Fig. 6.

The transient electrical activity shown in Fig. 7 is similar to that observed in islets after their initial exposure to physiological concentrations of glucose (5). In our simulations, this transient is caused by the delay in the increase of the rate of ATP hydrolysis when glucose is added (Eq. 39). This delay compensates for a slow rise in [Ca2+]m that leads to what may be bistability between bursting and the depolarized state when the rate of ATP hydrolysis is treated as a parameter. This is only one of several explanations of the glucose-induced transient that include a potential role for ER depletion-dependent Ca2+ influx (6, 60).

The simulated dependencies of [Ca2+]i and [ATP]i on the level of added D-glucose are summarized in Fig. 8. The low subthreshold value of [Ca2+]i, which more than doubles at high D-glucose concentrations, and its K0.5 of ~7 mM are both plausible experimentally. The greatest increase in [ATP]i is simulated to occur at the onset of electrical activity, with little average change demonstrated at higher D-glucose concentrations; these results are also consistent with experimental results (31).


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Fig. 8.   A: dependence of [Ca2+]i on D-glucose, as simulated by full beta -cell model with standard parameter settings. Top and bottom lines are bounds of burst and continuous spiking oscillations (square ) and include discontinuities that correspond to variable, abrupt transition between types of electrical activity. Center line is defined by calculated points of time-averaged values (bullet ). B: [ATP]i with respect to D-glucose concentration, with time averages shown for D-glucose concentrations above threshold for beta -cell excitability. Relatively small burst oscillation amplitude for [ATP]i is indicated by line thickening.

Our calculations make a number of predictions that are possible to test experimentally. First and foremost, the conductance of the KATP channels should oscillate during bursting electrical activity. During the active phase in Fig. 9, a gradual rise in conductance of ~20 pS occurs that is associated with a ~10% increase in [ADP]i. Recently, experimental measurements with clusters of beta -cells from ob/ob mice have revealed oscillations of the KATP conductance of 196 ± 109 pS during bursting (27). As in our simulations, the oscillations are associated with oscillations in [Ca2+]i, which our calculations predict should be out of phase with the increases in the KATP conductance. Our calculated value of 20 pS for the change in conductance is 5-10 times smaller than that found in these measurements. This may be explained by differences between ob/ob and normal mice, which are the focus of our model, or the fact that recordings were made from clusters of cells, whereas our results reflect the properties of a single beta -cell.


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Fig. 9.   Simulated V (A ), KATP channel conductance (gKATP, B ), and [ADP]i time series (C ). Glucose concentration = 4.9 mM and Delta Jhyd,ss = 4.5 nmol · min-1 · mg protein-1 have been selected to produce [ADP]i and gKATP oscillations with greatest amplitude possible using model. All other parameter settings are standard.

The experiments with clusters of ob/ob beta -cells raise other questions dealing with the duration of a burst. In islets the active phase has a duration on the order of 10-20 s, whereas in clusters the active phase is several minutes long ("long" bursts) (27). Thus it is possible that the bursting mechanism may be different in islets and clusters. Nonetheless, by changing the glucose sensitivity in the model (via the parameter Delta Jhyd,ss), it is possible to generate bursts with very long active phases. For example, with Delta Jhyd,ss set to 88% of the value of Delta Jhyd,max in Table 3 and [Glc] = 16.7 mM, the period is 78 s and, when set to 102% of Delta Jhyd,max with [Glc] = 22.3 mM, the period is 220 s. Therefore, the current model may have relevance for bursting both in islets and clusters of beta -cells. On the other hand, the model does not reproduce experiments with ob/ob cells (27) that show a rapid return to the active phase after a short hyperpolarization.

                              
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Table 3.   Other model parameters

Our simulations predict oscillations in [Ca2+]m with amplitudes in the range of 0.4-0.6 µM. Even larger values are possible if our choice of mitochondrial buffering parameters is too high. The peak value of [Ca2+]m is predicted to occur at the end of the active phase in a burst and to be followed by a steady, slow decline. Recently, it has become possible to measure [Ca2+]m in suspensions of insulin-secreting INS-1 cells using aequorin targeted to the inner mitochondrial membrane (47). Those measurements demonstrate that, in response to depolarization of the plasma membrane, [Ca2+]m increases substantially on the time scale of a burst. Thus our predictions regarding [Ca2+]m appear to be in line with current measurements. Whether such measurements can be extended to clusters of INS-1 cells remains to be seen.

The calculations described here also provide information about a number of other mitochondrial variables, e.g., [NADH]m and [ADP]m, as well as mitochondrial fluxes, which exhibit characteristic changes during bursting. A detailed presentation of those results, along with an analysis of how mitochondria respond during periodic Ca2+ uptake, is the subject of the companion article (33).

    APPENDIX A
Top
Abstract
Introduction
Discussion
Appendix A
Appendix B
References

Organelle fluxes and concentrations are usually measured in nanomoles per milligram of protein. For rat hepatocytes and heart cells, the mitochondrial protein density is ~1.25 g protein/ml. Because the mouse beta -cell is compartmentalized into 3.9 and 53.2% mitochondrial and cytosolic volumes, a factor of ~0.09 converts measured nanomoles per milligram of protein to cytosolic millimolar terms (31).

The intracellular H2O space, using [14C]urea and [3H]sucrose as markers, has been determined to be 2.25 ± 0.10 nl/islet. By use of estimates of 763 µm3 for mouse beta -cell cytoplasm and 201 µm3 for other islet cell types and assuming 65-80% beta -cell content per islet, a factor of 0.31 ± 0.06 will convert picomoles per islet typical of metabolism experiments to the cytoplasmic millimolar terms of a single beta -cell (31).

Single-atom oxygen consumption rates for perfused islets can be used to estimate Jred,basal, the steady-state rate of beta -cell NADH production in the absence of added D-glucose. Respiration levels reported for unstimulated mouse islets include 5.6 nmol O2 · h-1 · µg DNA-1 and 5.7 pmol O2 · min-1 · µg dry wt-1 (43, 57). These fluxes correspond to 4.8-7.6 pmol O · min-1 · islet-1 or 16-26 nmol O · min-1 · mg protein-1, assuming the equivalences of 38.5 ng DNA/µg dry wt and 0.67 µg dry wt/islet (31).

    APPENDIX B
Top
Abstract
Introduction
Discussion
Appendix A
Appendix B
References

For completeness we summarize the kinetic expressions for the plasma membrane currents IKdr, ICaf , and ICas. For the delayed rectifier K+ channel we use
<IT>I</IT><SUB>K<SUB>dr</SUB></SUB> = <OVL><IT>g</IT></OVL><SUB>K<SUB>dr</SUB></SUB><IT>nI</IT>(<IT>V</IT> − <IT>V</IT><SUB>K</SUB>) (40)
where <OVL><IT>g</IT></OVL>Kdr is the maximal whole cell conductance, n is the activation gating variables, and I is the inactivation. The two gating variables satisfy
d<IT>n</IT>/d<IT>t</IT> = −[<IT>n</IT> − <IT>n</IT><SUB>∞</SUB>(<IT>V</IT>)]/&tgr;<SUB><IT>n</IT></SUB>(<IT>V</IT>) (41)
d<IT>I</IT>/d<IT>t</IT> = −[<IT>I</IT> − <IT>I</IT><SUB>∞</SUB>(<IT>V</IT>)]/&tgr;<SUB><IT>I</IT></SUB> (42)
where V has units of millivolts and
<IT>n</IT><SUB>∞</SUB>(<IT>V</IT>) = <FR><NU>1</NU><DE>1 + exp [(−20 − <IT>V</IT>)/5.3]</DE></FR> (43)
<IT>I</IT><SUB>∞</SUB>(<IT>V</IT>) = <FR><NU>1</NU><DE>1 + exp [(−36 − <IT>V</IT>)/4.5]</DE></FR> (44)
&tgr;<SUB><IT>n</IT></SUB>(<IT>V</IT>) = <FR><NU>50 ms</NU><DE>exp (<IT>V</IT> + 75)/65 − exp [(<IT>V</IT>+ 75)/20]</DE></FR> (45)
&tgr;<SUB><IT>I</IT></SUB> = 2,600 ms (46)
For the two Ca2+ currents we also follow previous work
<IT>I</IT><SUB>Ca<SUB>f</SUB></SUB> = 0.27<OVL><IT>g</IT></OVL><SUB>Ca</SUB>O<SUB>f</SUB> <FR><NU>[Ca<SUP>2+</SUP>]<SUB>o</SUB><IT>V</IT></NU><DE>1 − exp (2<IT>FV</IT>/<IT>RT</IT>)</DE></FR> (47)
<IT>I</IT><SUB>Ca<SUB>s</SUB></SUB> = 0.73<OVL><IT>g</IT></OVL><SUB>Ca</SUB>O<SUB>s</SUB> <FR><NU>[Ca<SUP>2+</SUP>]<SUB>o</SUB><IT>V</IT></NU><DE>1 − exp (2<IT>FV</IT>/<IT>RT</IT>)</DE></FR> (48)
where <OVL><IT>g</IT></OVL>Ca is the maximal whole cell Ca2+ conductance and Of and Os are the open fractions for the fast and slow Ca2+ channels, respectively. The open fraction for the fast channel is calculated using the kinetic scheme for domain Ca2+ inactivation
<IT>C</IT> <LIM><OP><ARROW>↔</ARROW></OP><UL>1</UL></LIM> O <LIM><OP><ARROW>↔</ARROW></OP><UL>2</UL></LIM> O − Ca<SUP>2+</SUP> <LIM><OP><ARROW>↔</ARROW></OP><UL>3</UL></LIM><IT> B</IT> (49)
where O and O - Ca2+ are open states. By use of rapid binding assumptions, Of is determined by the condition C + B + Of = 1 and the equations
d<IT>C</IT>/d<IT>t</IT> = <IT>k</IT><SUB>−1</SUB><FENCE><FR><NU><IT>k</IT><SUB>−2</SUB></NU><DE><IT>k</IT><SUB>+2</SUB>Ca<SUB>d</SUB>(<IT>V</IT>) + <IT>k</IT><SUB>−2</SUB></DE></FR></FENCE>O<SUB>s</SUB> − <IT>k</IT><SUB>+1</SUB><IT>C</IT> (50)
d<IT>B</IT>/d<IT>t</IT> = <IT>k</IT><SUB>+3</SUB><FENCE><FR><NU><IT>k</IT><SUB>+2</SUB>Ca<SUB>d</SUB>(<IT>V</IT>)</NU><DE><IT>k</IT><SUB>+2</SUB>Ca<SUB>d</SUB>(<IT>V</IT>) + <IT>k</IT><SUB>−2</SUB></DE></FR></FENCE>O<SUB>s</SUB> − <IT>k</IT><SUB>−3</SUB><IT>C</IT> (51)
where k-1, k-2, k+1, and k+3 are rate constants for the three processes in Eq. 49 and the domain Ca2+ concentration [Cad(V )] is calculated from
Ca<SUB>d</SUB>(<IT>V</IT>) = <FR><NU>−3.02/<IT>k</IT><SUB>+2</SUB></NU><DE>mM⋅mV⋅ms</DE></FR> <FENCE><FR><NU>[Ca<SUP>2+</SUP>]<SUB>o</SUB><IT>V</IT></NU><DE>1 − exp (2<IT>FV</IT>/<IT>RT</IT>)</DE></FR></FENCE> (52)
The open fraction for the slow Ca2+ channel is written
O<SUB>s</SUB> = <FR><NU><IT>J</IT></NU><DE>1 + exp (−<IT>V</IT>/3.6)</DE></FR> (53)
where the slow voltage-dependent inactivation satisfies
d<IT>J</IT>/d<IT>t</IT> = −[<IT>J</IT> − <IT>J</IT><SUB>∞</SUB>(<IT>V</IT>)]/&tgr;<SUB><IT>J</IT></SUB>(<IT>V</IT>) (54)
<IT>J</IT><SUB>∞</SUB>(<IT>V</IT>) = <FR><NU>1</NU><DE>1 + exp [(<IT>V</IT> + 50)/6.3]</DE></FR> (55)
&tgr;<SUB><IT>J</IT></SUB>(<IT>V</IT>) = <FR><NU><IT>T</IT><SUB><IT>J</IT></SUB></NU><DE>exp [(<IT>V</IT> + 50)/6.3]− exp [−(<IT>V</IT> + 50)/6.3]</DE></FR> (56)

    ACKNOWLEDGEMENTS

We thank Dr. A. Sherman for constructive criticism and careful reading of the manuscript.

    FOOTNOTES

This work was supported in part by National Science Foundation Grants BIR-9214381 and BIR-9300799, National Institutes of Health Grant R01-RR-10081, and the Agricultural Experiment Station of the University of California, Davis.

Address for reprint requests: J. Keizer, Institute of Theoretical Dynamics, University of California, Davis, CA 95616.

Received 30 June 1997; accepted in final form 15 December 1997.

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