Identifying the targets of the amplifying pathway for insulin secretion in pancreatic beta-cells by kinetic modeling of granule exocytosis

Abstract

A kinetic model for insulin secretion in pancreatic beta-cells is adapted from a model for fast exocytosis in chromaffin cells. The fusion of primed granules with the plasma membrane is assumed to occur only in the "microdomain" near voltage-sensitive L-type Ca(2+)-channels, where [Ca(2+)] can reach micromolar levels. In contrast, resupply and priming of granules are assumed to depend on the cytosolic [Ca(2+)]. Adding a two-compartment model to handle the temporal distribution of Ca(2+) between the microdomain and the cytosol, we obtain a unified model that can generate both the fast granule fusion and the slow insulin secretion found experimentally in response to a step of membrane potential. The model can simulate the potentiation induced in islets by preincubation with glucose and the reduction in second-phase insulin secretion induced by blocking R-type Ca(2+)-channels (Ca(V)2.3). The model indicates that increased second-phase insulin secretion induced by the amplifying signal is controlled by the "resupply" step of the exocytosis cascade. In contrast, enhancement of priming is a good candidate for amplification of first-phase secretion by glucose, cyclic adenosine 3':5'-cyclic monophosphate, and protein kinase C. Finally, insulin secretion is enhanced when the amplifying signal oscillates in phase with the triggering Ca(2+)-signal.

FIGURE 1

Two-compartment model for intracellular calcium ion dynamics. (A) Ca²⁺-channels and the “microdomain” of a β-cell model. The influx of Ca²⁺ is handled by two types of voltage-gated Ca²⁺-channels: the L-type and the R-type. A microdomain is defined as the half-sphere surrounding the inner mouth of an L-type Ca²⁺-channel with a diameter of ∼0.3 μm to provide a phenomenological compartment with elevated Ca²⁺; the region of elevated Ca²⁺ in a real cell would in fact be much smaller, perhaps 10 nm. When the cell membrane is depolarized, Ca²⁺ passing through the L-type channel first goes into the microdomain and then diffuses to the cytosol, whereas only the contribution of Ca²⁺ passing through the R-type channel to cytosolic Ca²⁺-concentration is accounted for. (B) The two-compartment kinetic model describing the dynamic distribution of Ca²⁺ between the microdomain and the cytosol compartments. and denote the concentration of Ca²⁺ in the microdomain and the cytosol compartment, respectively, B is the rate of exchange between the microdomain and the cytosol, and denotes the rate of Ca²⁺ clearance from the cell, which is handled by four kinds of Ca²⁺-transporters (see text).

FIGURE 2

Effect of B on [Ca²⁺] distribution. The time course of [Ca²⁺] in the microdomain (C_md) and cytosolic (C_i) compartments after a train of alternating square pulses between and with a period of 16 s, 8 on and 8 off for B = 200 (solid curves) and 250 (dotted curves).

FIGURE 3

(A) Schematic drawing of the EC proposed for pancreatic β-cells. The thick long line represents the plasma membrane of the cell, and the two shaded blocks on the membrane represent an L-type Ca²⁺-channel where a microdomain is formed. The state of the granule-membrane complex is schematically represented by the shape of the drawing between the granule and the membrane: state 6 consists of vesicles “docked” but not yet primed for fusion; state 5 is the primed vesicles outside the microdomain; state 1 is the primed vesicles bound to the microdomain; state 4 is the prefusion state; F represents the fused state; and R represents the insulin releasable state. The Ca²⁺-triggering step involves the Ca²⁺ in the microdomain, whereas both the resupply and the priming steps involve the Ca²⁺ in the cytosol. (B) Kinetic scheme proposed for the EC in (A). is the concentration of Ca²⁺ in the microdomain. (C) Expressions showing the dependency of and on the concentration of Ca²⁺ in the cytosol compartment ().

FIGURE 4

Fitting the model with the experimental data of Henquin et al. (39). For the two panels on the left, the insulin secretion rate (ISR) is plotted as a function of time after a train of five alternating square membrane potentials between −70 mV and −20 mV with a period of 12 min is applied at time 0, whereas a step depolarization is applied at time 0 for the right two panels. G0 means no glucose is present in the bathing solution, and G3 means the concentration of glucose is 3 mM as in the experiments. In the model calculations, the value of in Table 2 is set to 0 for the G0 case, whereas is multiplied by a factor of 1.2 for the G3 case. The lines with filled circles are the experimental curves of Henquin et al., with basal secretion removed, and those with open squares are calculated with the model.

FIGURE 5

Fitting the model with the experimental data of Fig. 4 C (8). (A) The total capacitance of a single cell after the cell is depolarized from −70 mV to −20 mV at time 0 is shown as a function of time. The experimental data are shown as filled squares and the calculated data as the solid line. Capacitance is calculated with Eq. 10a, with each fused granule assumed to increase the membrane capacitance by 3.5 fF. (B) The total number of granules reaching the releasable state R (Fig. 3 B) is obtained using Eq. 10b as a function of time after the cell is depolarized at time 0 with a step to −20 mV of 0.5-s duration followed by return to rest (−70 mV). The data are normalized with that measured at the 7-s time point after the onset of the depolarization.

FIGURE 6

Simulated effects of amplification on first- and second-phase secretion. Insulin secretion rate (ISR) is shown for the model run with the standard parameters and initial conditions in Tables 1–3 but with V_m stepped to −20 mV (solid line) at time 0 to represent glucose-induced depolarization. A rising second phase is observed if is increased by a factor of 2 (dashed line) or 3 (dot-dashed line) to represent the hypothetical amplifying effect of glucose on the rate of resupply of vesicles from the reserve pool to the docked pool. If glucose is assumed to amplify priming and resupply equally by a factor of 2 (dotted line), both first- and second-phase secretion are increased, but second-phase secretion is flat, not rising.

FIGURE 7

Simulation of potentiation. Stimulatory glucose is applied at time 0, maintained for 60 min, removed for 10 min, and then restored. Stimulation is modeled as an increase in V_m to −20 mV combined with a threefold increase in The response to the second step of glucose is potentiated due to a buildup in the docked pool that persists during the low-glucose gap and consequent refilling of the primed pool as exocytosis of vesicles ceases.

FIGURE 8

Ablation of first phase by ramped glucose. When glucose is ramped up, represented by a linear increase in V_m from −70 to −20 mV and in the factor multiplying from 1 to 3, the first phase is nearly completely abolished. Compare the ramped increase (solid line) to the stepped increase (dashed line), which is equivalent to the dot-dashed line in Fig. 6. All other parameters and initial conditions are as in Tables 1–3.

FIGURE 9

Simulation of R-channel knockouts. Depolarization stimulated insulin secretion rate (solid curve) is first calculated for the standard cell model; the amplifying effect of increased glucose is modeled by increasing the value of by a factor of 3 at time 0. The knockout is simulated by setting g_R in Eq. 4b to 0 (dashed curve). Removal of the R-type channels decreases the second-phase insulin secretion rate. If the L-type channels are removed by setting g_L in Eq. 4a to 0 and R-type channels are upregulated by increasing g_R fourfold, the second phase is relatively more restored than the first phase (dotted curve).

FIGURE 10

Amplifying the signal sensitivity test of the EC steps. The thick solid line in each of the six figures is the usual biphasic insulin secretion rate calculated for the model at rest by applying a step depolarization from −70 to −20 mV at time 0. The thin solid line with filled circles in each figure is obtained after the rate constant indicated in the figure is increased 10 min after the onset of the depolarization. A threefold increase of can generate the second-phase pattern observed in the experiment of Fig. 4 (13), and a threefold increase of can generate a large transient reminiscent of first-phase secretion. None of the other parameters can produce a significant, long-lasting increase in secretion despite a 10-fold increase.

FIGURE 11

Effect of the phase shift between bursting membrane potential and oscillating rate constant. The period of the bursting potential and the period of the oscillating rate constant are assumed to be identical. For each oscillating rate constant (), the “averaged” insulin secretion rate is calculated for the model at steady state as a function of the angular shift, where and are, respectively, the time shift and the period of the two oscillations, for the case Oscillations have a significant impact only when applied to the priming () and resupply () steps. The averaged ISR at steady state is obtained by discarding the first 20,000 min of simulations and averaging over the last five periods. Identical curves are also obtained for ranging from 60 to 600 s; i.e., the phasic effect is found to be insensitive to the frequency of the oscillation.

FIGURE 12

Accounting for reduced fusion in situ. Solid curves are run with the same parameters and initial conditions as the dot-dashed curve of Fig. 6, dashed with r₁ reduced to 0.02 and initial values recalculated accordingly. This 30-fold reduction in r₁ reduces the peak fusion rate 15-fold (A). The net capacitance increase has almost caught up by the end of the first minute (B), and response of insulin secretion rate to a 1-h step of glucose is minimally affected.