A control-theoretic law of human glucose homeostasis

A control-theoretic law of human glucose homeostasis | medRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-P4HH5NV'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search A control-theoretic law of human glucose homeostasis View ORCID Profile Hikaru Sugimoto , Ken-ichi Hironaka , Tomoko Yamada , Kazuhiko Sakaguchi , Wataru Ogawa , Shinya Kuroda doi: https://doi.org/10.1101/2025.09.30.25337012 Hikaru Sugimoto 1 Department of Biological Sciences, Graduate School of Science, The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Hikaru Sugimoto Ken-ichi Hironaka 1 Department of Biological Sciences, Graduate School of Science, The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site Tomoko Yamada 2 Division of Diabetes and Endocrinology, Department of Internal Medicine, Kobe University, Graduate School of Medicine , Kusunoki-cho 7-5-1, Chuo-ku, Kobe, Hyogo 650-0017, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site Kazuhiko Sakaguchi 2 Division of Diabetes and Endocrinology, Department of Internal Medicine, Kobe University, Graduate School of Medicine , Kusunoki-cho 7-5-1, Chuo-ku, Kobe, Hyogo 650-0017, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site Wataru Ogawa 2 Division of Diabetes and Endocrinology, Department of Internal Medicine, Kobe University, Graduate School of Medicine , Kusunoki-cho 7-5-1, Chuo-ku, Kobe, Hyogo 650-0017, Japan 3 Section of Metabolic Diseases, Division of Translational Research, Kobe University Graduate School of Medicine , 1-5-6 Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650-0047, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: ogawa{at}med.kobe-u.ac.jp skuroda{at}bs.s.u-tokyo.ac.jp Shinya Kuroda 1 Department of Biological Sciences, Graduate School of Science, The University of Tokyo , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: ogawa{at}med.kobe-u.ac.jp skuroda{at}bs.s.u-tokyo.ac.jp Abstract Full Text Info/History Metrics Supplementary material Data/Code Preview PDF Abstract Glucose homeostasis is a fundamental component of human physiology, and its failure underlies diabetes. Existing measures, such as mean glucose levels, provide limited insight into the underlying physiological processes because a quantitative law linking glycemic trajectories directly to regulatory capacity has yet to be established. We derived an equation, F = K · G , that connects glucose inputs ( F ) and measurable features of glucose dynamics ( G ) to the system’s regulatory capacity ( K ). K includes proportional-integral-derivative (PID)-like components from control theory, while G integrates glucose-trajectory characteristics, including area under the curve, amplitude, and temporal distortion. Analyses of clamp and continuous glucose monitoring data from more than 2,000 individuals showed that K was identifiable from glucose dynamics and explained interindividual variation in glucose tolerance and diabetes complication risks beyond conventional metrics. This framework identified four subtypes of impaired glucose regulation, including an underappreciated subtype characterized by deficient insulin-independent glucose-lowering. These results provide a mechanistic interpretation of how biological systems achieve robust glucose regulation and a practical basis for stratifying glucose tolerance and disease risk. INTRODUCTION Glucose homeostasis is a fundamental component of human physiology, and its disruption leads to diabetes mellitus, which now affects over 500 million people worldwide ( 1 , 2 ). Current indices of glycemic control, such as single-point blood glucose and glycated hemoglobin (HbA1c), provide limited temporal resolution and little mechanistic insight into the physiological processes that maintain glucose homeostasis ( 3 – 9 ). These limitations have prompted efforts to establish theoretical frameworks that better explain glucose regulation. Over the past few decades, numerous mathematical models ( 10 – 15 ) and machine learning approaches ( 16 – 19 ) have been proposed to advance the understanding of glucose homeostasis. One example is the hyperbolic law ( 20 – 22 ), which states that as insulin sensitivity declines, insulin secretion increases to maintain glucose homeostasis, resulting in a compensatory shift on the hyperbolic curve defined by these two variables. Consequently, the disposition index (DI), defined as the product of insulin sensitivity and insulin secretion, has been widely used as an indicator of glucose regulatory capacity ( 20 ). However, this framework relies on invasive and labor-intensive testing, such as oral glucose tolerance tests (OGTT) or glucose clamp procedures, and does not provide a complete picture of glucose dynamics. While machine learning models are increasingly applied to predict diabetes-related outcomes, they are often constrained by complexity, lack of interpretability, and limited methodological transparency ( 23 ). These limitations reinforce the enduring value of simple, interpretable laws that provide mechanistically meaningful, robust insights—analogous to how Newton’s second law of motion (m a = F ) remains foundational in physics despite the advent of advanced computational tools ( 24 ). Here, we sought to derive a unifying and interpretable theoretical framework for human glucose homeostasis. Using hyperglycemic and hyperinsulinemic–euglycemic clamp data as a reference standard, we first constructed a compartment model of glucose, insulin, and C-peptide interactions. We then integrated the model and derived a governing equation, F = K · G , which links measurable features of glucose dynamics ( G ) and glucose input ( F ) to the system’s regulatory capacity ( K ). The vector K generalizes the DI and aligns conceptually with proportional–integral–derivative (PID) control systems used in control engineering to achieve robust regulation ( 25 ). We further provide evidence that this framework not only offers a simple yet comprehensive representation of glucose regulation, but also opens the possibility of using readily obtainable continuous glucose monitoring data from clinical practice to infer the underlying physiological functions and risks of diabetes complications. RESULTS Mathematical modeling of human glucose homeostasis To generate a governing equation of glucose homeostasis, we first analyzed hyperglycemic and hyperinsulinemic–euglycemic clamp data ( 26 ) in 121 individuals, including 50 with normal glucose tolerance (NGT), 18 with impaired glucose tolerance (IGT) and 53 with type 2 diabetes mellitus (T2DM) ( Fig. 1A ). Clamp tests, which measure blood glucose, insulin and C-peptide concentrations after controlled infusions of glucose and insulin, have been the gold standard for assessing glycemic control abilities ( 10 ). Building on previous studies ( Supplementary Texts 1, 2, Fig. S1 ), we formulated a compartment model capturing the interactions among six compartments: plasma glucose ( G ), remote glucose ( Q ), remote insulin ( X ), serum insulin ( I ), serum C-peptide ( C ), and remote C-peptide ( Y ) ( Fig. 1B, C ). The equations are: Download figure Open in new tab Fig. 1. Estimation of glucose control abilities from clamp data. ( A ) Schematic of the study design. Glucose control abilities and glucose dynamics were assessed in 121 individuals using clamp tests and mathematical modeling. ( B, C ) Model diagrams. ( B ) Insulin-C-peptide regulatory model and ( C ) glucose regulatory model. Circular nodes denote variables, arrows indicate fluxes, and lines ending in circles represent regulatory activation. φ denotes a fixed value. ( D ) Time courses of plasma glucose, serum insulin, and serum C-peptide during clamp tests. Data are mean ± s.d. for each group. ( E ) Time courses of normalized residuals showing discrepancies between the measured and simulated values (mean ± s.d.). ( F ) Schematic linking glucose control abilities and glucose dynamics features. ( G ) Geometric representation of feasible G values for given F and K , colored by tolerance group (NGT, green; IGT, red; T2DM, blue). Axes correspond to G AUC , G Distortion and G Amplitude for G and to DI /clep , DI /clei , and DI /cled for K . ( H, I ) Scatter plots of individual values for K ( H ) and G ( I ). ( J ) Principal component analysis of K showing the cumulative variance explained by each component. ( K ) Box plots of DI /clep , DI /clei , and DI /cled in NGT, IGT, and T2DM groups. * Q < 0.05; Welch’s t-test with Benjamini– Hochberg correction. ( L ) Predicted versus measured K values using linear regression with G components as predictors. ( M ) Predicted versus measured K values using oral DI as a predictor. Shaded areas denote 95% confidence intervals. R : Spearman’s correlation coefficient; AIC: Akaike information criterion. Here, k Glub and k GluG ⁄ k GluQ represent distinct aspects of glucose effectiveness. The parameter k Glub captures an irreversible process in which glucose exits the bloodstream to extravascular compartments under hyperglycemia. By contrast, k GluG and k GluQ govern the reversible exchange of glucose between the plasma ( G ) and the remote compartment ( Q ): glucose flows from G to Q when G rises, and it returns from Q to G when G falls. Larger values of k Glub and k GluG ⁄ k GluQ indicate stronger G -lowering effects in response to hyperglycemia. The parameter k Sen quantifies insulin sensitivity. The parameters k Sp , k Si , and k Sd represent the basal, potentiated, and first-phase components of insulin secretion, respectively, conceptually corresponding to the proportional (P), integral (I), and derivative (D) elements of a PID controller, which is widely used in control systems engineering ( 27 , 28 ). The parameters 1 − k r , k Ic , and k Cc denote hepatic insulin clearance, peripheral insulin clearance, and C-peptide clearance, respectively. External infusion rates of glucose and insulin are denoted by f G and f I , respectively. Ġ + equals Ġ when G is increasing and zero otherwise. Although the model behaves linearly around G , this does not rule out nonlinear interactions between G and I . Our objective was to test whether first-order approximations near equilibrium could adequately capture glucose dynamics, even under the extreme perturbations imposed by clamp conditions. This strategy parallels prior work showing that complex nonlinear processes, such as gene–gene interactions, can often be approximated effectively with linear models ( 29 ). The model accurately reproduced experimental glucose, insulin and C-peptide trajectories, even at the physiological upper limits of glucose and insulin imposed by the clamps ( Fig. 1D, E ). Correlations between simulated and observed data were 0.96 (95% CI, 0.95–0.96) for glucose, 0.97 (95% CI, 0.96–0.97) for insulin, and 0.95 (95% CI, 0.94–0.97) for C-peptide. Moreover, the insulin sensitivity index, derived from a nonlinear model ( 10 , 26 ), was significantly correlated with k Sen (R = 0.92, 95% CI: 0.89–0.95). These results indicate that this simple formulation remains valid across a wide physiological range, even in the presence of underlying nonlinearities. A governing equation of human glucose homeostasis Building on the model, we next sought to derive a unifying equation linking observable glucose trajectories to regulatory capacity. Previous studies have indicated that features of glucose dynamics, such as mean, variance, and temporal autocorrelation, are independently associated with glycemic control and diabetes complications ( 5 – 9 ). Recent machine-learning analyses have also suggested that entire postprandial glucose trajectories can be reconstructed using these three statistics ( 9 ). However, these links are merely correlative and lack a theoretical basis. We therefore aimed to formulate an equation that connects these three measurable glucose dynamics to the system’s underlying regulatory capacity. By integrating the differential equations between steady states, we derived a single equation ( Fig. 1F , Supplementary Text 3 ): Here, F denotes the total amount of exogenously infused glucose. The K represents composite regulatory capacity and is defined as: K = ( k Glub + k Sen k r k Sp ⁄ k Ic , k Sen k r k Si ⁄ k Ic , k Sen k r k Sd ⁄ k Ic :. The first component represents the glucose-lowering effect proportional to the current blood glucose levels and is termed DI /clep . The second component captures glucose-lowering effects dependent on the integrated difference between current and basal glucose levels over time and is termed DI /clei . The third component reflects glucose-lowering effects related to the derivative of blood glucose and is termed DI /cled . Conceptually, these components align with the P, I, and D terms of a PID control system. The G was defined from plasma glucose dynamics: G = (∫( G − G b ) dt , ∫( G − G b ) tdt , G max − G min ). These components reflect the area under the curve ( G AUC ), temporal distortion ( G Distortion ), and amplitude ( G Amplitude ), respectively. Conceptually, these components align with previously reported features, such as mean, temporal autocorrelation, and variance, that have been independently associated with the DI and diabetes complications ( 5 , 6 , 9 ). Geometric representation of F = K · G reveals inter-individual heterogeneity in K We then applied the F = K · G framework to visualize regulatory capacity ( K ), thereby extending the classical DI into a multidimensional space. The classical DI is typically visualized as a hyperbolic curve defined by the trade-off between insulin sensitivity and secretion ( 20 , 30 ). Even among individuals with NGT, considerable inter-individual variability exists along this curve ( 20 ). To extend this concept, we developed a geometric representation of F = K · G ( Fig. 1G-I ). For fixed F and K , G is constrained to a plane defined by F = K · G ( Fig. 1G ). As glucose tolerance declines, G is expected to increase, while K is expected to decrease. We estimated K from clamp tests and G from OGTT data ( Fig. 1H, I ). Although G is ideally calculated using glucose profiles between steady states (approximately 0-180 min during the OGTT), we observed a strong correlation between estimates derived from the 0-120 min window and the extended interval ( Fig. S2A, B ). Accordingly, values derived from the 0-120 min OGTT data were used for reference. Three-dimensional visualization of K revealed substantial heterogeneity in the P, I and D components of glycemic control even within the NGT group ( Fig. 1H ). Principal component analysis (PCA) confirmed this diversity: in NGT individuals, the first and second components accounted for only 46% and 34% of the variance, respectively ( Fig. 1J ). In contrast, individuals with T2DM showed reduced dimensionality, with the first two components explaining a greater proportion of the variance. As expected, all three components of K were significantly reduced in IGT and T2DM groups compared with NGT ( Fig. 1K ). We then compared K with the conventional clamp-derived DI (clamp DI), which primarily reflects first-phase insulin secretion and insulin sensitivity. While clamp DI showed a strong correlation with DI /cled (R = 0.89, 95% CI: 0.84–0.92), its correlation with DI /clep was only moderate (R = 0.59, 95% CI: 0.46–0.70) ( Fig. S2C ), suggesting that the conventional index does not fully capture the multidimensional nature of glycemic regulation. We assessed the predictive utility of K for post-OGTT glucose levels relative to clamp DI. Across all time points, linear regression models based on K consistently showed stronger correlations and lower Akaike Information Criterion (AIC) values compared to those using clamp DI ( Fig. S2D, E ). Collectively, these results indicate that K provides a more comprehensive assessment of an individual’s ability to control blood glucose by capturing multiple dimensions of glucose regulation that are not fully captured by the traditional DI. Prediction of K from OGTT profiles within the F = K · G framework To assess the clinical applicability of F = K · G , we tested whether glucose regulatory capacity ( K ) could be predicted from glucose profiles ( G ) during OGTT. If successful, this approach could eliminate the need for insulin measurements, which are required to calculate the conventional OGTT-derived DI (oral DI). Given that the oral DI primarily reflects first-phase insulin secretion, we hypothesized that it would correlate most strongly with DI /cled , while providing limited information on DI /clep and DI /clei . Consistent with the hypothesis, models based solely on G during OGTT achieved prediction accuracies comparable to those obtained using the oral DI ( Fig. 1L, M ). The G -based models outperformed oral DI in estimating DI /clep , suggesting that this regulatory dimension cannot be fully captured by oral DI alone. Collectively, these results suggest that under standardized conditions (fixed F and near-uniform ingestion rate), glucose dynamics features ( G ) alone can empirically reconstruct the full vector of glucose regulatory capacities ( K ). Prediction of K from glucose dynamics using F = K · G Given that K exists in three-dimensional space, it should, in principle, be uniquely determined by G and F , provided that data from at least three distinct glucose input conditions are available. Although the equation F = K · G was originally derived from clamp experiments, it is important to note that F represents the total amount of glucose entering the circulation, independent of its temporal administration pattern. Furthermore, the DIs estimated from the OGTT profiles were significantly correlated with those derived from the clamp data ( Fig. 1L, M ). Building on this, we hypothesized that a similar structural relationship would hold under physiological conditions, such as during a meal when glucose enters the circulation via alternative routes. Specifically, despite differences in the route of administration and changes in the absolute values of the parameters ( 31 ), the structure of the mathematical model and the relative ordering of the parameter values would remain consistent. To test this hypothesis and to assess the consistency of our simplified formulation with an established, more complex mathematical model of daily postprandial glucose dynamics ( 32 ), we performed large-scale in silico simulations ( Fig. 2A ). As previously described ( 32 ), parameters for insulin sensitivity (V mx ) and first-phase insulin secretion (K) were systematically varied to cover a physiological range including both NGT and T2DM. Each simulated individual underwent both rapid (bolus) and prolonged (continuous) glucose administration ( Fig. 2B , Fig. S2F ). For each individual, we estimated the K vector from the blood glucose dynamics ( G ) and evaluated the correlation between the predicted K and V mx K values. Given that K mathematically corresponds to k Sd , we hypothesized that V mx K would exhibit the strongest correlation with DI /cled . Download figure Open in new tab Fig. 2. Relationship between glucose control abilities, administered glucose, and glucose dynamics. ( A ) Schematic of in silico simulations evaluating the relationship between glucose control abilities and glucose dynamics across 10,000 human individuals. ( B ) Examples of simulated blood glucose trajectories following the oral administration of 20 g, 30 g, or 40 g glucose, delivered either as a bolus or continuously. ( C ) Correlations between predicted K components (DI /clep , DI /clei , and DI /cled ) and ground-truth VₘₓK values (product of insulin sensitivity and first-phase insulin secretion). Spearman’s correlation coefficients ( R ) with 95% confidence intervals are shown. ( D ) Schematic of oral glucose tolerance test experiments with varying glucose loads and administration patterns. ( E ) Examples of plasma glucose trajectories following the oral ingestion of 25 g, 50 g, or 75 g glucose as a bolus or continuous input. ( F ) Cumulative contribution rate of principal component analysis of G / F . Mean ± s.d. across individuals. ( G ) The relation between the administered glucose dose ( F ) and the predicted F , using either F = K · G AUC (AUC) or F = K · G (All). Shaded areas indicate 95% confidence intervals. ( H ) Schematic of humans undergoing mixed meal tolerance tests. ( I ) Examples of plasma glucose trajectories following the ingestion of meals with different macronutrient compositions: PB (bread and peanut butter), CF (cornflakes and milk), and Bar (protein bar). ( J ) Cumulative contribution rate of the principal component analysis of G / F . Mean ± s.d. across individuals undergoing mixed meal tolerance tests. ( K ) Comparison of the predicted and measured oral disposition index (oral DI) using K as predictors. Each point represents one participant. The shaded band indicates the 95% confidence interval. ( L ) Comparison of the predicted and measured oral DI using fasting plasma glucose (Glucose (0)), fasting serum insulin (Insulin (0)) and HbA1c levels as predictors. ( M ) Schematic of humans undergoing continuous glucose monitoring (CGM). Relationships between estimated K (DI /clep , DI /clei , and DI /cled ) and clinical measures were assessed. These measures included carotid intima-media thickness (IMT) and liver ultrasound-derived measures, such as attenuation, elasticity, and dispersion. ( N ) Multiple regression analysis of associations between K and the clinical parameters. Bars represent 95% confidence intervals for regression coefficients. As expected, DI /cled , derived from the F = K · G , showed a strong correlation with the true V mx K values ( Fig. 2C ), supporting the accuracy of the model and its utility in inferring regulatory capacity from glucose dynamics and dosing. Furthermore, the temporal distortion feature of G was significantly correlated with the temporal autocorrelation-derived index (AC_Mean) ( Fig. S2G ), suggesting that the three components of G (area under the curve, temporal distortion and amplitude) are conceptually consistent with previously reported glucose dynamics-derived measures (mean, temporal autocorrelation and variance), which have been independently associated with the DI and diabetes complications ( 5 , 6 , 9 ). Prediction of F from glucose dynamics using F = K · G To further validate the equation, we analyzed a previously published human dataset ( 33 ) reporting plasma glucose responses to three oral glucose loads (25 g, 50 g and 75 g) administered as either bolus or continuous infusion ( Fig. 2D, E ). Although G is defined in three dimensions, the geometric interpretation of F = K · G ( Fig. 1G ) predicts that G / F for each individual should reside on a two-dimensional plane. This prediction was confirmed by PCA of G / F , with the first two components accounting for 99.5% of total variance ( Fig. 2F ), providing empirical support for the geometric structure of the model. We next tested whether the framework could be used to infer total glucose input ( F ) from observed glucose dynamics. Linear regression models using only G AUC yielded weak and non-significant predictions ( R = 0.34; 95% CI: –0.06 to 0.68), whereas models including the full G vector achieved much higher predictive accuracy ( R = 0.85; 95% CI: 0.70 to 0.96) ( Fig. 2G ). These results suggest the necessity of considering multiple dimensions of glucose dynamics to accurately infer both regulatory capacity and input glucose load. Prediction of K from CGM data using F = K · G We then investigated whether glucose regulatory capacity could be estimated from continuous glucose monitoring (CGM) data during mixed meal tolerance tests ( 34 ) using the F = K · G equation. Although the equation was developed for glucose-specific inputs and does not explicitly account for lipid or amino acid effects, we tested whether the K vector derived from interstitial glucose profiles under real-life meal conditions could approximate the oral DI. Interstitial glucose trajectories ( G ) and carbohydrate intake ( F ) were measured after three meals with different macronutrient compositions ( Fig. 2H, I ). PCA of G / F for each individual showed that the first two components accounted for 96.7% of the variance ( Fig. 2J ), providing empirical support for the theoretical framework even under free-living conditions. In linear regression models, the CGM-derived K vector had comparable predictive performance to the model that used markers requiring blood sampling (fasting plasma glucose, fasting serum insulin, and HbA1c) in predicting oral DI ( Fig. 2K, L ), suggesting that the F = K · G framework enables minimally invasive estimation of glucose regulatory capacity using CGM data. CGM-derived K predicts vascular and hepatic conditions independently of conventional measures To examine the clinical significance of the F = K · G framework, we tested whether glucose control capacity ( K ), estimated under free-living conditions using CGM, is associated with vascular and hepatic health independently of conventional clinical measures by analyzing data from the Human Phenotype Project (HPP) cohort ( 18 , 35 , 36 ) ( Fig. 2M ). Using the equation F = K · G , we estimated individual K values from CGM-recorded glucose dynamics and dietary carbohydrate intake. We then assessed the relationship between K values and vascular/hepatic measures, including intima-media thickness (IMT) and ultrasound-based hepatic attenuation, elasticity and dispersion. Systolic and diastolic blood pressure (SBP, DBP), triglycerides (TG), high- and low-density lipoprotein cholesterol (HDL, LDL) and mean glucose levels (Mean) were included as covariates. Multiple linear regression analysis showed that DI /cled was significantly associated with both IMT (β = –0.0040, 95% CI: –0.0080 to 0.0000) and liver attenuation (β = –0.0054, 95% CI: – 0.0110 to 0.0000) independently of other clinical measures ( Fig. 2N ). In contrast, mean glucose levels were independently associated with liver elasticity and dispersion. These results indicate that the integration of K , estimated from CGM glucose dynamics and dietary intake, with routinely measured clinical indices can improve the prediction of vascular and hepatic conditions. Biphasic glucose dynamics is associated with high glucose effectiveness ( k GIuG ⁄ k GIuQ ) The equation F = K · G holds for any value of k GluG ⁄ k GluQ , indicating that this parameter does not directly influence the outcome of the equation and cannot be inferred from the formulation alone. To address this limitation, we sought to identify an interpretable feature of the OGTT glucose trajectory that could serve as a surrogate marker for k GluG ⁄ k GluQ . Given that k GluG ⁄ k GluQ reflects glucose flux from G to Q during the ascending phase, and from Q to G after the peak, we hypothesized that individuals with higher k GluG ⁄ k GluQ would show smoother and more attenuated glucose excursions in both the ascending and descending phases, resulting in a more prominent biphasic pattern ( 15 ). To test this hypothesis, we applied hierarchical clustering to standardized OGTT glucose trajectories from 121 individuals, aiming to group participants by the shape of their glucose response curves ( Fig. 3A ). Cluster quality was assessed by silhouette analysis, which identified a four-cluster solution as optimal ( Fig. 3B ). Among these, cluster 3 was characterized by a biphasic glucose profile ( Fig. 3C, D ). Individuals in cluster 3 had significantly higher values of k GluG ⁄ k GluQ compared to other groups ( Fig. 3E ), suggesting that biphasic glucose dynamics may serve as a less invasive phenotypic marker of increased k GluG ⁄ k GluQ . Download figure Open in new tab Fig. 3. Associations between OGTT-derived glucose response profiles and clamp-derived glucose control capacities. ( A ) Schematic overview of the study design. Oral glucose tolerance tests (OGTT) and clamp tests data from 121 individuals were analyzed to examine the relationship between OGTT response patterns and underlying glucose control abilities derived from clamp measurements. ( B ) Determination of the optimal cluster number by silhouette coefficient analysis. The dashed line marks the number of clusters that maximized the silhouette score. ( C ) Heatmap showing the standardized plasma glucose responses during the OGTT (0–120 min) obtained by hierarchical clustering. Rows represent individual participants and columns represent time points. The color scale indicates z-scored glucose concentrations. ( D ) Time courses of normalized plasma glucose (left), absolute plasma glucose (middle), and serum insulin (right) across the four identified clusters. Data are mean ± s.e.m. ( E ) Box plots of k Sp , k Si , k Sd , 1 − k r , k Ic , k Cc , k Glub , k GluG ⁄ k GluQ , and k Sen across clusters. Boxes show median and interquartile range; Q values were calculated using Welch’s t-tests with Benjamini-Hochberg correction. * Q < 0.05. Inter-relationship between glucose regulatory capacity We next examined how the components of K interrelate across different states of glucose tolerance to clarify compensatory mechanisms and their disruption. While both insulin sensitivity and secretion are known to decline with progression to T2DM, previous studies have shown that in individuals with NGT, reduced insulin sensitivity is often offset by increased insulin secretion ( 20 , 30 ). Consequently, when analyzing a heterogeneous dataset, a positive correlation between insulin sensitivity and secretion would be expected, whereas within the NGT subgroup these parameters may instead display an inverse relationship. To systematically explore these interdependencies, we constructed correlation networks using clamp-derived estimates across the dataset and within stratified subgroups ( Fig. 4A, B ). We retained parameter pairs with absolute Spearman’s correlation coefficients > 0.21 ( Q < 0.05 for the entire dataset) to construct network edges. Download figure Open in new tab Fig. 4. Interrelationships among glucose control abilities and pathways to glucose intolerance. ( A ) A schematic overview of clamp-based assessments performed on 121 individuals across a spectrum of glucose tolerance to examine interrelationships among components of glucose regulatory capacity. ( B ) Correlation networks of regulatory parameters visualized using a spring layout. Edges connect parameter pairs with absolute Spearman’s correlation ≥ 0.21 ( Q < 0.05 for the entire dataset), with edge width reflecting correlation strength. Networks are shown for all participants (gray) and stratified by glucose tolerance group. For clarity, only negative correlations are annotated in subgroup networks. ( C ) Heatmap of factor loadings from exploratory factor analysis. Rows represent individual parameters, and columns represent latent factors. Color intensity and hue indicate the magnitude and direction of the parameters’ contributions to each factor. ( D ) Box plots of factor scores in the NGT, IGT, and T2DM groups. Boxes denote the median and interquartile range. Q values were determined using a Welch’s t-test with a Benjamini–Hochberg correction. * Q < 0.05. ( E ) Correlations of DI /clep , DI /clei , and DI /cled with factors 1-4. Bars represent 95% confidence intervals of the correlation coefficients. ( F ) A summary diagram showing the governing equation of glucose homeostasis, methods for parameter estimation, and major pathways leading to glucose intolerance. In the combined dataset, most correlations were positive, reflecting coordinated dysregulation with glucose intolerance. In contrast, stratified analyses revealed several negative correlations, particularly in the NGT and IGT subgroups. In NGT individuals, insulin sensitivity ( k Sen ) was negatively correlated with 1-hepatic insulin clearance ( k r ), suggesting that reduced insulin sensitivity may be partially compensated by suppressed hepatic insulin clearance to maintain systemic insulin levels. Furthermore, k r showed negative correlations with both basal insulin secretion ( k Sp ) and glucose effectiveness ( k GluG ⁄ k GluQ ), suggesting additional possible compensatory adjustments involving hepatic handling of insulin. In IGT, negative correlations became more widespread, with k Sen inversely correlated not only with k r but also with k Sp . In contrast, individuals with T2DM showed fewer and weaker negative associations, particularly between insulin sensitivity and secretion parameters. Collectively, these results suggest that in IGT, multiple compensatory mechanisms involving insulin secretion, sensitivity, and clearance may exist, whereas in T2DM, such relationships appear to be disrupted, potentially contributing to progressive hyperglycemia. Four different types of glucose intolerance Group-wise comparisons confirmed that insulin sensitivity ( k Sen ), insulin secretion ( k Sp , k Si , and k Sd ), and glucose effectiveness ( k Glub , k GluG ⁄ k GluQ ) were significantly reduced in IGT or T2DM compared to NGT ( Fig. S2H ). Insulin-dependent glucose-lowering effects were particularly decreased in these groups ( Fig. S2I, Supplementary Text 4 ). Despite these average trends, previous studies ( 20 , 30 ) have indicated heterogeneity within each group, with some individuals showing disproportionately impaired insulin secretion, while others show more severe deficits in insulin sensitivity. To investigate whether the F = K · G framework could identify latent patterns underlying this heterogeneity, we performed exploratory factor analysis on the glucose regulatory parameters ( Fig. 4C ). The Very Simple Structure complexity criteria indicated that a four-factor solution best captured the structure of the data. Factor 1 primarily loaded on k Sen , k Ic , and k Glub . Factor 2 captured insulin secretion capacities ( k Sp , k Si , and k Sd ). Factor 3 was affected by glucose effectiveness ( k GluG ⁄ k GluQ ), while factor 4 was characterized by hepatic insulin clearance and first-phase secretion (1 − k r , k Sd ). The Kaiser-Meyer-Olkin measure of sampling adequacy was 0.61, and Bartlett’s spherical test indicated statistically significant intercorrelations among the variables ( P < 0.01), confirming the suitability of the dataset for factor analysis. All four factor scores were significantly reduced in IGT or T2DM compared to NGT ( Fig. 4D ), suggesting the presence of four mechanistically distinct subtypes of glucose intolerance. We next examined the correlations between the factors and DI /clep , DI /clei , and DI /cled ( Fig. 4E ). Factor 1, reflecting insulin sensitivity ( k Sen ) and glucose effectiveness ( k Glub ), was significantly correlated with all three indices (DI /clep , DI /clei , and DI /cled ), most strongly with DI /clep ( R = 0.60; 95% CI: 0.46 to 0.71), which is consistent with its strong loading on k Glub . Factor 2, representing k Si and k Sd , showed strong associations with DI /clei ( R = 0.72; 95% CI: 0.62 to 0.80) and DI /cled ( R = 0.85; 95% CI: 0.77 to 0.90). Factor 3, reflecting glucose effectiveness ( k GluG ⁄ k GluQ ), showed minimal correlation with any DI indices, as expected since it is not directly included in their calculation. Factor 4, characterized by high loadings on k Sd , was associated with DI /cled ( R = 0.62; 95% CI: 0.48 to 0.72). These distinct correlation patterns suggest that the impairment in DI /clep , DI /clei , and DI /cled may reflect mechanistically divergent pathways of metabolic deterioration across individuals. DISCUSSION Despite decades of efforts in mathematical modeling ( 10 – 14 ) and machine learning ( 16 , 17 , 37 ), a unified, mechanistically interpretable equation linking observable blood glucose dynamics to the underlying physiological processes governing glucose homeostasis has remained elusive. Here we propose the relations underlying glucose homeostasis ( Fig. 4F ): (i) glucose dynamics and its regulatory parameters satisfy the equation F = K · G , and (ii) individual differences exist within the range that satisfies this equation. K consists of three components reflecting the proportional (P), integral (I) and derivative (D) elements in control theory, while G is a three-dimensional vector consisting of the area under the curve, amplitude, and temporal distortion of glucose profiles. Although simple in form, this equation was derived from a system of six differential equations with thirteen parameters and accurately reproduced the dynamics of glucose-insulin interactions. This model theoretically enables minimally invasive estimation of regulatory capacity ( K ) from CGM data ( G ) and glucose input ( F ), without requiring blood sampling or insulin measurements. Previous studies have shown that conventional indices, such as fasting blood glucose levels, HbA1c or 2-hour OGTT values, primarily reflect mean glucose levels and are less predictive of glucose tolerance and risks of diabetes complications than CGM-derived indices including glucose variability and temporal autocorrelation ( 5 , 6 , 9 ), which are all included in G . However, these approaches have been correlative in nature and have not accounted for the administered glucose load. In contrast, the F = K · G equation provides a theoretically grounded method to directly infer the physiological parameters underlying glucose regulation by integrating both glucose dynamics and dosing. From a clinical perspective, the framework offers immediate translational potential because it relies on CGM data, which are already widely used in daily diabetes care. It may complement existing diagnostic criteria and enable earlier detection of impaired glucose tolerance or specific subtypes that would otherwise remain unrecognized. It may also improve risk stratification for vascular and hepatic complications and enhance the predictive accuracy of clinical outcomes. Furthermore, by distinguishing mechanistic subtypes—such as secretion-deficient, resistance-dominant, or glucose-effectiveness-deficient phenotypes—the model may inform the choice of individualized therapeutic strategies and facilitate the identification of novel treatment targets. Because CGM can be repeatedly and relatively easily performed in daily clinical practice, changes in K components can be monitored before and after lifestyle interventions or pharmacological therapies, thereby allowing treatment effects to be evaluated and optimized in real time. These applications suggest that the F = K · G framework could contribute directly to the realization of precision medicine in diabetes. We confirmed the utility of the framework using multiple independent datasets, including clamp studies, OGTT data, mixed-meal tests, and CGM profiles. Nonetheless, several datasets had modest sample sizes, underscoring the need for larger, prospective studies to establish generalizability and clinical utility. Although the observed correlations in the HPP cohort were statistically significant, they were modest. This is likely because the study mainly included healthy individuals with minimal inter-individual differences. Including individuals with diabetes may produce stronger correlations. Building on this foundation, the model generates several testable predictions. For example, simultaneous measurement of glucose trajectories and clamp-derived parameters under variable glucose loads could provide empirical validation of the relationship ( Supplementary Text 3 ). Moreover, while genetic and omics studies have traditionally focused on insulin sensitivity or secretion, the identification of a subtype of impaired glucose effectiveness ( k GluG ⁄ k GluQ ) points to previously unexplored physiological mechanisms. Although previous studies have shown that glucose effectiveness decreases as glucose tolerance worsens ( 38 ), most studies have quantified glucose effectiveness using the minimal model ( 39 ), which mathematically approximates k Glub . While k Glub correlated strongly with insulin sensitivity, k GluG ⁄ k GluQ was largely independent of insulin sensitivity and secretion. Additionally, compensatory pathways, including insulin clearance, may also warrant further investigation. Genetic and omics investigations stratified by these factors could deepen our understanding of glucose homeostasis. In conclusion, the F = K · G equation provides a simple and mechanistically interpretable theoretical basis for estimating both glucose regulatory capacity and the risk of diabetes-related complications. This framework has the potential to transform the conventional diagnosis and management of diabetes, refine risk prediction, and guide individualized therapy. Beyond its clinical applications, it may also serve as a foundation for integrative studies combining physiology, omics, and systems biology to further unravel the complexity of human glucose regulation. MATERIALS AND METHODS Study design and ethical approval This study includes exploratory analyses of five independent observational studies. In this paper, these studies are referred to as the Kobe University Study ( 26 ), the University of Tokyo Study ( 33 ), the San Francisco Bay Area Study1 ( 34 ), the San Francisco Bay Area Study2 ( 16 ), and the Human Phenotype Project (HPP) cohort ( 18 , 35 , 36 ). These studies were conducted in accordance with the Declaration of Helsinki and were approved by their institutional ethics committees. Written informed consent was obtained from all participants before enrollment. The Kobe University Study ( 26 ) is an observational study investigating the associations between glucose regulatory capacity and glucose dynamics. The primary objective of the analysis was to develop a mathematical model linking characteristics of glucose dynamics to underlying physiological mechanisms, such as insulin sensitivity, insulin secretion and clearance. The study was approved by the ethics committee of Kobe University Graduate School of Medicine and registered with the University Hospital Medical Information Network (UMIN000002359). The University of Tokyo Study ( 33 ) is an observational study investigating the relationship between administered glucose load and glucose dynamics. The primary objective of the analysis was to validate the mathematical model developed in this study by predicting the administered glucose amount from glucose dynamics data. The study was approved by the ethics committees of the Life-Science Committee of the University of Tokyo ( 16 –265) The San Francisco Bay Area Study1 ( 34 ) is an observational study investigating the relationship between continuous glucose monitoring (CGM)-derived measures and glucose control ability quantified by an oral glucose tolerance test (OGTT). The primary objective of the analysis was to predict the OGTT-derived disposition index from CGM-measured glucose dynamics. The study was approved by the Stanford Institutional Review Board (IRB 37141). The San Francisco Bay Area Study2 ( 16 ) is an observational study investigating the relationship between glucose levels during OGTT and glucose control abilities. The primary objective was to investigate the consistency of glucose-derived indices calculated from OGTT data within the time frames of 0-120 minutes and 0-180 minutes. The study was approved by the Stanford University School of Medicine Human Research Protection Office (Institutional Review Board #43883). The HPP cohort ( 18 , 35 , 36 ) is an observational study investigating glucose fluctuations in non-diabetic individuals using CGM. The primary objective of the analysis was to assess the associations between CGM-predicted glucose control ability and carotid/liver ultrasound-derived measures. The study was approved by the Institutional Review Board of the Weizmann Institute of Science. Study population and inclusion criteria Detailed inclusion and exclusion criteria, as well as comprehensive data acquisition protocols, have been reported previously ( 16 , 26 , 33 – 36 , 40 ). A summary of the main criteria for each dataset is provided below. The Kobe University Study enrolled individuals aged 20-80 years between October 2008 and December 2011. Exclusion criteria included renal impairment (creatinine clearance 50 IU/mL), and ongoing insulin therapy. Of the 121 participants recruited, 50, 18, and 53 individuals met the criteria of normal glucose tolerance (NGT), impaired glucose tolerance (IGT), and type 2 diabetes mellitus (T2DM), respectively. The University of Tokyo Study enrolled individuals aged 20-50 years with HbA1c levels between 4.9% and 5.4% who had not been diagnosed with diabetes. A total of three participants were included in the study. The San Francisco Bay Area Study1 included individuals aged 25-76 years with no prior diagnosis of diabetes. Exclusion criteria included major organ diseases, chronic inflammatory diseases, malignancy, uncontrolled hypertension, eating disorders, a history of bariatric surgery, current use of weight-loss or diabetogenic medications. Of the 57 participants screened, 22 who completed both CGM during three standardized meals and an OGTT were included in the analysis. The San Francisco Bay Area Study2 included individuals aged 30-70 years with a body mass index of 23-40 kg/m 2 . Exclusion criteria included a history of diabetes, major organ diseases, uncontrolled hypertension, malignancy, chronic inflammatory diseases, use of any medications known to alter blood glucose or insulin sensitivity, hematocrit < 30%, serum creatinine above the normal reference range, and alanine aminotransferase levels more than twice the upper limit of normal. Individuals with missing values were excluded from the analysis. Of the 56 participants screened, 51 met the inclusion criteria. After screening tests, 8, 27, and 16 individuals met the criteria of T2DM, IGT, and NGT, respectively. The HPP cohort enrolled individuals aged 40-70 years between January 2019 and May 2023. Participants who underwent an ultrasound examination and had a food record were included in the analysis. Exclusion criteria included self-reported diabetes and use of diabetes-related medications. Of the 9,529 participants at baseline, 1,899 met the inclusion criteria. Clinical measures and assessments Detailed descriptions of the clinical assessments have been published previously ( 16 , 26 , 33 – 36 , 40 ). The main methods used for each dataset are summarized below. In the Kobe University Study , a hyperglycemic clamp was performed by intravenous infusion of a glucose bolus for 15 minutes, followed by a variable-rate glucose infusion to maintain a plasma glucose concentration of 200 mg/dL. This was followed, after a 10-minute interval, by a hyperinsulinemic-euglycemic clamp in which insulin was infused to achieve a plasma glucose concentration of 90 mg/dL or a fasting plasma glucose level in participants with NGT or IGT whose baseline glucose was below 90 mg/dL. Both clamp procedures and a 75 g OGTT were performed within a 10-day window. Missing values were imputed using the k-nearest neighbor algorithm (k=2). Insulin secretion was quantified as the incremental area under the insulin curve during the first 10 minutes of the hyperglycemic clamp. The insulin sensitivity index ( 26 ) was calculated as the average glucose infusion rate during the last 30 minutes of the hyperinsulinemic-euglycemic clamp, normalized to plasma glucose and serum insulin concentrations. The clamp-derived disposition index (Clamp DI) was defined as the product of insulin secretion and sensitivity. The OGTT-derived disposition index (Oral DI) was calculated as the product of the insulinogenic index and the Matsuda index, as previously described ( 6 , 41 , 42 ). In the University of Tokyo Study , a glucose solution containing 25 g, 50 g, or 75 g of glucose was administered orally after a 10-hour fast. Blood samples were collected from a superficial forearm vein at 0 minutes and every 10 minutes up to 240 minutes after ingestion. Two ingestion patterns were used: rapid bolus ingestion (completed within 1 minute) and continuous ingestion over two hours. For the latter, a non-contact micro dispensing robot delivered the glucose solution through tubes held in the participants’ mouths and ingested continuously over the two-hour period. All glucose solutions were diluted with distilled water to a final volume of 225 mL to standardize ingestion. Each participant completed all three-glucose dose and method combinations, with test sessions separated by at least one month. To characterize the dynamics of blood glucose following administration, three distinct indices ( G AUC , G Distortion and G Amplitude ) were computed for each condition. G AUC was calculated by integrating the linearly interpolated, baseline-corrected glucose curve using Simpson’s rule. G Distortion was calculated by the product of the baseline-corrected glucose values and their corresponding time points. The time-weighted curve was then linearly interpolated and integrated using Simpson’s rule. G Amplitude represents the amplitude of the glucose response. In the San Francisco Bay Area Study1 , participants consumed three standardized breakfast meals on separate occasions: (i) bread with peanut butter, (ii) an energy bar, and (iii) cereal with milk and raisins. Each meal was eaten twice on non-consecutive days, and participants wore continuous glucose monitors throughout. Participants were instructed to eat each meal twice on two separate days, and to record the time of each meal. Data from those with incomplete records or who consumed non-standardized items were excluded. The Oral DI was calculated as the product of the insulinogenic index and the Matsuda index, as previously described ( 6 , 41 , 42 ). In the San Francisco Bay Area Study2 , participants were instructed to maintain their usual diet and physical activity for three days prior to the OGTT, to fast for 10-12 hours, to abstain from strenuous physical activity after 17:00 the previous evening, and to abstain from smoking or nicotine use on the morning of the test. Blood samples were taken at 16 time points (−10, 0, 10, 15, 20, 30, 40, 50, 60, 75, 90, 105, 120, 135, 150 and 180 minutes) via an antecubital intravenous catheter. Plasma glucose concentrations were measured by the oximetric method. Analyses in this study were limited to 15 of these time points (0-180 minutes, excluding the −10-minute baseline). In the HPP cohort , ultrasound assessments were performed to measure carotid intima-media thickness (IMT), liver attenuation, elasticity, and dispersion. Systolic blood pressure (SBP), diastolic blood pressure (DBP), triglycerides (TG), low-density lipoprotein cholesterol (LDL-C), and high-density lipoprotein cholesterol (HDL-C) measured at the time of the CGM assessment were also included in the analyses. Mathematical modeling of plasma glucose, serum insulin and C-peptide dynamics To characterize the temporal dynamics of plasma glucose ( G ), remote glucose ( Q ), remote insulin ( X ), serum insulin ( I ), serum C-peptide ( C ), and remote C-peptide ( Y ), the following system of differential equations, based on previously published models, was used ( Supplementary Texts 1, 2, Fig. S1 ): Here, k Glub and k GluG ⁄ k GluQ represent different aspects of glucose effectiveness. The parameters k Sp , k Si , and k Sd represent the basal, potentiated, and first-phase components of insulin secretion, respectively. The parameters 1 − k r , k Ic , and k Cc denote hepatic insulin clearance, peripheral insulin clearance, and C-peptide clearance, respectively. Meanwhile, f G and f I represent externally infused glucose and insulin rates, which were calculated as previously described ( 40 , 41 ). Ġ + equals zero when G is decreasing and equals Ġ when G is increasing. Parameter estimation was performed using a meta-evolutionary programming algorithm combined with non-linear least squares optimization, minimizing the normalized residual sum of squares between the measured clamp data and the model predictions, as previously described ( 40 , 41 , 43 ). This simulation was conducted using SciPy 1.10.1 ( 44 ). The following equation was derived by integrating the differential equations between steady states ( Supplementary Text 3 ): where F denotes the total amount of exogenously infused glucose, and K represents glucose regulatory capacity and is defined as: K = ( k Glub + k Sen k r k Sp ⁄ k Ic , k Sen k r k Si ⁄ k Ic , k Sen k r k Sd ⁄ k Ic :. The G is a composite vector summarizing blood glucose dynamics: G = (∫( G − G b ) dt , ∫( G − G b ) tdt , G max − G min ). Mathematical model simulating postprandial plasma glucose dynamics A previously validated meal simulation model of human glucose-insulin regulation ( 32 ) was used to simulate glucose-insulin interactions under varying glucose intake profiles. This physiologically based model consists of modular subsystems representing glucose absorption, endogenous glucose production, insulin secretion and action, and peripheral glucose uptake. Parameters were estimated using data sets from individuals with NGT and T2DM undergoing the triple tracer tolerance test. The detailed formulation has been described previously ( 32 ). A dataset of 10,000 human individuals was generated by systematically varying parameters across their known physiological ranges. Specifically, insulin sensitivity (Vmx) and first-phase insulin secretion (K) were each assigned 100 evenly spaced values across their respective ranges, and the Cartesian product of these grids (100 × 100) yielded 10,000 unique parameter combinations representing phenotypes from NGT to T2DM ( 32 , 45 ). Simulations were performed for both bolus ingestion (20, 30, or 40 g glucose ingested within 4 minutes) and continuous ingestion over 1 hour. Using administered glucose dose ( F ), glucose dynamics ( G ), and the equation, F = K · G , the K vector was estimated for each individual. The estimated K values were then compared with the product of Vmx and K. Additionally, the temporal distortion of glucose dynamics in G was compared with glucose autocorrelation-based index (AC_Mean ( 6 )). Statistical analyses Group comparisons were performed using a Welch’s t-test. Associations between variables were assessed using either Spearman’s correlation coefficients or linear regression models. Multiple comparisons were adjusted using the Benjamini-Hochberg procedure. Confidence intervals were estimated using bootstrap resampling with 10,000 iterations. Statistical significance was defined as Q < 0.05 or 95% confidence intervals excluding zero. Prior to the linear regression analyses, all data were standardized using z-score normalization. To evaluate and compare the predictive performance of statistical models including different sets of input variables, the Akaike Information Criterion (AIC) was calculated for each model. These analyses were performed using statsmodels 0.13.2 ( https://www.statsmodels.org/stable/index.html ). To evaluate the predictive power of the blood glucose response indices ( G ) on the administered glucose dose ( F ), two linear regression models ( F = K · G AUC or F = K · G ) were evaluated using data from the University of Tokyo Study . For each participant, the K was assumed to remain constant throughout all observations and was estimated using the regressions. The predictive accuracy was evaluated by comparing the predicted F with the observed F . Using data from the HPP cohort and the F = K · G equation, individual K values were estimated from CGM-recorded glucose dynamics and dietary carbohydrate intake. For each annotated meal, CGM values were extracted within a four-hour postprandial window starting at the recorded mealtime. These free-living profiles included real-world complexities, such as periods of exercise, mixed (non-glucose-load) meals, and discrepancies between actual and logged mealtimes. Rather than excluding these sources of variability, analyses were conducted to test whether the simple formulation was informative for predicting IMT and hepatic status despite these challenges. The model parameters ( K or K ) were estimated using non-linear least squares fitting with the constraint that all values were greater than zero. This estimation was conducted using SciPy 1.10.1 ( 44 ). Hierarchical clustering analyses and principal component analyses were performed after z-score normalization of the input features. Euclidean distance metrics and Ward linkage were used for clustering. The quality of the hierarchical clustering analysis was assessed using silhouette analysis ( 46 ). These analyses were performed using scikit-learn 1.0.2 ( https://scikit-learn.org/stable/ ). Exploratory factor analysis was used to explore the latent structure between correlated measures, following the previously reported approach ( 47 – 50 ) with some modifications. Variables with absolute factor loadings of at least 0.35 were used for interpretation. Oblimin rotation was used to improve interpretation. To evaluate the applicability of the factor analysis, the Kaiser-Meyer-Olkin and Bartlett’s spherical test were performed. Prior to the analysis, all data were standardized using z-score normalization. These analyses were performed using factor-analyzer 0.5.1 ( https://pypi.org/project/factor-analyzer/ ). Data Availability Data from the Kobe University Study are available at https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0143880#pone.0143880.ref031 . Data from the University of Tokyo Study are available at https://www.nature.com/articles/s41540-019-0108-1#Sec9 . Data from the San Francisco Bay Area Study1 are available at https://pmc.ncbi.nlm.nih.gov/articles/PMC6057684/#sec011 . Data from the San Francisco Bay Area Study2 are available at https://www.nature.com/articles/s41551-024-01311-6#Sec13 . Data from the HPP cohort are available at https://humanphenotypeproject.org/ . Funding Japan Society for the Promotion of Science (JSPS) KAKENHI (JP21H04759) (SK) CREST, the Japan Science and Technology Agency (JST) (JPMJCR2123) (SK) The Uehara Memorial Foundation (SK) The Takeda Science Foundation (HS) Author contributions Conceptualization: HS, SK Methodology: HS Investigation: HS Visualization: HS Funding acquisition: SK Project administration: WO, SK Supervision: WO, SK Writing – original draft: HS Writing – review & editing: KH, TY, KS, WO, SK Competing interests Authors declare that they have no competing interests. Data and materials availability The data supporting the findings of this study originate from the Kobe University Study ( 26 ), the University of Tokyo Study ( 33 ), the San Francisco Bay Area Study1 ( 34 ), the San Francisco Bay Area Study2 ( 16 ), and the HPP cohort ( 18 , 35 , 36 ). Data from the Kobe University Study are available at https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0143880#pone.0143880.ref031 ( 41 ). Data from the University of Tokyo Study are available at https://www.nature.com/articles/s41540-019-0108-1#Sec9 ( 33 ). Data from the San Francisco Bay Area Study1 are available at https://pmc.ncbi.nlm.nih.gov/articles/PMC6057684/#sec011 ( 34 ). Data from the San Francisco Bay Area Study2 are available at https://www.nature.com/articles/s41551-024-01311-6#Sec13 ( 16 ). The HPP cohort dataset used in this study is governed by strict privacy and ethical protocols, as approved by the relevant Institutional Review Boards. Due to data protection and participant confidentiality considerations, data access is limited and is granted only for approved research proposals ( 18 , 35 , 36 ). The code that simulates blood glucose, insulin, and C-peptide levels is available at our GitHub repository ( https://github.com/HikaruSugimoto/Glucose_simulation ). Supplementary Materials Supplementary Texts 1 to 4 Figs. S1 to S2 References (51-60) Acknowledgments We thank Nancy R. Gough (BioSerendipity, LLC) for critical reading and editorial assistance. We thank the Human Phenotype Project (HPP) and Pheno.AI for providing access to the cohort data used in this study. The images in this study are from TogoTV (© 2016 DBCLS TogoTV, CC-BY-4.0) and NIH BioArt ( https://bioart.niaid.nih.gov/ ). References 1. ↵ H. Sun , P. Saeedi , S. Karuranga , M. Pinkepank , K. Ogurtsova , B. B. Duncan , C. Stein , A. Basit , J. C. N. Chan , J. C. Mbanya , M. E. Pavkov , A. Ramachandaran , S. H. Wild , S. James , W. H. Herman , P. Zhang , C. Bommer , S. Kuo , E. J. Boyko , D. J. 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Share A control-theoretic law of human glucose homeostasis Hikaru Sugimoto , Ken-ichi Hironaka , Tomoko Yamada , Kazuhiko Sakaguchi , Wataru Ogawa , Shinya Kuroda medRxiv 2025.09.30.25337012; doi: https://doi.org/10.1101/2025.09.30.25337012 Share This Article: Copy Citation Tools A control-theoretic law of human glucose homeostasis Hikaru Sugimoto , Ken-ichi Hironaka , Tomoko Yamada , Kazuhiko Sakaguchi , Wataru Ogawa , Shinya Kuroda medRxiv 2025.09.30.25337012; doi: https://doi.org/10.1101/2025.09.30.25337012 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Endocrinology (including Diabetes Mellitus and Metabolic Disease) Subject Areas All Articles Addiction Medicine (568) Allergy and Immunology (863) Anesthesia (300) Cardiovascular Medicine (4435) Dentistry and Oral Medicine (444) Dermatology (382) Emergency Medicine (608) Endocrinology (including Diabetes Mellitus and Metabolic Disease) (1509) Epidemiology (15227) Forensic Medicine (30) Gastroenterology (1124) Genetic and Genomic Medicine (6597) Geriatric Medicine (668) Health Economics (997) Health Informatics (4534) Health Policy (1368) Health Systems and Quality Improvement (1613) Hematology (540) HIV/AIDS (1264) Infectious Diseases (except HIV/AIDS) (15916) Intensive Care and Critical Care Medicine (1103) Medical Education (623) Medical Ethics (146) Nephrology (667) Neurology (6599) Nursing (346) Nutrition (998) Obstetrics and Gynecology (1144) Occupational and Environmental Health (957) Oncology (3332) Ophthalmology (974) Orthopedics (369) Otolaryngology (420) Pain Medicine (436) Palliative Medicine (130) Pathology (663) Pediatrics (1693) Pharmacology and Therapeutics (691) Primary Care Research (711) Psychiatry and Clinical Psychology (5447) Public and Global Health (9230) Radiology and Imaging (2198) Rehabilitation Medicine and Physical Therapy (1370) Respiratory Medicine (1196) Rheumatology (593) Sexual and Reproductive Health (712) Sports Medicine (530) Surgery (712) Toxicology (99) Transplantation (289) Urology (265) (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'a0034d88abe1300f',t:'MTc3OTUzMTM2Mg=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();