Amino acids (AAs) are well known to be involved in the regulation of glucose metabolism and especially insulin secretion (1). Different AAs may exert distinct effects on postprandial glucose and insulin concentrations, which can be affected by their co-ingestion with glucose (2). Recently, much attention has been devoted to the branched-chain amino acids (BCAAs), comprising leucine, isoleucine, and valine. They are essential AAs that can be taken only with diet, meat and dairy products being the main sources (3). Whole-body disposal of BCAAs mainly results from their catabolism in the skeletal muscle and altered BCAAs catabolism has been associated with several disorders, such as obesity and type 2 diabetes (4). Moreover, fasting and postprandial plasma levels of BCAAs have been found associated with greater insulin secretion and reduced insulin clearance in healthy humans (5). Also, short-term dietary reduction of BCAAs reduced meal-induced insulin secretion (6). On the other hand, in recent-onset diabetes, BCAAs were associated negatively with meal-induced insulin secretion (7). Thus, understanding the effects of different AAs on insulin release and kinetics under different metabolic conditions may help to gain further insight on the development of several diseases, including type 2 diabetes, and to develop therapeutic strategies based on appropriate dietary regimen.

Mathematical models may help to analyze these effects. However, to the best of our knowledge, only one attempt has been done in this direction, considering the effects of AAs on the electrical activity leading to insulin granule exocytosis, based on in-vitro experiments (8). A model describing the AAs effect on the insulin kinetics at whole body level during a physiological metabolic test, such as a mixed-meal tolerance test (MMTT), is still lacking. Thus, our aim is to propose a new mathematical model, describing insulin kinetics in plasma during a MMTT, which includes the effect of AAs. To this end, a series of mathematical models was developed, analyzed and compared, on the basis of previously published data (7). The resulting chosen model may help in characterizing the different role of AAs (or specific categories of them, such as BCAAs) on insulin secretion.

The top model resulted Model 1, which was the best compromise between model parsimony and performance. Indeed, Model 1 showed global identifiability and the better AICc (20.53), as compared to other models with similar global identifiability (25.36, 25.86 and 27.07 for Model 5, 2, 3, respectively. Model 4 showed the best AICc, with very low value (-135.51) due to extremely accurate model fit, but it showed only local rather than global identifiability. Considering the relative likelihood, Model 4 would be again the best one, but for the previously mentioned identifiability issues Model 1 was preferable. When computing the relative likelihood of Model 1 vs. Model 2, 3 and 5, we found values remarkably higher than 1 (relative likelihood equal to 11.19, 14.39, 26.35, vs. Model 5, 2 and 3, respectively), this providing further support in considering Model 1 as the best model. For these reasons, we assumed that the model to be selected and hence analyzed further over the virtual populations was in fact Model 1. In any case, some results from the other models are reported in the Appendix A.

Results of model validation on the experimental average data of the study (7) provided the best fit reported in Figure 4 . In terms of the model parameters, for k_GL the estimated value was 0.0974 in CNT and 0.2087 in T2D (units: μU·ml^-1/(mg·dl^-1·min)); for k_AA, the estimated value was 0.0247 in CNT and -0.0048 in T2D (units: μU·ml^-1/(μmol·l^-1·min)). It is worth noting that, differently from k_GL, which has been constrained to be positive, k_AA was free to get possible negative values, since we could not a priori exclude a negligible or even deleterious effect of BCAAs on insulin secretion. Interestingly, in CNT the k_AA value was estimated positive (95% CI: 0.0168 – 0.0325), whereas based on the 95% CI, in T2D we obtained k_AA not significantly different from zero (95% CI: -0.0281 – 0.0185), suggesting a negligible contribution to insulin secretion from BCAAs.

Insulin, glucose and BCAA curves for the two virtual populations are shown in Figures 5 , 6 . Model validation on the two virtual populations provided the mean best fit and related residuals shown in Figures 7 , 8 . Distributions of k_GL and k_AA over the virtual CNT and T2D subjects are reported in Figure 9 . In terms of mean and SD, the values for k_GL were 0.3480 ± 0.1778 and 0.1134 ± 0.0654 μU·ml^-1/(mg·dl^-1·min), in CNT and T2D, respectively. In CNT, k_AA was 0.0134 ± 0.0112 μU·ml^-1/(μmol·l^-1·min), and according to the individual 95% CIs it was positive in the 53% of cases, negative in the 2%, and negligible in the 45% (i.e., CI crossing the zero value). In T2D, k_AA was 0.0075 ± 0.0142 μU·ml^-1/(μmol·l^-1·min), and it was positive in only 25% of cases, negative in the 65%, and negligible in the 10%. An inverse correlation was detected between k_GL and k_AA in CNT (r = -0.63, p<0.001) and T2D (r = -0.92, p<0.001). The inverse relationship between k_GL and k_AA was confirmed in CNT and T2D pooled (p<0.0001), with subjects’ type (CNT and T2D) being significant factor in the relationship (p<0.0001).

Furthermore, the unpaired t-test showed for both k_GL and k_AA a significant difference between virtual CNT and T2D (p<0.0001 for both parameters). In addition, we found that the percentages of positive, negative and negligible k_AA values (see percentage values reported above) were different between the two groups (p=0.0015 or lower). Consistently, in CNT the percentage of positive k_AA values was higher than that of negative values, whereas in T2D it held the opposite condition (p<0.0001 for both tests).

In this study, we propose a modelling approach to determine effects of AAs on insulin secretion in humans with normal glucose tolerance and T2D. The importance of our model strategy lies in the opportunity to estimate in single individuals the possible effects of AAs on insulin secretion, in addition to the known role of glucose. Specifically, the model assesses AAs effects on insulin secretion under dynamic (stimulated) conditions, but on the other hand physiologically consistent, as determined by food ingestion. The model includes glucose as the major determinant of insulin secretion, but also AAs during the mixed-meal ingestion. The k_AA model parameter in the single person may then be seen as the individual sensitivity of the beta cell to AAs stimulation (in short, beta-cell AAs sensitivity).

Based on the experimental data in the glucose tolerant people that we exploited for our model development (7), we found a significant contribution of AAs in explaining insulin secretion (i.e., positive k_AA, with positive 95% CI). Instead, in T2D, the model essentially indicated a negligible k_AA (CI including zero), and hence negligible contribution of AAs to insulin secretion. This result is consistent with that of the study by Karusheva et al. (7) in T2D (based on the analysis of the patients’ experimental data without model support). With regard to the virtual population, we found that in the control participants k_AA was positive in more than half of cases, whereas in T2D k_AA was positive only in 1/4 of cases. Accordingly, we found that k_AA was higher in virtual CNT than T2D. We also found that the relationship between k_AA and k_GL was significant, but different between the two virtual groups. These results need further confirmation in future studies on real data, but they appear reasonable and physiologically plausible, thus further suggesting appropriateness and reliability of our modelling approach.

One aspect of our results that, in contrast, appears somehow questionable relates to the k_GL values in the experimental data. Indeed, k_GL was lower in the CNT than in T2D. However, it has to be considered that the average data in each of the two participants’ group represent kind of further single subject, different from the real ones. On the other hand, the virtual populations analysis showed that parameters value distributions in the two groups partially overlap, thus meaning that there are in fact some virtual CNT subjects with k_GL lower than in some virtual T2D subjects. This aspect emphasizes the relevance of the virtual analysis as well. Thus, the indicated result, found in the experimental-based average data analysis, is not unaccountable. It is also worth noting that the lower k_GL in CNT than in T2D appears compensated by the higher k_AA, which is significant in CNT while negligible in T2D.

The remarkable importance of studying the stimulatory effect of AAs on insulin secretion has been recognized already decades ago (1, 16–18). Among the AAs, arginine has been clearly shown as a potent insulin secretagogue (18, 19). Notably, the insulinotropic effect of arginine was shown in both normal glucose tolerance (20) and T2D (21). On the other hand, apart for arginine, the possible insulinotropic effect of specific AAs has been investigated in several studies. Already in one of the earlier studies, it was reported that in healthy people intravenous administration of leucine was able to induce a robust release of insulin that conversely was not observed when injecting a solution of eight amino acids not containing leucine (22). Those findings were confirmed and corroborated some years later in an animal model by another study, which analyzed the effects of intravenous infusion of 17 amino acids. It was found that leucine was the most effective amino acid in stimulating insulin secretion, followed by alanine, glycine, and serine (23). Generally, it was shown that neutral straight-chain amino acids stimulated both insulin and glucagon secretion, with greater secretory response than shorter C-chain amino acids, whereas branched-chain amino acids tended to enhance insulin and suppress glucagon secretion (23). Another study showed in a perfused rat pancreas model that, in the presence of glucose, first-phase insulin secretion was potentiated by both cationic AAs (arginine and lysine) and BCAAs (leucine). Conversely, arginine and lysine determined inhibition of second-phase insulin secretion, whereas leucine had no significant inhibitory effect, this suggesting that those amino acids mediate their effects on first- and second-phase insulin release via different mechanisms of action, which may reflect differences in charge and/or metabolic fates within the beta cell (24). The strong insulinotropic characteristics of leucine were also reported in other studies, as outlined in a review report (25). Some studies also analyzed the increase in insulin response following AAs ingestion (with special focus on leucine) in both healthy control and T2D subjects (26, 27). The increase in insulin response was observed in both control and T2D groups, but, interestingly, in T2D such increase was typically more robust, with insulin release nearly tripled following the AAs ingestion (26).

The indicated review (25) also emphasized that, beyond the insulinotropic action, leucine may increase AAs availability for muscle protein synthesis and reduce muscle protein breakdown, this being important for prevention of muscle-related syndromes, such as sarcopenia (25). This may be of special relevance in T2D, in relation to the bidirectional relationship between T2D and sarcopenia (28, 29), remarkably linked to nutritional aspects including AAs supplementation (30, 31). It should also be observed that AAs may potentiate insulin secretion through indirect mechanisms involving incretin hormones. In fact, the effect of incretins in promoting insulin secretion has been clearly established (32, 33), and accurately quantified by appropriate methodology (34, 35). Interestingly, one study investigated the effects of combined AAs and incretin hormones administration by performing experiments in healthy humans with intravenous infusion of an AA mixture, and of human GIP or GLP-1-(7–36) amide, and found that AAs affected insulin secretion indirectly, by improving the potentiation of insulin secretion determined by GIP and GLP-1 (36).

The wide set of studies on the possible effect of AAs on insulin secretion and kinetics emphasizes the heterogeneity of the AAs action on insulin, which may involve several different mechanisms (direct and indirect), may be different for the different AAs, and may depend on the phenotypic characteristics of the studied subjects (for instance, their glucose tolerance category). This may explain the different findings of some studies, which at first sight may appear contradictory. In fact, one study in Chinese healthy subjects showed that, following a MMTT, fasting and postprandial plasma BCAA levels were consistently associated with greater fasting and postprandial insulin secretion (5). Instead, in the study from our research group, similarly based on a MMTT (7), in T2D we found postprandial isoleucine higher than in controls, but neither isoleucine nor the other BCAAs correlated with C-peptide levels (assumed as marker of insulin secretion). On the other hand, it should be noted that in another study in T2D, we found that short-term dietary reduction of BCAAs decreased postprandial insulin secretion (6). In addition, in the study (7), we also investigated a group of type 1 diabetic patients, and found increased BCAA levels that were associated with lower C-peptide (i.e., lower insulin secretion). Nonetheless, as stated above, we hypothesize no contradiction between those studies (5–7), as the observed differences in the association between BCAAs and insulin secretion among those studies are likely mainly due to the different categories of the subjects under investigation (healthy, type 1 and type 2 diabetes), and partly to the different experimental conditions and study purposes. Since our proposed model may be applied to data related to different AAs and different categories of patients (ranging from normal glucose tolerance to type 2 diabetes), we are confident that it will contribute to clarify the possible specific effects of various AAs on insulin secretion, also likely depending on the studied population. Notably, single AA may be also analyzed, in the hypothesis that the study providing the AA data has appropriate design. As an example, the model can be applied to the analysis of data derived from experiments consisting in administration of a glucose-leucine mixture (37). In Appendix B we reported some results related to the analysis of the glucose-leucine data of that study (37).

Several mathematical models describing the effects of different compounds on insulin secretion and beta-cell function have been proposed (38). Surprisingly, in contrast to other compounds carefully studied with mathematical models (especially the incretin hormones), for AAs we found only one model pertinent to our scientific literature analysis (8), despite the wide cluster of experimental studies showing effects of AAs on insulin secretion, as outlined above. This further motivated us to develop the models presented in this study. Of note, in the indicated previous model (8), the main aim was to gain insights into the role played by alanine in both amino acid-stimulated insulin secretion and in glucose stimulated insulin secretion. The model described the kinetics of core metabolic processes leading to ATP production and Ca²⁺ handling in the pancreatic β-cells, and related these processes to insulin secretion. Experimental data were collected over a rat insulin-secreting cell line and used for validating the model and for fine-tuning of the parameters. The model simulations suggested that alanine produces a potent insulinotropic effect via both a stimulatory impact on beta-cell metabolism, and as a result of the membrane depolarization due to Ca²⁺ influx triggered by alanine and Na⁺ co-transport. In summary, the model (8) was developed for simulation purposes aimed at the investigation of the mechanisms responsible for the AAs effects on the beta cell, in an in-vitro context. Our model aim, instead, is the assessment of the possible effects of AAs in single individuals, in an in-vivo context, with no expectation to get insight into cellular mechanisms. Thus, given the major differences in terms of approach and purposes, no real comparison is possible between the model (8) and our model.

In this study, we analyzed the performance of different models, and eventually selected the specific model showing the best balance between accuracy in model fit, parsimony in the number of model parameters, and plausibility of the model parameter values. Of note, in some cases only the equations describing glucose behavior were changing among models, but nonetheless we were expecting effects on both glucose and AAs parameters. Indeed, each model somehow “distributes” to glucose and AAs part of the effect on the output variable (insulin). Thus, changing the formulation of the glucose model section would affect the assessment of both glucose and AAs effect on insulin, and vice-versa.

We finally selected the model that showed the best corrected Akaike’s information criterion (AICc) value (13) except for the model with time variant k_AA parameter. In fact, the time variant model yielded the best AICc (very low value), but our analysis of model identifiability indicated the model as locally and not globally (i.e., univocally) identifiable. This means that the model parameters can assume a limited set of values, which may still be acceptable under certain circumstances, but it represents a potential weakness of the model. Thus, since Model 1 showed the best AICc compared to all remaining models, it was globally identifiable, and provided reasonable estimated parameter values, we selected it as the most appropriate and convenient for our applications. Nonetheless, we reported in Appendix A the formulation of the excluded models, since they may be useful for possible future studies in other populations or in different experimental conditions. Notably, the time-variant model may be improved by possible application of regularization strategies, as done in some of our previous studies (15, 34). It is also worth noting that we tested both a two-step and a single-step modelling approach. In the two-steps approach, we first run the model without AAs contribution, thus determining the effect of glucose on insulin secretion. In the second step, AAs was introduced, thus allowing AAs to yield the residual contribution for the full explanation of insulin secretion, if needed. Instead, in the one-step approach both glucose and AAs role are estimated concomitantly. In terms of modelling strategy both approaches have potential advantages and pitfalls, but we eventually selected the one-step approach being more consistent with the considered physiological processes, because both glucose and AAs are expected to act concomitantly on insulin release and kinetics.

Our study has some limitations. First, the model is currently not applicable in people with type 1 diabetes, because it has not been yet validated using type 1 diabetes data. Indeed, the different findings of the study (7) with regard to type 2 and type 1 diabetes indicated that the two types of diabetes may require somehow customized modelling approach. Thus, we agreed that for the moment it was convenient to focus only on T2D. As a matter of fact, the model should not be used in data from populations with metabolic abnormalities different from simple impaired glucose regulation (prediabetes) or overt T2D. Another limitation may be seen in the fact that the experimental data used for modelling validation was restricted to average rather than individual AAs data. However, this was counteracted by the extensive analysis in virtual subjects (generated with wide variations around the experimental reference AA values of study (7)), this amending for the possible limitation. Of note, we have been permissive in the generation of the virtual patients (to avoid possible unphysiological constraints), but on the other hand this provided some cases with particularly high sample values (i.e., hardly observable in real subjects). This may explain the imperfect fit in those cases, as also observable in the average fit at the high insulin values in the virtual subjects with normal glucose tolerance. Furthermore, another limitation of the study is that the selected model, as conceived, determines the AAs effect on insulin kinetics rather than on insulin secretion and insulin clearance separately. On the other hand, the scientific literature on the AAs contribution to insulin kinetics rarely showed AAs effects on insulin clearance, and hence we are confident that the insulin kinetics effects described by our model are mainly attributable to insulin secretion, rather than to insulin clearance.

Furthermore, in the present model we did not consider the potentiating effect of incretin hormones on insulin secretion, which was instead analyzed in several previous models from both our research group and others, as recently reviewed (38). This is a limitation, but it was necessary to avoid excessive model complexity, which would prevent applicability in the clinical context. In fact, currently it is not extremely common that metabolic datasets include glucose, insulin and amino acids measures, though we expect that more datasets will be available in the future for clarification of still unsolved aspects of individual amino acids actions, in the light of the general recommendations for precision medicine in metabolism and diabetes (39). Thus, metabolic datasets including incretin hormones measure in addition to that of glucose, insulin and amino acids are even rarer. Developing a model requiring incretin hormones in addition to glucose, insulin and amino acids would render a model hardly usable in the clinical practice. Since potential clinical applicability of our model was a pivotal goal of the study, we did not explicitly model the incretin effect. It has also to be noted that the model is not overfitted (model residual has shown to be not high but not negligible either). Thus, we expected that the effect of incretin hormones, and possible other variables not explicitly modelled that may affect insulin secretion, is likely captured (i.e., included) in the model residuals. On the other hand, we acknowledge that we cannot exclude a bias in the determination of the model parameters, though we hypothesize it modest for the reasons explained. This is especially true in the case of experiments with robust administration of proteins (including amino acids), as in the MMTT originating the data that we have first analyzed (7).

It should also be acknowledged that such MMTT (Boost High Protein, Nestlé S.A., Vevey, Switzerland) included several amino acids (i.e., not only BCAAs). Since we analyzed BCAAs only, the possible effect of the other AAs was not taken into account. However, similarly to the incretin effect, as reported above we expect the possible effect of other AAs being likely captured in model residuals.

Finally, some caveats are necessary to prevent misuse of the model. In fact, when exploiting our model to analyze the possible effect of one AA (or a pool of AAs, as in this study), it is necessary to ensure that the study design is adequate for our purposes. The studied AA/AAs must be appropriately stimulated by the experimental procedure, thus determining sufficient variation from the basal, unstimulated value. In addition, it should be already established from previous knowledge that the AA/AAs under investigation can affect insulin secretion. Furthermore, the experiments should ensure that the effect of the AA/AAs of interest is not much lower than that of other AAs or different compounds. Indeed, in such conditions the model would hardly be able to reliable “extract” from the data the specific small effect of the AA/AAs under analysis.

In conclusion, this study proposes a model to determine the effects of amino acids on insulin secretion in people with or without type 2 diabetes. The model provides the opportunity to estimate amino acids-induced insulin secretion in single individuals, possibly contributing to setting up patient’s tailored dietary prescriptions to delay the onset of diabetes or prevent further metabolic derangement in patients already suffering from diabetes.

The original contributions presented in the study are included in the article/ Supplementary Material . Further inquiries can be directed to the corresponding author.

MM, MCP, and AT conceived and designed the study. MM, MCP, and AT analyzed and interpreted the data. CG, LB, YK, MR, and GP validated the analysis. MM and AT wrote the first draft of the manuscript. MCP, CG, LB, YK, MR, and GP critically revised the manuscript. All authors contributed to the article and approved the submitted version.

This study was partly funded by a research contract for a scientific project commissioned to the CNR Institute of Neuroscience (Padova, Italy) by the German Diabetes Center (Düsseldorf, Germany) (April 2018 – April 2019; CNR protocol no. 0001219). The experimental average data by YK et al. (7) that we analyzed in the present study derives from the German Diabetes Study, initiated and financed by the German Diabetes Center, which is funded by the German Federal Ministry of Health (Berlin, Germany), the Ministry of Culture and Science of the state North Rhine-Westphalia (Düsseldorf, Germany), and the German Federal Ministry of Education and Research (Berlin, Germany). The study by YK et al. (7) was also supported by grants of German Research Foundation (DFG, CRC 1116/2) and of the Schmutzler Stiftung.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.