We develop a simplified, but useful, mathematical model to describe Ca2+ handling by mitochondria in the pancreatic beta-cell. The model includes the following six transport mechanisms in the inner mitochondrial membrane: proton pumping via respiration and proton uptake by way of the F1Fzero-ATPase (adapted from D. Pietrobon and S. Caplan. Biochemistry 24: 5764-5778, 1985), a proton leak, adenine nucleotide exchange, the Ca2+ uniporter, and Na+/Ca2+ exchange. Each mechanism is developed separately into a kinetic model for the rate of transport, with parameters taken from experiments on isolated mitochondrial preparations. These mechanisms are combined in a modular fashion first to describe state 4 (nonphosphorylating) and state 3 (phosphorylating) mitochondria with mitochondrial NADH and Ca2+ concentrations as fixed parameters and then to describe Ca2+ handling with variable mitochondrial Ca2+ concentration. Simulations are compared to experimental measurements and agree well with the threshold for Ca2+ uptake, measured mitochondrial Ca2+ levels, and the influence of Ca2+ on oxygen uptake. In the absence of Ca2+ activation of mitochondrial dehydrogenases, the simulations predict a significant reduction in the rate of production of ATP that involves a "short circuit" via Ca2+ uptake through the uniporter. This effect suggests a potential role for mitochondrial Ca2+ handling in determining the ATP-ADP ratio in the pancreatic beta-cell.

We continue our development of a kinetic model of bursting electrical activity in the pancreatic β-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 whend-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 β-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 KATPcurrent 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.

In the preceding article [ Am. J. Physiol. 274 ( Cell Physiol. 43): C1158–C1173, 1998], we describe the development of a kinetic model for the interaction of mitochondrial Ca2+ handling and electrical activity in the pancreatic β-cell. Here we describe further results of those simulations, focusing on mitochondrial variables, the rate of respiration, and fluxes of metabolic intermediates as a function of d-glucose concentration. Our simulations predict relatively smooth increases of O2consumption, adenine nucleotide transport, oxidative phosphorylation, and ATP production by the tricarboxylic acid cycle asd-glucose concentrations are increased from basal to 20 mM. On the other hand, we find that the active fraction of pyruvate dehydrogenase saturates, due to increases in matrix Ca2+, near the onset of bursting electrical activity and that the NADH/NAD+ ratio in the mitochondria increases by roughly an order of magnitude as glucose concentrations are increased. The mitochondrial ATP/ADP ratio increases by factor of <2 between thed-glucose threshold for bursting and continuous spiking. According to our simulations, relatively small changes in mitochondrial membrane potential (∼1 mV) caused by uptake of Ca2+ are sufficient to alter the cytoplasmic ATP/ADP ratio and influence ATP-sensitive K+ channels in the plasma membrane. In the simulations, these cyclic changes in the mitochondrial membrane potential are due to synchronization of futile cycle of Ca2+ from the cytoplasm through mitochondria via Ca2+ uniporters and Na+/Ca2+exchange. Our simulations predict steady mitochondrial Ca2+concentrations on the order of 0.1 μM at low glucose concentrations that become oscillatory with an amplitude on the order of 0.5 μM during bursting. Abrupt increases in mitochondrial Ca2+concentration >5 μM may occur during continuous electrical activity.

Based on the existence of ATP-sensitive potassium channels in the plasma membrane of pancreatic beta cells, we develop a quantitative explanation of the electrical activity observed in pancreatic islets. The proposed mechanism involves the voltage-dependent inward calcium and outward potassium currents described by Rorsman and Trube (1986), which are voltage-activated when an increase in the cytoplasmic ATP/ADP ratio decreases the conductance of the ATP-sensitive potassium channels. It is proposed that modulation of the ATP/ADP ratio occurs through calcium inhibition of oxidative phosphorylation. In this picture the mitochondria serve as a transducer of metabolic activity whose sensitivity is modulated by cytosolic calcium. Solution of the differential equations that describe this mechanism gives rise to both bursting and continuous spiking electrical activity similar to that observed experimentally. While the mechanism for bursting in this model involves the ATP/ADP ratio, the feedback is still provided by calcium, as originally proposed by Chay and Keizer (1983) using a Ca2+-activated potassium conductance. A mixed-model, which includes both ATP-sensitive and Ca2+-activated potassium conductances, also reproduces the experimentally observed electrical activity and may correspond more closely to the actual situation in vivo.

In the mammalian cerebellum, deep cerebellar nuclear (DCN) cells convey all information from cortical Purkinje cells (PCs) to premotor nuclei and other brain regions. However, how DCN cells integrate inhibitory input from PCs with excitatory inputs from other sources has been difficult to assess, in part due to the large spatial separation between cortical PCs and their target cells in the nuclei. To circumvent this problem we have used a Cre-mediated genetic approach to generate mice in which channelrhodopsin-2 (ChR2), fused with a fluorescent reporter, is selectively expressed by GABAergic neurons, including PCs. In recordings from brain slice preparations from this model, mammalian PCs can be robustly depolarized and discharged by brief photo stimulation. In recordings of postsynaptic DCN cells, photo stimulation of PC axons induces a strong inhibition that resembles these cells' responses to focal electrical stimulation, but without a requirement for the glutamate receptor blockers typically applied in such experiments. In this optogenetic model, laser pulses as brief as 1 ms can reliably induce an inhibition that shuts down the spontaneous spiking of a DCN cell for ∼50 ms. If bursts of such brief light pulses are delivered, a fixed pattern of bistable bursting emerges. If these pulses are delivered continuously to a spontaneously bistable cell, the immediate response to such photostimulation is inhibitory in the cell's depolarized state and excitatory when the membrane has repolarized; a less regular burst pattern then persists after stimulation has been terminated. These results indicate that the spiking activity of DCN cells can be bidirectionally modulated by the optically activated synaptic inhibition of cortical PCs.