INTRODUCTION

Top Abstract Introduction Discussion Appendix A Appendix B References

WHEN SUBJECTED to >5-8 mM D-glucose, pancreatic -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 -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 (I_KATP) in the -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 -cell by inactivating I_KATP (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 I_KATP 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 -cell mitochondria suppresses the rate of production of ATP via oxidative phosphorylation, which subsequently activates I_KATP and repolarizes the burst. Previously, we argued the plausibility of this hypothesis using a kinetic model of electrical activity in the -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 I_KATP using the data of Hopkins et al. (22).

Recently, the endoplasmic reticulum (ER) has been implicated in agonist-induced electrical activity in -cells (6, 24, 60). However, glucose itself appears to induce only a transient increase in ER Ca²⁺ uptake in -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 I_KATP should accompany electrical activity and [Ca²⁺]_i oscillations during bursting. Experimental evidence for this type of behavior in clusters of -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 -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 (J_gly,total where gly represents glycolysis), the lactate dehydrogenase rate (J_{gly,anaerobic}), the PDH rate (J_PDH), and an average rate for the tricarboxylic acid (TCA) cycle (J_TCA). (Here the "" 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.

View larger version (50K): [in this window] [in a new window]   Fig. 1.   Fluxes of D-glucose metabolism (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. JTCA, rate for tricarboxylic acid cycle; 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 -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

(1)

Similarly the rate of production of FADH₂ and FMNH₂ for complex II due to mitochondrial succinate dehydrogenase (SDH) and cytosolic GPDH is

(2)

where J_shunt 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 (J_SDH = 2J_TCA). Making the approximations that 1) all -cell D-glucose is metabolized to either pyruvate or lactate (J_gly,total = J_{gly,anaerobic} + J_PDH) and 2) the NADH produced in the cytosol at steady state either reduces pyruvate or enters the glycerol phosphate shunt (J_shunt = 2J_gly,total  2J_{gly,anaerobic}) allows Eq. 2 to be rewritten as

(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 J_TCA/J_PDH remains constant at substimulatory and maximum concentrations of labeled D-glucose (7)

(4)

Thus writing J_TCA in terms of J_PDH in Eqs. 1 and 3 gives

(5)

(6)

We can then eliminate J_PDH in these expressions in favor of J_gly,total, the D-glucose utilization rate. Indeed, labeling experiments show that J_PDH is ~0.28 of J_gly,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

(7)

or using Eqs. 5 and 6

(8)

(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 -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 J_gly,total to the empirical expression for the rate of glucokinase (1, 16)

(10)

where [ATP]_i is intracellular ATP concentration. The parameter _max in Eq. 10 is the maximum rate, which we base on experimental values for J_gly,total (~130 pmol · h¹ · islet¹) (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 J_gly,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 J_gly,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 -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 -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

(11)

The enzyme PDH, which catalyzes the initial decarboxylation step, has an active form (PDH_a), which becomes completely inactivated when phosphorylated. Interconversion between these two forms is controlled by a kinase and phosphatase, i.e.

(12)

where the relative rates of PDH_a kinase and PDH phosphate (PDH-P) phosphatase determine the fraction of activated PDH ( f_PDHa) 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 PDH_a kinase and competitively inhibit PDH_a. 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 PDH_a kinase by pyruvate is negligible in the -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  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 -cell (21). This is expressed through the D-glucose dependence of J_gly,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 K_0.5 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 ( J_phos) is then first order in the PDH-P concentration, with the rate constant determined by the rapid equilibrium binding of Mg²⁺. If J_phos,max is the maximum reaction velocity when f_PDHa = 0, then

(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 J_phos can be expressed through an elevation of the affinity for Mg²⁺ (14, 56). If K_Mg²⁺,max corresponds to the absence of Ca²⁺ and K_Ca²⁺ is the concentration constant producing half the maximum Ca²⁺-dependent increase in the affinity of PDH-P phosphatase for Mg²⁺, then

(14)

Defining

(15)

and combining Eq. 15 with Eqs. 13 and 14 gives

(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 -cells, also acts independently of its role in the regulation of inner membrane Ca²⁺ transport (31, 32), lowering the range of K_Mg²⁺,max (Eq. 15) to 1-2 mM in mitochondrial extracts (56). Because [Mg²⁺]_m  0.35 mM (11), it is reasonable to set u₁ = 15 in Eq. 16. Also in Eq. 16, a hypothetical value for K_Ca²⁺ of 0.04 pmol/mg protein  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⁺.

PDH_a kinase, unlike PDH-P phosphatase, is tightly bound to the PDH complex and not affected by Mg²⁺ or Ca²⁺ (56). Although PDH_a 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 PDH_a 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 PDH_a kinase rate is then

(17)

where J_kin,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 J_kin and J_phos to obtain

(18)

Equation 18 does a good job of fitting experimental data from heart mitochondria if the parameter u₂ = J_kin,max/J_phos,max = 1.1 (Fig. 2). The data points from heart mitochondria are typical, where the substrate-dependent maximum for f_PDHa 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 K_0.5 for matrix Ca²⁺ activation of PDH, ~0.1 µM in Fig. 2, is also in good agreement with experiments (36, 42).

View larger version (13K): [in this window] [in a new window]   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, f_PDHa rises from ~16 to 50% as the D-glucose concentration increases from 2 to 12 mM (39). Such results parallel the increase in the J_PDH/J_gly,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

(19)

The production rate for NADH and its equivalents now depends on the level of added D-glucose only through the glucokinase rate law, J_gly,total (Eq. 10). Because J_PDH 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 f_PDHa([Ca²⁺]_m) rather than on f ([Glc]) implies accelerations of J_TCA and J_shunt similar to that described explicitly for J_PDH. Such an approximation is not unreasonable, because submicromolar matrix Ca²⁺ is known to stimulate the -ketoglutarate and NAD-isocitrate dehydrogenases of the TCA cycle by increasing the affinity of these enzymes for subsaturating levels of their respective substrates -ketoglutarate and threo-D_S-isocitrate (36-38, 53). Also, although it faces the extramitochondrial side of the inner membrane, the GPDH that determines J_shunt undergoes a similar increase in its substrate affinity at the rising [Ca²⁺]_i typical of electrically excited -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 ( J_red) is written in terms of a basal, D-glucose-independent term ( J_red,basal) and the two D-glucose-dependent terms derived in MITOCHONDRIAL REDUCING EQUIVALENTS FROM D-GLUCOSE, i.e.

(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

(21)

with f_PDHa and J_gly,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).

Thus the balance equation for [NADH]_m* becomes

(22)

where J_o 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

(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 ( J_p,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 J_red,basal/3 to J_p,TCA. The glucose-dependent contribution to J_p,TCA can be obtained by multiplying Eq. 4 by 2 to account for the fact that J_TCA is expressed in D-glucose concentration units. Adding these terms together gives

(24)

The balance equation for mitochondrial ATP + ADP may now be expressed as

(25)

where J_ANT is the exchange rate of cytosolic ADP³ for matrix ATP⁴ mediated by the adenine nucleotide translocator, J_p,F1 is the flux of ATP production by oxidative phosphorylation, and

(26)

is a conservation condition for mitochondrial adenine nucleotides (32).

The balance for the matrix free Ca²⁺ concentration has the form

(27)

where f_m is the fraction of unbound mitochondrial Ca²⁺ and J_uni and J_{Na⁺/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 -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 (v_max) and the dissociation constant for the influx of Ca²⁺ ( J_max,uni and K_trans) 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 J_{Na⁺/Ca²⁺} to the ordinary differential equation for the inner membrane voltage gives

(28)

where C_mito is the membrane capacitance (in the empirical units "nmol · mV¹ · mg protein¹"), J_H,res is the respiration-driven H⁺ ejection, H_H,F1 is the H⁺ uptake through the F₁F_o-ATPase, is inner membrane voltage and J_H,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 -cells have been proposed (54), we have chosen here to simulate the primary currents in mouse -cells (55). The currents in the plasma membrane used in our model of the -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 (I_n in fA, where the subscript defines the current type). The plasma membrane potential (V ) then satisfies the usual differential equation

(29)

where I_Kdr is the delayed rectifier K⁺ current, I_Caf and I_Cas are the fast Ca²⁺-inactivated and slow voltage-inactivated Ca²⁺ currents, and I_NS is a nonselective cation current that is activated by D-glucose, which for simplicity we assume carries only Ca²⁺. The dependence of I_Kdr, I_Caf, and I_Cas 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.

View larger version (29K): [in this window] [in a new window]   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 (). Heavy lines and arrows, ion fluxes; thin lines and arrows, activation () or inactivation () of membrane transport and other key processes by increasing values of indicated effectors.

We have updated the description of I_KATP 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

(30)

where g_KATP is the whole cell K_ATP conductance and is its maximal value, V_K is the K⁺ Nernstian reversal potential, and O_KATP 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

(31)

The dissociation constants (K_dd, K_td, and K_tt) 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.

View larger version (16K): [in this window] [in a new window]   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 O_KATP, 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

(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 -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 (g_NS) increases hyperbolically with the total [ATP]_i. Thus

(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 -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

(34)

where ₁ converts mitochondrial rate units to millimolar per millisecond (see APPENDIX A), J_hyd is the rate of hydrolysis of cytosolic ATP, and J_p,gly = 2J_gly,total is the net glycolytic rate of ADP phosphorylation. Although compared with oxidative phosphorylation in the -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 J_hyd as a sum of two components: one that represents the basal ATP hydrolysis rate in the cytosol of the unexcited -cell and another that depends on the concentration of added D-glucose at steady state

(35)

where J_hyd,ss is the steady-state hydrolysis rate of cytosolic ATP. An empirical expression for J_hyd,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 J_hyd,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 (J_ANT) and for the regulation of the K_ATP channel conductance, we use fixed fractional values of the total [ADP]_i and [ATP]_i determined from data for -cells, as described elsewhere (31, 32). The numerical values are summarized in Table 2.

The balance equation for [Ca²⁺]_i consists of six terms

(36)

In this expression,  = 1,000/2FV_cyt converts between the plasma membrane Ca²⁺ current and the rate of change of Ca²⁺ concentration, where V_cyt is the cytosolic volume of a -cell (treated as a sphere with a radius of 7 µm). The factor ₂ = 1.53 × 10³ converts mitochondrial rate units to micromolar per millisecond (see APPENDIX A), and f_i = 0.01 is the fraction of Ca²⁺ that is free in the cytosol. The currents I_NS, I_Caf, and I_Cas are defined in PLASMA MEMBRANE KINETIC EQUATIONS, and J_uni and J_{Na⁺/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 -cell model for 8.3 mM D-glucose and a D-glucose-dependent cytosolic ATP hydrolysis rate of 14 nmol · min¹ · mg protein¹ (J_hyd,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 (J_ANT + J_p,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 (J_hyd 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 K_ATP 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.

View larger version (24K): [in this window] [in a new window]   Fig. 5.   -Cell oscillations concurrent with bursting, as simulated using whole cell model for glucose concentration = 8.3 mM and Jhyd = 14 nmol · min1 · mg protein1; 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 -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 -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 -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 -cell electrical activity and a hypothetical D-glucose dependence for J_hyd,ss, the second term in Eq. 35. Values of J_hyd,ss that will generate bursts, indicated by vertical lines in Fig. 6, have been determined from the simulations. Values of J_hyd,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.

View larger version (27K): [in this window] [in a new window]   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 (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 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 (J_hyd,ss) and the D-glucose concentration, using the Hill relation defined by J_{hyd, max} = 30.1 nmol · min¹ · mg protein¹, K_Glc = 8.7 mM, and n_hyd = 2.7 

(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 J_hyd,max = 30-31.5 nmol · min¹ · mg protein¹, K_Glc = 8.7-9.0 mM, and n_hyd = 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 -cell excitability along with the corresponding values of J_hyd,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 J_PDH 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 ri