Defective Amplifying Pathway of β-Cell Secretory Response to Glucose in Type 2 Diabetes: Integrated Modeling of In Vitro and In Vivo Evidence

Impaired insulin secretion from pancreatic β-cells is a long-recognized hallmark of type 2 diabetes (T2D) (1) and plays a crucial role in the pathogenesis of the disease, as documented in countless in vivo studies. In parallel, in vitro investigation of islets or cells from both animals and humans has formed the basis of our current knowledge of insulin secretory mechanisms (2–4).

Despite a tremendous increase of knowledge, our understanding of β-cell dysfunction in T2D remains limited. One reason for this underwhelming outcome is the impossibility to use in vivo the sophisticated approaches developed for islet research; in addition, the availability of human islets is limited, and the studies are difficult to perform. Thus, a gap persists between our understanding of the mechanisms that regulate insulin secretion, mostly derived from in vitro studies, and their relevance to in vivo physiology and β-cell dysfunction of T2D. For example, the crucial and independent role of calcium and glucose (the so-called triggering and amplifying pathways) is well characterized in vitro but not in vivo. In addition, the involvement of these pathways in T2D is unclear.

We used mathematical modeling to address these issues. Mathematical models have been used to shed light on the mechanisms of insulin secretion since the early days of islet research (5–7). This approach has provided a robust mathematical representation for a variety of experimental conditions from ex vivo or in vivo studies. New knowledge from experimental studies has been incorporated into new models, with a special focus on the role of calcium in exocytosis (8,9), but because the model representation of the glucose-calcium-insulin secretion relationship is often incomplete, only limited extrapolation from in vitro to in vivo findings has been attempted, and β-cell dysfunction in diabetes has been marginally addressed.

We propose a relatively simple model to explain in vitro and in vivo findings with consistent mechanisms and to gain insight into β-cell dysfunction in T2D. The mathematical model has three aims: 1) to provide a parsimonious and physiologically based description of key in vitro findings, 2) to describe a variety of experimental conditions from in vivo studies in both healthy volunteers and patients with T2D, and 3) to identify the mechanisms underlying the secretory defects in human T2D.

We used data from three studies carried out in mouse islet preparations in which intracellular calcium concentrations were measured. Their designs are summarized below, and more details can be found in the original publications (10–12).

Henquin et al. (10) characterized the amplifying pathway by decoupling the effects of calcium and glucose on insulin secretion. K_ATP channels were held open by diazoxide, and either a step increase in extracellular potassium chloride or a sequence of pulses were used to depolarize β-cells and steadily or intermittently raise calcium concentration. The protocol was repeated at extracellular glucose concentrations of 0 and 3 mmol/L to identify the amplifying effect of glucose.

Henquin et al. (11) included a hyperglycemic clamp performed on islets at three glucose levels (11.1, 16.7, and 30.0 mmol/L), starting from a basal glucose concentration of 8.5 mmol/L.

Jonkers and Henquin (12) performed a hyperglycemic clamp that raised glucose in sequential steps in mouse β-cell clusters and showed that although second-phase insulin secretion follows the glucose increase, first-phase insulin release is progressively blunted.

We used data from seven of our previous studies in human subjects with normal glucose tolerance or T2D. The study designs are summarized below; more details can be found in the original publications. Glucose, insulin secretion (reconstructed by using C-peptide deconvolution [13]), and other parameters were obtained from the original databases. A summary of the subject characteristics is provided in Table 1.

A glucose bolus (0.3 g/kg) was injected, and the test lasted 240 min. We report data from the first 90 min because insulin secretion had returned to the basal level by this time (14).

A 120-min hyperglycemic clamp was performed in young obese subjects, raising glucose to ∼12.5 mmol/L (15).

Glucose was infused intravenously at increasing rates for 180 min to create a glucose ramp, starting from the basal level and reaching a maximum value of ∼15 mmol/L above basal (16).

This test was performed as part of studies on the incretin effect (17,18). An oral glucose tolerance test (OGTT) was performed first, and subsequently, the same glucose profile was generated by intravenous glucose infusion. The study lasted 180 min. We used the data obtained with intravenous glucose infusion.

Glucose was raised in three 30-min sequential steps from the basal value to ∼15 mmol/L. The test was performed in normal subjects (19).

First-phase insulin secretion was evaluated by using a short (8-min) intravenous glucose tolerance test (IVGTT), and insulin sensitivity was evaluated by using the euglycemic glucose clamp (20). We considered the data from subjects with normal glucose tolerance only.

First-phase insulin secretion was evaluated by using an IVGTT (21).

Figure 1 shows a schematic representation of the β-cell. An immediately releasable pool (IRP) accounts for the insulin granules that undergo rapid exocytosis when intracellular calcium increases (triggering pathway). Over a longer time, insulin secretion is sustained by the refilling of the IRP (amplifying pathway). Refilling is controlled by both calcium and glucose (through the refilling function). Calcium concentration is an input to the model shown in Fig. 1 and is required to calculate insulin secretion.

IRP dynamics are represented by:

where Q(t) is the insulin mass in the IRP (measurement units depend on the experimental data [e.g., picograms of insulin per islet]), k(t) is the calcium-controlled exocytosis rate constant (per minute), and r(t) is the refilling rate that depends on both calcium and glucose (units are also experiment specific [e.g., picograms per min per islet]).

To describe the dependence of k(t) and r(t) on calcium [C(t) in nanomoles per liter] and glucose [G(t) in millimoles per liter], we used two functions, respectively: f_k(C(t)) and f_r(C(t),G(t)), which have a similar structure. Both f_k and f_r depend on calcium in a quasilinear way above a given threshold; below this threshold, the dependence is still virtually linear but much flatter (Supplementary Fig. 1). For f_r, the slope above the threshold is increased by glucose concentration. These functions are described in detail in the (Supplementary Data, Eqs. 1–4).

The model assumes that a delay occurs between a change in glucose and calcium concentrations and the corresponding changes in k(t) and r(t). The delay is a combination of a time-shift and a first-order delay (Supplementary Data, Eq. 5), and is represented here in compact form with the operator ∂{·}. Thus, Eqs. 2 and 3 for k(t) and r(t), respectively, are:

where the subscript of ∂{·} indicates whether the delay refers to exocytosis (k) or refilling (r). It should be noted that the delay has a different physiological interpretation for the calcium effect on k(t) and for the refilling. For k(t), the role of the time-shift is to accommodate for possibly imprecise alignment of the calcium and secretion data; the potential delayed effect of calcium on k(t) is accounted for by the first-order delay component. Thus, ∂_k plays a minor role. For r(t), the delay is an essential component of the description because a delayed rise of the refilling is crucial to explaining typical features of insulin secretion.

The instantaneous insulin secretion rate S(t) (units depend on the experiment [e.g., picograms per minute per islet]) is shown in Eq. 4 as follows:

Equations 1–4, with initial steady-state conditions, provide the complete description of the model; insulin secretion can be predicted when both glucose and calcium are known.

Cytosolic calcium concentration is available from key in vitro studies in mice, but it is lacking from in vivo studies in humans. For simulation of human studies, we have developed a simplified model that predicts calcium concentration from glucose concentration (Supplementary Fig. 2) on the basis of known relationships in mice (11). The model expresses calcium concentration as a sum of a static and a dynamic component:

The static component is a sigmoidal function of glucose concentration C_s(G(t)); the dynamic component (C_d(t)) exhibits a calcium peak when glucose concentration is abruptly increased, as experimentally observed, and generated with a simple zero-gain linear model. The calcium model is described in detail in Supplementary Data, Eqs. 6–9.

We considered the average data in the analysis. Parameters were determined by fitting the model to the data by using least-squares methods or empirically. The parameters of the calcium model were obtained from a mouse study (11) and used with minimal changes in all tests in humans. In mouse studies, all simulations were performed by using experimental calcium traces.

Figure 2A shows insulin secretion rates (measured and calculated by the model) for the experiment by Henquin et al. (10). The data and the model consistently show that calcium is a key regulator of insulin secretion and that at 3 mmol/L glucose compared with 0 mmol/L, the secretory response is more sustained as a result of the amplifying pathway.

The model reproduces the experimental data because glucose influences the refilling function. When glucose is 3 mmol/L, the slope of the relationship between calcium and refilling is steeper, and refilling and insulin secretion are more sustained. When glucose is 0 mmol/L, refilling is blunted, and calcium-stimulated exocytosis depletes the IRP.

Figure 2B shows calcium concentrations and the insulin secretion rates (measured and calculated by the model) in response to three glucose steps of the hyperglycemic clamp in mouse islets (11). The first-phase insulin secretion peak requires a rapid and marked elevation of calcium and corresponds to the release of insulin granules from the IRP. The second phase is determined by the elevation of calcium but also requires the amplifying action of glucose, which influences the refilling of the IRP. The model accurately describes basal and first-phase secretion and the flatter or steeper second phase under the three different conditions. The delay in the refilling equation determines the slowly ascending second phase. Of note, model parameters are the same for all three glucose steps.

Figure 2C shows calcium concentrations and insulin secretion rates (measured and calculated by the model) during the sequential glucose steps of the experiment by Jonkers and Henquin (12). The model explains the loss of first-phase secretion by a progressive IRP depletion; the reduction of the calcium peak at the beginning of the glucose steps also contributes to the effect.

Figure 2D reproduces the potentiating effect of glucose on the relationship between calcium concentration and insulin secretion, which becomes steeper at higher compared with lower glucose concentrations, as shown by Henquin (22). The key determinant of this hallmark effect of the amplifying pathway is the dependence of the refilling function on both calcium and glucose.

Figure 3 shows how the calcium model reproduces the calcium response in the study by Henquin et al. (11). One notable aspect is the transient initial peak in calcium concentration, which has relevance in shaping the peak of first-phase insulin secretion (Supplementary Fig. 3).

Figure 4 (left) shows the measured glucose concentrations and insulin secretion (measured and calculated by the model) in the tests performed in subjects with normal glucose tolerance. The model-predicted calcium concentration in these tests and the stepped hyperglycemic clamp are shown in Supplementary Fig. 4. The model parameters for the simulations (Supplementary Table 1) were similar across all studies. The parameters of the calcium model were the same in all tests and similar to those derived from the mouse study (11). The relatively less accurate fit in the hyperglycemic clamp in young subjects (Fig. 4B) is attributable to a markedly higher first-phase secretion in this study compared with the other studies.

To simulate the insulin secretion response in the same set of tests as applied in normal subjects, we hypothesized an isolated defect of glucose action on refilling, with the magnitude of the defect depending on the specific study. As shown in Fig. 4 (right), this defect is sufficient to consistently explain all the impaired insulin secretion responses typical of diabetes. Similarly to normal subjects, first-phase insulin secretion is underestimated by the model in the hyperglycemic clamp (Fig. 4B). However, both in the experimental data and in the simulations, the secretion peak is decreased by two-thirds in subjects with T2D compared with normal subjects.

Figure 5B and C focus on the difference between normal subjects and subjects with T2D in the refilling function of the glucose ramp test (Fig. 4C); other conditions are shown in Supplementary Fig. 5. The impaired refilling in subjects with T2D represents a defective amplifying pathway. In all simulations in subjects with T2D, the relationship between glucose and calcium concentrations (i.e., the model shown in Supplementary Fig. 2) and the exocytosis function (Fig. 5A and Supplementary Fig. 5) were the same as in normal subjects.

Because calcium also affects the refilling function and because decreased calcium levels in diabetes cannot be ruled out, we investigated whether a decrease in calcium could explain the data in subjects with T2D. As detailed in Supplementary Fig. 6 and discussed below, the results are inconsistent and suggest that the diabetic response cannot be explained by a sole defect in calcium changes.

The increase in first-phase insulin secretion observed with insulin resistance and its decrease with basal hyperglycemia are simulated in Figs. 6 and 7. Because insulin resistance increases fasting insulin secretion, the increased refilling rate augments the IRP size as long as glucose concentration remains normal, thereby enhancing first-phase insulin secretion upon glucose stimulation (Figs. 6 [gray line] and 7A). When fasting glucose is increased above the normal level, exocytosis is markedly stimulated, the IRP is decreased, and first-phase insulin secretion consequently is reduced (Figs. 6 [black line] and 7B).

The IRP dynamics underlying first-phase insulin secretion also are operative in the test mimicking the OGTT (Fig. 4D), where the glucose increase is more gradual. The fast release of insulin granules from the IRP generates an anticipated initial rise in insulin secretion, which is impaired in subjects with T2D in whom the IRP is reduced (Supplementary Fig. 7).

We sought a unifying explanation for a series of characteristic features of insulin secretion by using mathematical modeling. On the basis of the formal representation of both in vitro and in vivo processes, we conclude that one key β-cell function defect in T2D is the amplifying pathway (i.e., the role of glucose per se to enhance insulin secretion). Although the use of mathematical models in the study of insulin secretion is a time-honored approach, the current development is the first in our knowledge to show a wide agreement with a wealth of experimental data in vitro and in vivo.

In the current model, cytosolic calcium controls exocytosis by mediating the emptying of the IRP, thus simulating the triggering pathway activated by an increase in extracellular glucose. A crucial aspect is that refilling is modulated by both glucose and calcium, which is necessary and sufficient to explain the results of the experiments shown in Fig. 2A (10) and the fact that at near steady state, insulin secretion increases linearly with calcium concentration (22) (Fig. 2D). The model also describes the first-phase response and its modulation as a balance between exocytosis and refilling, whereas the second phase is determined by refilling.

The IRP in the model can be identified with the pool of membrane-docked granules that undergo rapid exocytosis upon elevation of cytosolic calcium; however, the model does not identify refilling with a specific mechanism. Thus, processes taking place both at the cell membrane (change of granule status) and in the cytosol (granule translocation) may be involved. These may include an augmented recruitment of reserve granules by the cytoskeleton (2,4), an acceleration of the priming process that confers release competence to insulin granules (2,23) or a combination of such effects mediated by a variety of metabolic pathways activated by glucose (24,25). Although the lack of a more elaborate cellular description of the refilling might be regarded as a limitation, our representation effectively shows how the refilling process modulates the insulin secretion phases in vitro and in vivo, regardless of the details of the subprocesses.

The model predicts a gradual increase in refilling when the glucose stimulus is maintained, thereby explaining the ascending second-phase response of the hyperglycemic clamp (Figs. 2B and 4B; in Fig. 4B, the ascending second-phase is less visible because of the axis scales). This phenomenon is modeled as a simple first-order delay; other models have used more complex representations with an analogous rationale and similar results (i.e., an effect on refilling that is delayed compared with the glucose stimulus) (8,9). An exception is the model by Dehghany et al. (26) in which the ascending second-phase response is explained by a time-dependent increase of the number of docking sites at the cell membrane. Straub et al. (27) reported an increase in docking sites upon glucose stimulation, but showed that the increase is not related to insulin secretion. In vivo, an ascending second phase is a typical finding, and in vitro, it is observed in mouse islets when prestimulus glucose is in the physiological range (11), whereas it appears to be lacking in human islets (28). The mechanisms underlying this phenomenon thus remain elusive, and we deliberately kept the model simple in this respect. We interpret this slowly increasing response to hyperglycemia as a manifestation of adaptation to increased insulin demands, an important function of the β-cell in the presence of insulin resistance. Such adaptation is operative in several conditions, particularly in vivo (29–31), and therefore deserves additional investigation.

The empirical model of the relationship between glucose and calcium concentration (Supplementary Fig. 2) is based on experimental steady-state relationships between the two variables in mouse islets (11) and a simplified representation of the calcium peak typically observed when glucose is increased abruptly. The model does not attempt to capture the complexity underlying the onset of the early calcium peak, which may involve synchronous initial membrane depolarization followed by asynchronous oscillations (3). However, the model attributes first-phase insulin secretion to the high sensitivity of the exocytotic mechanisms to calcium. Without this mechanism, the early insulin secretion peak on a hyperglycemic clamp (or IVGTT) would be less sharp than observed (Supplementary Fig. 3), and the insulin secretion peaks in response to each glucose step in the stepped hyperglycemic clamp would not be adequately reproduced. This phenomenon has not been considered by other models.

The representation of first-phase insulin secretion as a calcium-mediated rapid release of granules from a single IRP, to which an initial calcium spike also contributes, is an approximation that does not fully reproduce the subtle features of the sharp peak, particularly in vivo. However, the limited precision of available experimental data does not permit testing of more sophisticated dynamics. Intracellular calcium, a crucial player, is not available in humans, and in mice, it is measured with limited temporal accuracy. Nevertheless, regardless of the subtleties of the first-phase response, its main features, particularly modulation in sequential hyperglycemic clamps (Fig. 2C and Supplementary Fig. 4E), dependence on insulin resistance (Figs. 6 and 7A), progressive loss with hyperglycemia (Fig. 6 and 7B), and even its role during the test mimicking the OGTT (Supplementary Fig. 7), are faithfully represented. The overall implication from the modeling, therefore, is that first-phase secretion, despite its importance as a marker of β-cell dysfunction, is likely a consequence of other primary players.

The parameters of the in vitro hyperglycemic clamp (11) (Fig. 2B) and of the in vivo studies are remarkably homogeneous, except for those determining the insulin secretion scale (Supplementary Table 1). Such differences, which are clearly identified in the model, are expected because in the various studies, insulin secretion is expressed by different metrics. Of relevance are the parameters characterizing the dose responses and the refilling delay. Such parameters are homogeneous across all the in vivo studies but show sizeable differences across the in vitro studies (Supplementary Table 1). This heterogeneity possibly reflects the diversity of the experimental preparations, although model limitations cannot be excluded.

One of the main goals of the study was to explore the potential mechanisms of β-cell dysfunction in T2D. This analysis was made possible by the availability of in vivo data in patients with T2D by using the same study protocols as used in subjects with normal glucose tolerance (Fig. 4). A main finding is that to reproduce diabetic responses (Fig. 4, right), decreasing the effects of glucose in the refilling dose response was sufficient (Fig. 5 and Supplementary Fig. 5). Diabetic fasting hyperglycemia per se decreased the fasting IRP size because it increased calcium-mediated exocytosis, producing a markedly reduced first-phase secretion. This feed-forward mechanism underlies the progressive loss of first-phase response, with fasting hyperglycemia preceding diabetes (Fig. 7B). Although that the impaired insulin secretion response in T2D can be explained by a single defect in the refilling mechanism is remarkable, the model does not identify such a defect with a physical mechanism. However, the model does describe the diabetic condition through mechanisms independent of differences in the calcium signal.

We also evaluated the alternative hypothesis of a defective increase in calcium. As detailed in Supplementary Fig. 6, on the assumption of an impaired calcium signal, refilling and second-phase insulin secretion both are decreased, which is similar to what we obtain by postulating a defective refilling function (Figs. 4 and 5). However, decreased basal calcium would lower basal exocytosis, increase the IRP size, and increase rather than decrease first-phase insulin secretion (Supplementary Fig. 6A). Therefore, to impair first- and second-phase insulin secretion, calcium must be decreased only at suprabasal glucose concentrations (Supplementary Fig. 6B). This condition would imply that in the absence of an additional defect in the refilling function, basal insulin secretion (relative to glucose) is normal in subjects with T2D, which is contrary to experimental findings (18,32).

The cellular mechanisms underlying β-cell dysfunction in T2D are still under intense investigation. On the basis of indirect findings, Henquin et al. (28) proposed that fasting hyperglycemia can reduce the rapid, large increase in cytosolic calcium in response to a hyperglycemic clamp. However, calcium measurements were not performed in this study. Our knowledge about β-cell calcium levels in T2D is incomplete and only based on animal data, where the degree of alteration of calcium influx upon glucose stimulation has been reported to be highly variable across studies, spanning from no differences to a delayed or reduced response (33). Supporting the hypothesis of a defective amplifying pathway in T2D, Ferdaoussi et al. (24) identified a pathway linking glucose metabolism to the amplification of insulin secretion and demonstrated that the pathway is defective in T2D cells from both humans and mice and that restoration of this pathway rescues β-cell function. Recently, Henquin et al. (34) confirmed the important contribution of the amplifying pathway in both mouse and human islets and proposed that this pathway is a plausible cause of insulin secretion defects in T2D. The current results support the hypothesis of a defective amplifying pathway as the key dysfunctional mechanism in T2D on the basis of the dynamic characteristics of insulin secretion in vivo. We also show that within the model assumptions, an isolated defect in calcium levels is not compatible with the observations in T2D.

Our modeling approach provides a unified view of the processes underlying several key features of insulin secretion in vitro and in vivo. The analysis identifies the alterations underlying some typical findings, such as the increased first-phase response in insulin resistance and the β-cell dysfunction of T2D. However, although our approach highlights the role of the amplifying pathway, it cannot, and was not aimed to, identify the corresponding cellular alterations. In addition, the scarce knowledge on the possible alterations of calcium in insulin resistance and glucose intolerance (33) limits the possibility to validate the proposed mechanisms.

In conclusion, the current modeling analysis offers a unifying explanation of a series of characteristic features of insulin secretion and posits that one major defect in the β-cell response to glucose in T2D is at the level of the supply of granules to the IRP, which we relate to the amplifying pathway. The loss of the first-phase response is a consequence of this dysfunction, and it is not necessary to postulate a defect of the calcium-mediated triggering pathway.

Acknowledgments. The authors thank Valentina Nofrate (formerly at the Institute of Neuroscience, National Research Council) for contributions to the early development of the model.

Funding. This study has received support from the Innovative Medicines Initiative Joint Undertaking under grant agreement No. 115156, resources of which comprise financial contributions from the European Union’s Seventh Framework Programme (FP7/2007-2013) and in-kind contributions from companies of the European Federation of Pharmaceutical Industries Associations. The Drug Disease Model Resources project also is financially supported by contributions from academic and small/medium-sized enterprise partners.

Duality of Interest. No potential conflicts of interest relevant to this article were reported.

Author Contributions. E.G. analyzed the data. E.G., T.G., and A.M. developed the model and wrote the manuscript. E.G., T.G., S.A., A.N., E.F., and A.M. reviewed the manuscript. S.A., A.N., and E.F. contributed the experimental data. A.M. planned the study. A.M. is the guarantor of this work and, as such, had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

Prior Presentation. Parts of this study were presented in oral form at the 25th Meeting of the European Group for the Study of Insulin Resistance, Pisa, Italy, 12–14 May 2016, and in abstract form at the 52nd Annual Meeting of the European Association for the Study of Diabetes (EASD), Munich, Germany, 12–16 September 2016, and the EASD Islet Study Group Meeting, Garmisch-Partenkirchen, Germany, 16–18 September 2016.