MODELING IN PHYSIOLOGY
Model of
-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 |
We continue our development of a kinetic
model of bursting electrical activity in the pancreatic
-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
-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
-cell; electrical activity; adenosine
triphosphate-sensitive potassium channel
 |
INTRODUCTION |
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
(IKATP) 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
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
-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
-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
-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
IKATP 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
(
Jgly,total where gly represents glycolysis),
the lactate dehydrogenase rate
(
Jgly,anaerobic), the PDH rate
(
JPDH), and an average rate for the
tricarboxylic acid (TCA) cycle (
JTCA). (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
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
(
JSDH = 2
JTCA). Making the approximations that 1) all
-cell
D-glucose is metabolized to either pyruvate or lactate
(
Jgly,total =
Jgly,anaerobic +
JPDH) and 2) the NADH produced in the cytosol at steady state
either reduces pyruvate or enters the glycerol phosphate shunt
(
Jshunt = 2
Jgly,total
2
Jgly,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
JTCA/
JPDH
remains constant at substimulatory and maximum concentrations of
labeled D-glucose (7)
|
(4)
|
Thus writing
JTCA in terms of
JPDH in Eqs. 1 and 3 gives
|
(5)
|
|
(6)
|
We can then eliminate
JPDH in these
expressions in favor of
Jgly,total, the
D-glucose utilization rate. Indeed, labeling experiments
show that
JPDH is ~0.28 of
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
|
(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
Jgly,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
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
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
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
-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 (PDHa), 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 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
-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
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
|
(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
|
(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
KMg2+,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 KCa2+ 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+.
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
|
(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
|
(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).

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, 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
JPDH/
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
|
(19)
|
The production rate for NADH and its equivalents now depends on the
level of added D-glucose only through the glucokinase rate
law,
Jgly,total (Eq. 10). Because
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
JTCA and
Jshunt
similar to that described explicitly for
JPDH.
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-DS-isocitrate (36-38, 53).
Also, although it faces the extramitochondrial side of the inner
membrane, the GPDH that determines
Jshunt
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 ( 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.
|
(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
fPDHa and
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).
Thus the balance equation for [NADH]m* becomes
|
(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
|
(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
JTCA 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 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
|
(26)
|
is a conservation condition for mitochondrial adenine
nucleotides (32).
The balance for the matrix free Ca2+ concentration has the
form
|
(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
-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
|
(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, 
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
-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 (In
in fA, where the subscript defines the current type). The plasma
membrane potential (V ) then satisfies the usual
differential equation
|
(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.

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
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
|
(30)
|
where gKATP is the whole
cell KATP conductance and
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
|
(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.

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
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
|
(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 (gNS) 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),
Jhyd is the rate of hydrolysis of cytosolic ATP,
and Jp,gly = 2
Jgly,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
Jhyd 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
Jhyd,ss is the steady-state
hydrolysis rate of cytosolic ATP. An empirical expression for
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
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
-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,
= 1,000/2FVcyt
converts between the plasma membrane Ca2+ current and the
rate of change of Ca2+ concentration, where
Vcyt is the cytosolic volume of a
-cell (treated as a
sphere with a radius of 7 µm). The factor
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
-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 (
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.

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 · 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
-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
Jhyd,ss,
the second term in Eq. 35. Values of
Jhyd,ss that will generate bursts, indicated
by vertical lines in Fig. 6, have been determined from the simulations.
Values of
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.

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
(
Jhyd,ss) and the D-glucose
concentration, using the Hill relation defined by
Jhyd, max = 30.1 nmol · min
1 · mg
protein
1, KGlc = 8.7 mM, and
nhyd = 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
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
-cell
excitability along with the corresponding values of
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
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
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
|
(38)
|
where
Jhyd is defined by the
relaxation equation
|
(39)
|
Simulated V and [Ca2+]i
time series generated by this augmented model for a
-cell excited by
8.3 mM D-glucose are shown in Fig.
7, where
hyd has been set at
50 s. The results duplicate characteristics of the prolonged transient
phase of
-cell excitability typically observed before the onset of
fixed-phase bursting (18, 41).

View larger version (14K):
[in this window]
[in a new window]
|
Fig. 7.
Simulated [Ca2+]i (A ) and
V time series (B ) with glucose concentration = 8.3 mM and hyd = 50 s; all other parameter settings are
standard. Version of full -cell model used here replaces Eq. 35 with Eqs. 38 and 39, allowing
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 |
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
-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
-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
-cells. This includes not only the currents in the plasma
membrane (55) but also the density of mitochondria in the
-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
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).

View larger version (14K):
[in this window]
[in a new window]
|
Fig. 8.
A: dependence of [Ca2+]i on
D-glucose, as simulated by full -cell model with
standard parameter settings. Top and bottom lines are bounds of burst
and continuous spiking oscillations ( ) 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 ( ). B: [ATP]i with
respect to D-glucose concentration, with time averages
shown for D-glucose concentrations above threshold for
-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
-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
-cell.

View larger version (14K):
[in this window]
[in a new window]
|
Fig. 9.
Simulated V (A ), KATP channel
conductance (gKATP, B ), and
[ADP]i time series (C ). Glucose concentration = 4.9 mM and 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
-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
Jhyd,ss), it is possible to generate bursts
with very long active phases. For example, with
Jhyd,ss set to 88% of the value of
Jhyd,max in Table
3 and [Glc] = 16.7 mM, the period is 78 s
and, when set to 102% of
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
-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.
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 |
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
-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
-cell cytoplasm and 201 µm3 for other islet
cell types and assuming 65-80%
-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
-cell (31).
Single-atom oxygen consumption rates for perfused islets can be used to
estimate Jred,basal, the steady-state rate of
-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 |
For completeness we summarize the kinetic expressions for the plasma
membrane currents IKdr,
ICaf , and
ICas. For the delayed rectifier
K+ channel we use
|
(40)
|
where
Kdr is the
maximal whole cell conductance, n is the activation gating
variables, and I is the inactivation. The two gating variables
satisfy
|
(41)
|
|
(42)
|
where
V has units of millivolts and
|
(43)
|
|
(44)
|
|
(45)
|
|
(46)
|
For
the two Ca2+ currents we also follow previous work
|
(47)
|
|
(48)
|
where
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
|
(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
|
(50)
|
|
(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
|
(52)
|
The
open fraction for the slow Ca2+ channel is written
|
(53)
|
where
the slow voltage-dependent inactivation satisfies
|
(54)
|
|
(55)
|
|
(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.
 |
REFERENCES |
1.
Ashcroft, S.
Metabolic controls of insulin secretion.
In: The Islets of Langerhans: Biochemistry, Physiology, and Pathology, edited by S. Cooperstein,
and D. Watkins. New York: Academic, 1981, p. 117-148.
2.
Ashcroft, S.,
and
M. Christie.
Effects of glucose on the cytosolic ratio of reduced/oxidized nicotinamide-adenine dinucleotide phosphate in rat islets of Langerhans.
Biochem. J.
184:
697-700,
1979[Medline].
3.
Ashcroft, F. M.,
D. E. Harrison,
and
S. J. H. Ashcroft.
Glucose induces closure of single potassium channels in isolated rate pancreatic
-cells.
Nature
312:
446-448,
1984[Medline].
4.
Ashcroft, F. M.,
and
P. Rorsman.
Electrophysiology of the pancreatic
-cell.
Prog. Biophys. Mol. Biol.
54:
87-143,
1989[Medline].
5.
Atwater, I.,
M. Kulkuljan,
and
E. Pérez-Armendariz.
Ion channels in pancreatic
-cells.
In: Molecular Biology of Diabetes. I. Autoimmunity and Genetics: Insulin Synthesis and Secretion, edited by B. Draznin,
and D. LeRoith. Totawa, NJ: Humana, 1994, p. 303-332.
6.
Bertram, R.,
S. Smolen,
A. Sherman,
D. Mears,
I. Atwater,
F. Martin,
and
B. Soria.
A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic
-cells.
Biophys. J.
68:
2323-2332,
1995[Abstract].
7.
Boschero, A.,
S. Bordin,
A. Sener,
and
W. Malaisse.
D-Glucose and L-leucine metabolism in neonatal and adult cultured rat pancreatic islets.
Mol. Cell. Endocrinol.
73:
63-71,
1990[Medline].
8.
Cook, D. L.,
and
C. N. Hales.
Intracellular ATP directly blocks K channels in pancreatic B-cells.
Nature
211:
269-271,
1984.
9.
Cook, D.,
L. Satin,
M. Ashford,
and
C. Hales.
ATP-sensitive K+ channels in pancreatic
-cells. The "spare channel" hypothesis.
Diabetes
37:
495-498,
1988[Abstract].
10.
Cooper, R.,
P. Randle,
and
R. Denton.
Regulation of heart muscle pyruvate dehydrogenase kinase.
Biochem. J.
143:
625-641,
1974[Medline].
11.
Corkey, B.,
J. Duszynski,
T. Rich,
B. Matschinsky,
and
J. Williamson.
Regulation of free and bound magnesium in rat hepatocytes and isolated mitochondria.
J. Biol. Chem.
261:
2567-2574,
1986[Abstract/Free Full Text].
12.
Damuni, Z.,
J. Humphreys,
and
L. Reed.
Stimulation of pyruvate dehydrogenase phosphatase activity by polyamines.
Biochem. Biophys. Res. Commun.
124:
95-99,
1984[Medline].
13.
Dean, P. M.,
and
E. K. Matthews.
Electrical activity in pancreatic islets.
Nature
219:
389-390,
1968[Medline].
14.
Denton, R.,
P. Randle,
and
B. Martin.
Stimulation by calcium ions of pyruvate dehydrogenase phosphate phosphatase.
Biochem. J.
128:
161-163,
1972[Medline].
15.
Duchen, M.,
P. Smith,
and
F. Ashcroft.
Substrate-dependent changes in mitochondrial function, intracellular free calcium concentration and membrane channels in pancreatic
-cells.
Biochem. J.
294:
35-42,
1993[Medline].
16.
Garfinkel, D.,
L. Garfinkel,
M. Meglasson,
and
F. Matschinsky.
Computer modeling identifies glucokinase as glucose sensor of pancreatic
-cells.
Am. J. Physiol.
247 (Regulatory Integrative Comp. Physiol. 16):
R527-R536,
1984[Medline].
17.
Ghosh, A.,
P. Ronner,
E. Cheong,
P. Khalid,
and
F. Matschinsky.
The role of ATP and free ADP in metabolic coupling during fuel-stimulated insulin release from islet
-cells in the isolated perfused rat pancreas.
J. Biol. Chem.
266:
22887-22892,
1991[Abstract/Free Full Text].
18.
Gilon, P.,
and
J.-C. Henquin.
Influence of membrane potential changes on cytoplasmic Ca2+ concentration in an electrically excitable cell, the insulin-secreting pancreatic B-cell.
J. Biol. Chem.
267:
20713-20720,
1992[Abstract/Free Full Text].
19.
Gunter, T.,
K. Gunter,
S.-S. Sheu,
and
C. Gavin.
Mitochondrial calcium transport: physiological and pathological relevance.
Am. J. Physiol.
267 (Cell Physiol. 36):
C313-C339,
1994[Abstract/Free Full Text].
20.
Gunter, T.,
and
D. Pfeiffer.
Mechanisms by which mitochondria transport calcium.
Am. J. Physiol.
258 (Cell Physiol. 27):
C755-C786,
1990[Abstract/Free Full Text].
21.
Hansford, R.,
and
L. Cohen.
Relative importance of pyruvate dehydrogenase interconversion and feed-back inhibition in the effect of fatty acids on pyruvate oxidation by rat heart mitochondria.
Arch. Biochem. Biophys.
191:
65-81,
1978[Medline].
22.
Hopkins, W.,
S. Fatherazi,
B. Peter-Riesch,
B. Corkey,
and
D. Cook.
Two sites for adenine-nucleotide regulation of ATP-sensitive potassium channels in mouse pancreatic
-cells and HIT cells.
J. Membr. Biol.
129:
287-295,
1992[Medline].
23.
Hougaard, D.,
J. Nielsen,
and
L.-I. Larsson.
Localization and biosynthesis of polyamines in insulin-producing cells.
Biochem. J.
238:
43-47,
1986[Medline].
24.
Keizer, J.,
and
G. DeYoung.
Effect of voltage-gated plasma membrane Ca2+ fluxes in IP3-linked Ca2+ oscillations.
Cell Calcium
14:
397-410,
1993[Medline].
25.
Keizer, J.,
and
G. Magnus.
The ATP-sensitive potassium channel and bursting in the pancreatic beta cell. A theoretical study.
Biophys. J.
56:
229-242,
1989[Abstract].
26.
Kröner, H.
Spermine, another specific allosteric activator of calcium uptake in rat liver mitochondria.
Arch. Biochem. Biophys.
267:
205-210,
1988[Medline].
27.
Larsson, O.,
H. Kindmark,
R. Bränström,
B. Fredholm,
and
P.-O. Berggren.
Oscillations in KATP channel activity promote oscillations in cytoplasmic free Ca2+ concentration in the pancreatic
cell.
Proc. Natl. Acad. Sci. USA
93:
5161-5165,
1996[Abstract/Free Full Text].
28.
Lehninger, A.
Biochemistry (2nd ed.). New York: Worth, 1977, p. 514-517.
29.
Longo, E. A.,
K. Tornheim,
J. T. Deeney,
B. A. Varnum,
D. Tillotson,
M. Prentki,
and
B. E. Corkey.
Oscillations in cytosolic free Ca2+, oxygen consumption, and insulin secretion in glucose-stimulated rat pancreatic islets.
J. Biol. Chem.
266:
9314-9319,
1991[Abstract/Free Full Text].
30.
MacDonald, M.
Calcium activation of pancreatic islet mitochondrial glycerol phosphate dehydrogenase.
Horm. Metab. Res.
14:
678-679,
1982[Medline].
31.
Magnus, G.
Mitochondria-Based Model for Bursting and Its D-Glucose Dependence in the Pancreatic Beta Cell (PhD thesis). Davis: University of California, 1995.
32.
Magnus, G.,
and
J. Keizer.
Minimal model of
-cell Ca2+ handling.
Am. J. Physiol.
273 (Cell Physiol. 42):
C717-C733,
1997[Abstract/Free Full Text].
33.
Magnus, G.,
and
J. Keizer.
Model of
-cell mitochondrial calcium handling and electrical activity. II. Mitochindrial variables.
Am. J. Physiol.
274 (Cell Physiol. 43):
C1174-C1184,
1998[Abstract/Free Full Text].
34.
Malaisse, W.,
and
A. Sener.
Glucose-induced changes in cytosolic ATP content in pancreatic islets.
Biochim. Biophys. Acta
927:
190-195,
1987[Medline].
35.
McCormack, J.
Effects of spermine on mitochondrial Ca2+ transport and the ranges of extramitochondrial Ca2+ to which the matrix Ca2+-sensitive dehydrogenases respond.
Biochem. J.
264:
167-174,
1989[Medline].
36.
McCormack, J.,
E. Bromidge,
and
N. Dawes.
Characterization of the effects of Ca2+ on the intramitochondrial Ca2+-sensitive dehydrogenases within intact rat kidney mitochondria.
Biochim. Biophys. Acta
934:
282-292,
1988[Medline].
37.
McCormack, J.,
and
R. Denton.
Role of calcium ions in the regulation of intramitochondrial metabolism. Properties of Ca2+-sensitive dehydrogenases within intact uncoupled mitochondria from the white and brown adipose tissue of the rat.
Biochem. J.
190:
95-105,
1980[Medline].
38.
McCormack, J.,
A. Halestrap,
and
R. Denton.
Role of calcium ions in regulation of mammalian intramitochondrial metabolism.
Physiol. Rev.
70:
391-425,
1990[Free Full Text].
39.
McCormack, J.,
E. Longo,
and
B. Corkey.
Glucose-induced activation of pyruvate dehydrogenase in isolated rat pancreatic islets.
Biochem. J.
267:
527-530,
1990[Medline].
40.
Meglasson, M.,
and
F. Matschinsky.
New perspectives on pancreatic islet glucokinase.
Am. J. Physiol.
246 (Endocrinol. Metab. 9):
E1-E13,
1984[Abstract/Free Full Text].
41.
Meissner, H.,
and
M. Preissler.
Ionic mechanisms of the glucose-induced membrane potential changes in B-cells.
In: Biochemistry and Biophysics of the Pancreatic B-Cell, edited by W. Malaisse. Stuttgart: Thieme Verlag, 1980, p. 91-99.
42.
Moreno-Sánchez, R.,
and
R. Hansford.
Dependence of cardiac mitochondrial pyruvate dehydrogenase activity on intramitochondrial free Ca2+ concentration.
Biochem. J.
256:
403-412,
1988[Medline].
43.
Panten, U.,
and
H. Klein.
O2 consumption by isolated pancreatic islets, as measured in a microincubation system with a Clark-type electrode.
Endocrinology
111:
1595-1600,
1982[Abstract].
44.
Reed, L.
Regulation of mammalian pyruvate dehydrogenase complex by a phosphorylation-dephosphorylation cycle.
Curr. Top. Cell. Regul.
18:
95-106,
1981[Medline].
45.
Rizzuto, R.,
P. Bernardi,
M. Favaron,
and
G. Azzone.
Pathways for Ca2+ efflux in heart and liver mitochondria.
Biochem. J.
246:
271-277,
1987[Medline].
46.
Rojas, E.,
J. Hidalgo,
P. Carroll,
M. Li,
and
I. Atwater.
A new class of calcium channels activated by glucose in human pancreatic
-cells.
FEBS Lett.
261:
265-270,
1990[Medline].
47.
Rutter, G.,
J. Theler,
M. Murgia,
C. Wollheim,
T. Pozzan,
and
R. Rizzuto.
Stimulated Ca2+ influx raises mitochondria free Ca2+ to supramicromolar levels in a pancreatic
-cell line. Possible role in glucose and agonist-induced insulin secretion.
J. Biol. Chem.
268:
22385-22390,
1993[Abstract/Free Full Text].
48.
Sala, S.,
R. Parsey,
A. Cohen,
and
D. Matteson.
Analysis and use of the perforated patch technique for recording ionic currents in pancreatic
-cells.
J. Membr. Biol.
122:
177-187,
1991[Medline].
49.
Sánchez-Andrés, J. V.,
A. Gomis,
and
M. Valdeolmillos.
The electrical activity of mouse pancreatic
-cells recorded in vivo shows glucose-dependence oscillations.
J. Physiol. (Lond.)
486:
223-228,
1995[Abstract].
50.
Santos, R.,
L. Rosario,
A. Nadal,
J. Garcia-Sancho,
B. Soria,
and
M. Valdeolmillos.
Widespread synchronous [Ca2+]i oscillations due to bursting electrical activity in single pancreatic islets.
Pflügers Arch.
418:
417-422,
1991[Medline].
51.
Sener, A.,
J. Hutton,
S. Kawazu,
A. Boschero,
G. Somers,
G. Devis,
A. Herchuelz,
and
W. Malaisse.
The stimulus-secretion coupling of glucose-induced insulin release. Metabolic and functional effects of NH+4 in rat islets.
J. Clin. Invest.
62:
868-878,
1978[Medline].
52.
Sener, A.,
J. Levy,
and
W. Malaisse.
The stimulus-secretion coupling of glucose-induced insulin release. XXIII. Does glycolysis control calcium transport in the
-cell?
Biochem. J.
156:
521-525,
1976[Medline].
53.
Sener, A.,
J. Rasschaert,
and
W. Malaisse.
Hexose metabolism in pancreatic islets. Participation of Ca2+-sensitive 2-ketoglutarate dehydrogenase in the regulation of mitochondrial function.
Biochim. Biophys. Acta
1019:
42-50,
1990[Medline].
54.
Sherman, A.
Contributions of modeling to understanding stimulus-secretion coupling in pancreatic
-cells.
Am. J. Physiol.
270 (Endocrinol. Metab. 34):
E362-E372,
1996.
55.
Smolen, P.,
and
J. Keizer.
Slow voltage inactivation of Ca2+ currents and bursting mechanisms for the mouse pancreatic
-cell.
J. Membr. Biol.
127:
9-19,
1992[Medline].
56.
Thomas, A.,
T. Diggle,
and
R. Denton.
Sensitivity of pyruvate dehydrogenase phosphatase phosphatase to magnesium ions. Similar effects of spermine and insulin.
Biochem. J.
238:
83-91,
1986[Medline].
57.
Welsh, M. The Importance of Substrate Metabolism in the
Regulation of Insulin Release by Mouse Pancreatic Islets (PhD
thesis). Uppsala, Sweden: Uppsala University.
58.
Wernette, M.,
R. Ochs,
and
H. Lardy.
Ca2+ stimulation of rat liver mitochondrial glycerophosphate dehydrogenase.
J. Biol. Chem.
256:
12767-12771,
1981[Free Full Text].
59.
Wilson, D.,
D. Nelson,
and
M. Ereci
ska.
Binding of the intramitochondrial ADP and its relationship to adenine nucleotide translocation.
FEBS Lett.
143:
228-232,
1982[Medline].
60.
Worley, J. F., III,
M. S. McIntyre,
R. J. Mertz,
M. W. Roe,
and
I. D. Dukes.
Endoplasmic reticulum calcium store regulates membrane potential in mouse islet
-cells.
J. Biol. Chem.
269:
14395-14362,
1994.
61.
Zawalich, W.,
and
F. Matschinsky.
Sequential analysis of the releasing and fuel function of glucose in isolated perifused pancreatic islets.
Endocrinology
100:
1-8,
1977[Abstract].
AJP Cell Physiol 274(4):C1158-C1173
0363-6143/98 $5.00
Copyright © 1998 the American Physiological Society