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The impact of mathematical modeling on the understanding of diabetes and related complications
review-article
Publication date (Electronic):
10 July 2013
Publisher:
Nature Publishing Group
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Abstract
Diabetes is a chronic and complex multifactorial disease caused by persistent hyperglycemia and for which underlying pathogenesis is still not completely understood. The mathematical modeling of glucose homeostasis, diabetic condition, and its associated complications is rapidly growing and provides new insights into the underlying mechanisms involved. Here, we discuss contributions to the diabetes modeling field over the past five decades, highlighting the areas where more focused research is required.
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BioModels Database: An enhanced, curated and annotated resource for published quantitative kinetic models
Chen Li, Marco Donizelli, Nicolas Rodriguez … (2010)
Background Quantitative models of biochemical and cellular systems are used to answer a variety of questions in the biological sciences. The number of published quantitative models is growing steadily thanks to increasing interest in the use of models as well as the development of improved software systems and the availability of better, cheaper computer hardware. To maximise the benefits of this growing body of models, the field needs centralised model repositories that will encourage, facilitate and promote model dissemination and reuse. Ideally, the models stored in these repositories should be extensively tested and encoded in community-supported and standardised formats. In addition, the models and their components should be cross-referenced with other resources in order to allow their unambiguous identification. Description BioModels Database http://www.ebi.ac.uk/biomodels/ is aimed at addressing exactly these needs. It is a freely-accessible online resource for storing, viewing, retrieving, and analysing published, peer-reviewed quantitative models of biochemical and cellular systems. The structure and behaviour of each simulation model distributed by BioModels Database are thoroughly checked; in addition, model elements are annotated with terms from controlled vocabularies as well as linked to relevant data resources. Models can be examined online or downloaded in various formats. Reaction network diagrams generated from the models are also available in several formats. BioModels Database also provides features such as online simulation and the extraction of components from large scale models into smaller submodels. Finally, the system provides a range of web services that external software systems can use to access up-to-date data from the database. Conclusions BioModels Database has become a recognised reference resource for systems biology. It is being used by the community in a variety of ways; for example, it is used to benchmark different simulation systems, and to study the clustering of models based upon their annotations. Model deposition to the database today is advised by several publishers of scientific journals. The models in BioModels Database are freely distributed and reusable; the underlying software infrastructure is also available from SourceForge https://sourceforge.net/projects/biomodels/ under the GNU General Public License.
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Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes.
Massimo Massi-Benedetti, H. E. Schaller, T Vering … (2004)
A nonlinear model predictive controller has been developed to maintain normoglycemia in subjects with type 1 diabetes during fasting conditions such as during overnight fast. The controller employs a compartment model, which represents the glucoregulatory system and includes submodels representing absorption of subcutaneously administered short-acting insulin Lispro and gut absorption. The controller uses Bayesian parameter estimation to determine time-varying model parameters. Moving target trajectory facilitates slow, controlled normalization of elevated glucose levels and faster normalization of low glucose values. The predictive capabilities of the model have been evaluated using data from 15 clinical experiments in subjects with type 1 diabetes. The experiments employed intravenous glucose sampling (every 15 min) and subcutaneous infusion of insulin Lispro by insulin pump (modified also every 15 min). The model gave glucose predictions with a mean square error proportionally related to the prediction horizon with the value of 0.2 mmol L(-1) per 15 min. The assessment of clinical utility of model-based glucose predictions using Clarke error grid analysis gave 95% of values in zone A and the remaining 5% of values in zone B for glucose predictions up to 60 min (n = 1674). In conclusion, adaptive nonlinear model predictive control is promising for the control of glucose concentration during fasting conditions in subjects with type 1 diabetes.
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Quantifying the Contribution of the Liver to Glucose Homeostasis: A Detailed Kinetic Model of Human Hepatic Glucose Metabolism
Introduction The human plasma glucose is kept in a narrow range between minimum values of ∼3 mM after prolonged fasting or extensive muscle activity and maximum values of ∼9 mM reached postprandially [1], [2]. Homoeostasis of plasma glucose is crucial for the organism: Hyperglycemia results in non-enzymatic glycosylation (glycation) and thus loss-of-function of proteins [3], glucose induced oxidative damage [4], [5] and other adverse effects [6], [7]. Hypoglycemia leads to an under-supply of tissues with glucose and is thereby of particular danger for neuronal cells, erythrocytes and fibroblasts, using glucose as dominant or even exclusive energy-delivering fuel under normal physiological conditions. The liver is a central player in buffering plasma glucose contributing either by net hepatic glucose utilization (HGU) or net hepatic glucose production (HGP) depending on the plasma glucose level exceeding or falling below a critical threshold value (in the following referred to as ‘set point’) of ∼6 mM. Switching between HGP and HGU is therefore a switch between positive (i.e. export of glucose) and negative (i.e. import of glucose) net hepatic glucose balance. This crucial metabolic function of the liver is performed by hepatocytes which exhibit high capacity of glycogenesis, glycogenolysis, glycolysis and gluconeogenesis enabling them to transiently store substantial amounts of glucose as glycogen, to synthesize glucose from lactate, glycerol and glucoplastic amino acids and to convert excess glucose into triglycerides [2], [8], [9]. Glucose homeostasis is controlled by several hormones, with insulin and glucagon being the main counteracting players [1], [10]. Insulin is the only known hormone lowering blood glucose, whereas multiple glucose increasing hormones exist. Glucagon plays the primary role in counter-regulation to hypoglycemia. Epinephrine has a secondary role, becoming critical under impaired glucagon responses, but with reduced effectiveness compared to glucagon [11]–[13]. Other counter-regulatory hormones like cortisol or thyroxin play only a minor role for the liver [10], [12], [13]. The plasma concentrations of insulin, glucagon and epinephrine change as direct response to varying blood glucose [1], [10]. In the liver insulin increases the activity of glucose utilizing pathways (HGU, glycolysis, glycogenesis) and decreases glucose producing pathways (HGP, gluconeogenesis, glycogenolysis), whereas glucagon and epinephrine have contrary effects. Main targets of the gluco-regulatory hormones are key interconvertible enzymes of glucose metabolism like pyruvate kinase or glycogen synthase. The kinetics of the interconvertible enzymes, and consequently also the hepatic glucose metabolism, depends on their phosphorylation state [2], [14] which is altered by the hormones. Despite the crucial role of the liver for glucose homeostasis, a detailed mathematical model of human glucose metabolism of the liver, indispensable for understanding the hepatic role under normal and impaired conditions like occurring in diabetes, has not been developed yet. Available kinetic models of hepatic glucose metabolism are either minimal models [15]–[18], concentrate on parts of the glucose metabolism [19], or lump reactions [20], [21]. Furthermore, all available models concentrate on the glucose-insulin system, and ignore the crucial insulin antagonists glucagon and epinephrine [15]–[19], or do not include hormonal regulation at all [20], [21]. All mentioned models with exception of [19] ignore the crucial dependency of enzyme kinetics on the respective phosphorylation state of the enzyme completely, an essential mechanism for short term regulation in hepatic glucose metabolism. Here, we present the first model of hepatic glucose metabolism in molecular detail, which includes the crucial control of hepatic glucose metabolism by insulin, glucagon and epinephrine via changes in phosphorylation state of key enzymes based on a new concept of linking hormonal regulation with metabolism. Methods The model of human hepatic glucose metabolism (Figure 1) consists of 49 localized metabolites (Text S1) and 36 reactions (Text S2) compartmentalized into cytosol, mitochondrion and blood. Abbreviations of metabolites and reactions are defined in the legend of Figure 1. All reactions and transporters are modeled with individual kinetics (Text S3) with parameters from literature research or by fitting to experimental metabolite and flux data (Equation 1, Dataset S1). An annotated SBML of the model is provided in the supplement (Dataset S2). The model includes regulation via allosteric mechanisms (Text S4) and regulation by hormones via phosphorylation and dephosphorylation of interconvertible enzymes (Text S5). Enzymes regulated by allosteric mechanisms are GK, GS, GP, PFK1, PK, PC, PDH, FBP1 and FBP2. Enzymes regulated via changes in phosphorylation state are GS, GP, PFK2, FBP2, PK and PDH. All model fluxes are given in (bw = bodyweight). 10.1371/journal.pcbi.1002577.g001 Figure 1 Overview of human hepatocyte model consisting of glycolysis, gluconeogenesis and glycogen metabolism. Metabolic model compartmentalized in blood, cytosol and mitochondrion. Key enzymes of HGP are shown in red, key enzymes of HGU in green. Enzymes regulated via allosteric mechanisms are marked with a red A, interconvertible enzymes with a yellow P. Insulin, glucagon and epinephrine regulate the glucose metabolism by changing the phosphorylation state of key interconvertible enzymes, with insulin ( +) decreasing and epinephrine ( +) and glucagon ( ) increasing . See Methods for additional information. Reactions: (ALD) aldolase, (AK) adenylate kinase, (CS) citrate synthase, (EN) enolase, (FBP1) fructose-1,6 bisphosphatase, (FBP2) fructose-2,6 bisphosphatase, (GAPDH) glyceraldehyde phosphate dehydrogenase, (G1PI) glucose-1 phosphate 1,6-phosphomutase, (G6P) glucose-6 phosphatase, (GK) glucokinase, (GLUT2) glucose transporter 2, (GP) glycogen phosphorylase, (GPI) glucose-6 phosphate isomerase, (GS) glycogen synthase, (LDH) lactate dehydrogenase, (MALT) malate transporter, (MDH) malate dehydrogenase, (NDK) nucleoside diphospate kinase, (PC) pyruvate carboxylase, (PEPCK) phosphoenolpyruvate cyrboxykinase, (PEPT) phosphoenolpyruvate transporter, (PDH) pyruvate dehydrogenase, (PFK1) phosphofructokinase 1, (PFK2) phosphofructokinase 2, (PGK) phosphoglycerate kinase, (PGM) 3-phosphoglycerate mutase, (PK) pyruvate kinase, (PPASE) pyrophosphate phosphohydrolase, (PYRT) pyruvate transporter, (TPI) triose phosphate isomerase, (UGT) UTP:glucose-1 phosphate uridylyltransferase. Metabolites: (13P2G) 1,3-bisphospho glycerate, (2PG) 2-phospho glycerate, (3PG) 3-phospho glycerate, (ACoA) acetyl-coA, (Cit) citrate, (CoA) coenzyme A, (DHAP) dihydroxyacetone phosphate, (Fru16P2) fructose-1,6 bisphosphate, (Fru26P2) fructose-2,6 bisphosphate, (Fru6P) fructose-6 phosphate, (Glc) glucose, (Glc1P) glucose-1 phosphate, (Glc6P) glucose-6 phosphate, (GRAP) glyceraldehyde 3-phosphate, (Lac) lactate, (OA) oxalacetate, (Mal) malate, (P) phosphate, (PEP) phosphoenolpyruvate, (PP) pyrophosphate, (Pyr) pyruvate, (UDP-Glc) UDP-glucose. Kinetic parameters The detailed kinetic equations for the reactions and transporters (Text S3) are specific for human liver and based on extensive literature research. All kinetic parameters except the maximal velocities are based on literature data with references given in the respective rate equation. All values in the model were determined by fitting simulated model fluxes to experimentally determined fluxes and simulated model metabolite concentrations to experimentally measured concentrations . The fit procedure minimizes the sum of quadratic relative differences in fluxes and concentrations, divided by the total number of experimental fluxes or respectively the total number of experimental concentrations (Equation 1). Relative differences were used to avoid a domination of the optimization by large absolute values and to have dimensionless quantities. The contributions of fluxes and concentrations to least-square fit function were weighed equally ( ) assigning same relevance to fluxes and concentrations. (1) The experimental flux data was human specific (Dataset S1), the metabolite concentrations were taken from human and rat liver and are given in (Text S1, Dataset S1). Equation 1 was minimized using the MATLAB® Optimization Toolbox (constraint nonlinear optimization) and resulted in a final value of , indicating that the overall remaining relative deviations of theoretical values from experimental ones were lower than a factor of 2. The fitted values are given in Text S2. Integration of differential equations For the time course and steady state simulations the differential equation system (Text S3) was integrated with a variable-order solver for stiff problems based on numerical differentiation formulas with absolute integration tolerance and relative integration tolerance (ode15s MATLAB® R2011a, MathWorks). Initial concentrations for all simulations are given in Text S1. Variation in blood glucose and glycogen are given in the respective figure legends. The external concentrations of blood glucose and lactate were kept at fixed values in all simulations as their time evolution in the blood depends on the metabolic activity of various organs not considered in this model. The cellular redox state (given by the concentrations of NAD and NADH) and cellular energy state (given by the concentrations of the adenine nucleotides) were kept constant during all simulations. Steady state solutions are defined as states with absolute changes in every concentration smaller than for a time interval . Steady state solutions were tested to be stable against small changes in initial concentrations (1%). All conservation entities were tested to be constant within the tolerances over the integration. Hormonal regulation (GHR) As the release of hormones and their elimination from the plasma are not part of the model, the relationships between plasma glucose level and hormone levels (Figure 2) are described by phenomenological functions called glucose-hormone responses (GHR). The GHRs are sigmoid functions ranging between basal hormone concentration hbase and maximal hormone concentration hmax, monotonically increasing with increasing blood glucose for insulin (Equation 2), monotonically decreasing for glucagon and epinephrine (Equation 3). The GHRs were determined by least-square fit to oral glucose tolerance tests and hypoglycemic, hyperinsulinemic clamp studies (Table 1, Dataset S1). The parameters correspond to , , the inflection point and the Hill coefficient which determines the steepness of the sigmoidal hormone response. (2) (3) Experimental data and standard deviations (Dataset S1) were extracted from figures, tables and supplemental information. Data points correspond to mean data for multiple subjects from the studies. No data points were omitted. 10.1371/journal.pcbi.1002577.g002 Figure 2 Glucose-Hormone Responses (GHR) of insulin, glucagon and epinephrine and phosphorylation state . Experimental data from multiple studies based on oral glucose tolerance tests and hypoglycemic, hyperinsulinemic clamp studies and the corresponding fitted GHR curves. Parameters for the GHR curves in Table 1 and experimental data in Dataset S1. (A) Glucagon GHR. Glucagon increases with decreasing blood glucose from a basal concentration of 37.9 pmol/l to a maximal concentration of 190.0 pmol/l with the inflection point at 3.01 mM (B) Epinephrine GHR. Epinephrine increases with decreasing glucose from a basal concentration of 100 pmol/l to a maximal concentration of 6090 pmol/l with the inflection point at 3.1 mM (C) Insulin GHR. Insulin increases with increasing glucose from a basal concentration of 0 pmol/l to a maximal concentration of 818.9 pmol/l with the inflection point at 8.6 mM (D) Resulting phosphorylation state (Equation 5) based on the GHRs for insulin, glucagon and epinephrine decreasing with increasing glucose from phosphorylated to dephosphorylated. 10.1371/journal.pcbi.1002577.t001 Table 1 Fit functions and parameters for the insulin, glucagon and epinephrine GHR. Hormone f max base k n Experimental Data [pmol/l] [pmol/l] [mM] insulin h1 818.9 0 8.6 4.2 [1], [28], [35]–[41] glucagon h2 190 37.9 3.01 6.4 [1], [28], [35], [37], [38], [40]–[46] epinephrine h2 6090 100 3.1 8.4 [40], [42]–[47] Phosphorylation state (γ) of interconvertible enzymes The short-term effects of insulin, glucagon and epinephrine result from changes in the phosphorylation state of key interconvertible enzymes of glucose metabolism, namely GS, GP, PFK2, FBP2, PK and PDH. The interconvertible enzymes exhibit different kinetic properties in the phosphorylated state (P) and dephosphorylated state (DP), thus carrying different fluxes in the phosphorylated state ( ) and dephosphorylated state ( ). The resulting effective kinetics (Equation 4) is the linear combination of and dependent on the phosphorylation state , with being the phosphorylated and the dephosphorylated fraction of the enzyme. (4) The phosphorylation state is a phenomenological function of insulin, glucagon and epinephrine (Equation 5). Insulin decreases , whereas glucagon and epinephrine increase . Epinephrine acts as a backup system for glucagon with reduced effectiveness . Only the currently dominating hormone (glucagon or epinephrine) was taken into account (max). The hormonal dependencies on the phosphorylation state were modeled by a Michaelis-Menten like hyperbolic function with half-saturation at whereby the saturation parameter was set to for all hormones. The maximal basal hormone concentrations of the three hormones ( ) result from the respective GHR curves and are given in Table 1. (5) We assumed that for given hormone concentrations the fraction of all interconvertible enzymes is equal. Hepatic glucose production, gluconeogenesis and glycogenolysis Experimental data (Table 2, Dataset S1) was extracted from figures and tables of 25 independent studies with different tracer methods. See [2] for review and Table 2 for detailed references, used methods and experimental data. Every data point is the mean for multiple subjects from one of the studies. No data points were omitted. 10.1371/journal.pcbi.1002577.t002 Table 2 Experimental data for HGP, gluconeogenesis (GNG), glycogenolysis (GLY). Method Reference Time HGP GNG GLY GNG/HGP [h] [µmol/kg(bw)/min] [µmol/kg(bw)/min] [µmol/kg(bw)/min] [%] 14C-acetate [48] 66 7.56 7.39 0.34 97 13C-NMR [22] 42–64 8.7 8.3 0.3 96 13C-glycerol MIDA [49] 60 7.87 7.71 1.58 98 13C-glucose MID [50] 40 9.8 9.1 1.1 92 2H2O [51] 42 - - - 93 2H2O [52] 40 7.93 7.13 0.8 90 14C -acetate [48] 14 12.5 3.6 9 28 14C -bicarbonate [53] 10–12 7.15 2.2 4.9 31 14C -bicarbonate [54] 12–14 8.3 2.6 5.5 31 14C -glucose [55] 12 8.8 4.5 4.3 51 13C-NMR [22] 22 12.2 7.9 4.3 64 13C-NMR [23] 23 8.9 6.1 2.8 70 13C-glycerol MIDA [49] 11 12.1 5.9 6.2 49 13C-glycerol [56] 10 11.7 4.9 6.8 41 13C-glucose MID [50] 12 12.9 5.3 7.7 41 13C-glucose [57] - 13.1 7.4 5.7 56 13C-glucose (a) [58] 18 8.5 5.9 2.6 59 13C-glucose (b) [58] 18 8.5 3.7 4.8 44 13C-glucose [59] 12 12.9 5.3 7.6 41 13C-glucose [59] 16 11.5 6.6 4.8 57 13C-glucose [59] 20 10 7.1 2.9 71 2H2O [51] 14 10.2 5.5 4.7 54 2H2O [51] 22 8.6 5.5 3.1 64 2H2O [60] 16 10 5.5 4.5 55 2H2O [60] 20 9 5.4 3.6 60 2H2O [60] 24 8.5 5.2 3.3 61 2H2O [61] 10 11.4 5.5 5.9 48 2H2O [62] 16 12.2 6.7 5.5 55 2H2O [63] 15 12.4 5.6 6.7 45 2H2O [56] 10 12.2 7.4 3.8 60 2H2O [64] 11 10.4 5.8 4.65 55 2H2O [65] 16 9.8 5 4.8 51 2H2O [66] 15 17.7 12.7 5 71 2H2O [67] 16 17.5 8.9 8.2 51 2H2O [68] 17 11.9 4.6 7.3 38 Glycogenolysis and glycogen synthesis Experimental data (Dataset S1) was extracted from figures and tables from [22] (individual data) and [23] (mean person data with STD) for the glycogenolysis and from [8], [24], [25] (mean person data) for the glycogen synthesis. The only difference between model simulations in states of either net glycogen synthesis or net glycogenolysis simulations are differences in the initial glycogen and blood glucose concentrations. Experimental data for the mixed meal and hyperglycemic, hyperinsulinemic clamp simulations were extracted from [26] using the experimental time courses of glucose as model input. Insulin and glucagon time courses were predicted with the respective hormone dose response curves (Equation 2 and Equation 3). Simulations were started the from initial glycogen concentrations reported in the experiments. Role of insulin and glucagon for HGP Experimental data (Dataset S1) were extracted from figures and tables from [27]. Basal hormone concentrations of dogs and humans differ due to difference in human and dog physiology. Therefore, in model simulations of the human liver typical insulin and glucagon concentrations for the basal glucose production rate in humans were used (insulin and glucagon ) and concentration changes for the hormones expressed relative to these basal values Changes in blood glucose were calculated from the simulated HGP, for a mean bodyweight of , blood volume of and whole-body basal glucose utilization rate of . In Simulation 1 no experimental data was taken into account, but only a drop in the respective hormones during the infusion period (135–200 min) assumed. In Simulation 2 the experimentally observed changes in hormones and GU relative to basal levels (mean of values during the pre-infusion period) were used. Hepatic glucose response coefficient (HGRC) The HGRC defined by Equation 6 measures the change in net hepatic glucose balance defined through the net rate of glucose exchange between hepatocytes and the blood ( ) elicited by a small change in blood glucose concentration. In model simulations, the HGRC was approximated by a difference quotient and numerically calculated from the steady state GLUT2 fluxes. For the HRGC corresponds to the changes in HGU, for to the change in HGP. (6) Coupling of metabolic pathways to the TCA cycle The reactions for citrate efflux, oxaloacetate influx and acetyl-CoA flux were included in the model, to test the model under various TCA cycle loads and changes in acetyl-CoA demand and production being a necessary condition in the model development of a functioning model of hepatic glucose metabolism under typical physiological TCA cycle loads. The malate-aspartate shuttle (MALT) including cytosolic and mitochondrial malate dehydrogenase (MDH) was not modeled in detail. Thus, the rate of the cytosolic isoform of the PEPCK was put to directly depend on mitochondrial oxaloacetate concentration. For the actual simulations these boundary fluxes where set to zero. Metabolic Control Analysis (MCA) To characterize the regulatory importance of various reactions in different states of hepatic glucose metabolism we used metabolic flux response coefficients (Equation 7), describing how the flux rate of an arbitrary reaction changes in response to a small change in the maximal activity of an arbitrary reaction . (7) Response coefficients with respect to the maximal activities of all enzymes were calculated at varying values of external glucose and cellular glycogen content for key fluxes of hepatic glucose metabolism (Figure S9): hepatic glucose uptake/release ( ), gluconeogenesis/glycolysis ( ) and glycogenolysis/gluconeogenesis ( ). Results We present the first detailed kinetic model of human hepatic glucose metabolism integrated with the hormonal regulation via insulin, glucagon and epinephrine (Figure 1 and Methods) via a new concept of changes in phosphorylation state (Figure 2 and Methods). The model was validated with quantitative and qualitative experimental data from a multitude of studies on hepatic glucose metabolism in overnight fasting, short term fasting and postprandial glycogen storage (Table 3, Figure 3, Figure 4, Dataset S1). In a next step the model was applied to analyze the central role of the liver in glucose homeostasis, especially the hepatic capability (i) to switch between an anabolic glucose mode (HGP) to a catabolic glucose mode (HGU) depending on the blood glucose concentration and hormonal signals (iii) to use glycogen as glucose buffer in this process (iii) to respond to counteracting changes in blood glucose with changes in glucose production (HGP) and glucose utilization, especially in the fasting and postprandial physiological state (Figure 5). 10.1371/journal.pcbi.1002577.g003 Figure 3 HGP, gluconeogenesis (GNG), glycogenolysis (GLY) and GNG/HGP in short term fasting over 70 h. Experimental data from 25 separately published studies based on a variety of different labeling approaches listed in Table 2 and given in Dataset S1. Simulation time courses (dashed lines) start at t = 10 h with glycogen partially filled glycogen stores (350 mM). Blood glucose is varied between 3.6 mM (red) and 4.6 mM (green) in steps of 0.2 mM. The blue curve corresponds to a situation where the plasma glucose concentration changes over the time of fasting from 4.2 mM to 3.6 mM in 70 h. (A) HGP decreases with increasing fasting time and reaches constant levels after around 25 h. With decreasing blood glucose HGP increases. (B) gluconeogenesis is almost constant over time. With decreasing blood glucose GNG increases (C) glycogenolysis decreases over time with decreasing glycogen. Lower blood glucose results in an initially higher GLY rate (D) GNG/HGP. The relative contribution of gluconeogenesis to the hepatic glucose production increases over time. After 40 h fasting over 90% of the produced glucose comes from de novo synthesis. 10.1371/journal.pcbi.1002577.g004 Figure 4 Glycogen metabolism and hormone perturbations. (A) Decrease in glycogen during short term fasting over 60 h. Experimental data from [22] (individual data) and [23] (mean data with STD) given in Dataset S1. Simulation time courses (dashed lines) start from filled glycogen stores (500 mM). Blood glucose is varied between 3.6 mM (red) and 5.0 mM (green) in steps of 0.2 mM (black). The blue curve corresponds to a situation where the plasma glucose concentration changes from initially 5 mM to 3.6 mM in 60 h. Higher blood glucose results in a slower decrease in glycogen. (B) Increase in glycogen after a glucose load. Experimental data from [8], [24], [25] given in Dataset S1. Simulation time courses (dashed lines) start from initial glycogen concentration of 200 mM. Blood glucose concentrations are varied between 5.5 mM (red) and 8.0 mM (green) in steps of 0.5 mM (black). The blue line corresponds to 7 mM. Higher blood glucose results in a faster increase in glycogen after a glucose load. (C) Change in HGP and plasma glucose in insulin deficiency. Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data from [27] (middle), and a simulation taking the experimental profile of hormones and glucose utilization into account are shown in the top row. Middle row depicts the time courses of HGP (red) and whole-body glucose utilization GU (black). Bottom row shows the resulting change in plasma glucose as a consequence of the changes in HGP. Insulin deficiency increases HGP relative to GU resulting in an increase in plasma glucose. After insulin normalization HGP decreases and falls below GU due to the increased blood glucose. (D) Changes in HGP during the infusion period (135–200 min) relative to the pre-infusion period (85–135 min) for the simulations and experiments and the various hormone perturbations. Saline infusion (Figure S1) and somatostatin infusion with insulin and glucagon restauration (Figure S5) have no effect on basal HGP. Insulin and glucagon deficiency (Figure S2) and glucagon deficiency alone (Figure S3) decrease HGP relative to basal values. Insulin deficiency increases basal HGP (Figure 4C, Figure S4). 10.1371/journal.pcbi.1002577.g005 Figure 5 HGP/HGU and HGRC. Steady state solution of time course simulations under constant glycogen and blood glucose concentration (every data points correspond to a single simulation to steady state). Glycogen is varied between 0 and 500 mM in steps of 5 mM, blood glucose is varied between 2 and 14 mM in steps of 0.05 mM. (A) HGP (red)/HGU (green) ([ ]) corresponding to GLUT2 flux. The liver switches between HGP and HGU depending on the blood glucose and glycogen. The set point varies between 5.4 mM for empty and 7.5 mM for completely filled glycogen stores (6.6 mM at glycogen 250 mM). (B) HGRC ([ ]). Response of the liver to changes in the blood glucose. Two main regions of high response exist: A hypoglycemic region below 4 mM, and a hyperglycemic region above 8 mM glucose. (C) gluconeogenesis (red)/glycolysis (green) ([ ]) corresponding to glucose-6p isomerase (GPI) flux. The gluconeogenesis/glycolysis set point varies between 8.8 mM for empty and 7.0 mM for filled glycogen stores. (D) glycogenolysis (red)/glycogenesis (green) ([ ]) corresponding to flux through glucose-1p isomerase (G1PI). 10.1371/journal.pcbi.1002577.t003 Table 3 Comparison of model predictions and experiments. Item Model Experimental data References A HGP at −5.5 mM blood glucose HGU at 8 mM blood glucose ∼4 µmol/kg(bw)/min HGU at 8 mM blood glucose ∼8.5 µmol/min/kg(bw) splanchnic glucose utilization (SGU) at 8 mM glucose (difference between 7 µmol/min/kg (bw) splanchnic glucose production and 15.5 µmol/min/kg (bw) splanchnic glucose uptake at physiological insulin of 300 pmol/l at 8 mM) [29] HGP/HGU set point: 6.6 mM glucose for half-filled glycogen; 7.3 mM glucose for filled glycogen stores; HGU 5 mM). A model is always an abstraction of reality describing a certain subset of biological phenomena. The underlining assumptions are crucial for to understand the range of application of the model and the limitations. Main model assumptions and simplifications are (i) usage of phenomenological functions to incorporate hormone-induced signal transduction, (ii) a constant cellular redox- and energy status (iii) modeling of an ‘average’ hepatocyte, i.e. neglecting metabolic zonation of hepatocytes along the sinusoid [32] and (iv) no inclusion of changes in the gene expression of metabolic enzymes. (i) A crucial part of this model is the phenomenological description of signal transduction ( ), which takes into account a substantial body of qualitative knowledge about the effects of the individual hormones on hepatic glucose metabolism. Due to the lack of quantitative experimental data on signaling processes as, for example, concentrations, activities and phosphorylation states of key kinases and phosphatases at varying concentrations of glucose, insulin, glucagon and combinations of these factors, a more detailed description was not possible. The model reproduces the experimentally observed dependency of glucose metabolism on the hormones. Especially, the reproduction of the classical insulin and glucagon perturbation experiments of [27], performed as independent validation without taking into account in the modeling process, underlines the validity of such a simplified treatment of the signaling network in the case of the hepatic glucose metabolism. Throughout the model the phosphorylation state of all regulatory enzymes is the same at given concentration of glucagon and insulin. This simplification could be the reason why the activation of the glycogen synthase was overestimated in mixed meal and clamp simulations. A second simplification was that the fraction ( ) of phosphorylated inter-convertible enzymes follow instantaneously the hormone concentrations in the plasma although signal transduction occurs in the range of some minutes [33]. Including time-dependent changes of proteins involved in the cAMP-dependent signaling cascade in a later and more advanced version of the model should result in slightly different time-courses of fluxes and metabolite concentrations elicited by abrupt changes of external metabolite- and hormone concentrations. However, the main results of this study refer to the normal physiological situation where significant changes of the blood glucose level occur in a time window of hours (Figure 3, 4 and 5) with the metabolic network always being in quasi-steady state. (ii) The model is limited to physiological states of the liver where changes in the energy- and redox state can be neglected. Normally, the hepatic energy state is decoupled from the hepatic glucose metabolism, with -oxidation of fatty acids providing ATP and reduced redox equivalents (NADH). Consequently, the model is not able to simulate conditions where this assumption of energy decoupling is not valid, for example, under hypoxic or ischemic conditions. (iii) The presented model describes the metabolic net behavior of the liver of a hepatocyte averaged over the liver and is therefore able to simulate the net effect of the liver on glucose balance, namely net HGP and net HGU. In reality the liver exhibits a zonated structure with varying capacities of gluconeogenesis and glycolysis of the hepatocytes along the sinusoid [32]. Some of the effects brought about by an ensemble of hepatocytes equipped with differing capacities of glucose metabolism cannot be simulated with a model of a ‘mean’ hepatocyte. For example, the remaining difference between absolute glucose production and net glucose production (Figure S6) are due to the simultaneous presence of hepatocytes with net HGP (peri-portal hepatocytes) and HGU (peri-central hepatocytes). (iv) Remarkably, the anabolic/catabolic switch was achieved by fast changes in the phosphorylation state and allosteric regulation of key enzymes of glucose metabolism without temporal changes in gene expression, i.e. changes in the protein levels of enzymes. In our opinion, changes in gene expression as observed to occur with different period lengths during the day [31] play no crucial role in short-term glucose homeostasis, a system which has to react to fast changes in the minute range of glucose supply (postprandial) and glucose demand (muscle activity) by an adaptation of metabolism. Changes in gene expression play a role in adapting hepatic metabolism on longer time scales to for example reduce futile cycles in counteracting pathways (like down-regulation of glucokinase and up-regulation of glucose-6 phosphatase under hypoglycemia) or match glucose metabolism to physiological requirements (up-regulating of glucose-6 phosphatase under hypoglycemic conditions to increase gluconeogenesis). The strong effects of such changes on hepatic glucose metabolism could be seen in the analysis of the response coefficients. Finally, it has to be noted, that our model comprises fairly active futile cycles in the glycogen metabolism, the PFK1/FBPase1 system and between PEPCK and PK. During model building it was tried to minimize these cycles. Reducing and removing individual cycles in the modeling process compromised the ability of the model to switch between glucose consuming and glucose producing pathways, namely HGP/HGU, gluconeogenesis/glycolysis and glycogen synthesis/glycogenolysis. An essential property of the model is the simultaneous activity of glycogen synthase and glycogen phosphorylase and the accompanied futile cycle, necessary to reproduce the experimental time-courses for glycogen synthesis in fasting and postprandially. Substrate cycling allows the system to react to fast changes in metabolite levels, the concomitant adaptation of key enzymes via phosphorylation or dephosphorylation causes an additional shift of the net flux in the right direction (see Figure S7). Taken together, metabolic regulation via phosphorylation/dephosphorylation of key enzymes in combination with futile cycles plays a key role in our model, in line with the results presented by Xu et al. [34]. Future work will apply the presented model to diseases of glucose homeostasis like diabetes, integrate the model in models of whole-body glucose metabolism and use individual patient data to generate individualized hepatic glucose models and analyze inter-individual differences in glucose metabolism. Supporting Information Dataset S1 Experimental data used in model building and validation. Data spreadsheets for model concentrations, the GHR dose response curves, the glycogen time courses in fasting and postprandial, HGP fluxes, and for the simulations reproducing experimental work by Cherrington and Krssak. (ZIP) Click here for additional data file. Dataset S2 Model SBML. Annotated SBML of the model. (XML) Click here for additional data file. Figure S1 Cherrington1976 Saline. Figure analog to Figure 4C. (top row) Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data [27] (center), and taking the experimental profile of hormones and glucose utilization into account (right). (middle row) Time course of HGP (red) and whole-body glucose utilization GU (black) for the respective hormone profiles. (bottom row) Resulting changes in plasma glucose as a consequence of the changes in HGP for the respective hormone profiles and HGP/HGU time courses. Experimental data is provided in Dataset S1. (TIF) Click here for additional data file. Figure S2 Cherrington1976 Insulin and glucagon deficiency. Figure analog to Figure 4C. (top row) Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data [27] (center), and taking the experimental profile of hormones and glucose utilization into account (right). (middle row) Time course of HGP (red) and whole-body glucose utilization GU (black) for the respective hormone profiles. (bottom row) Resulting changes in plasma glucose as a consequence of the changes in HGP for the respective hormone profiles and HGP/HGU time courses. Experimental data is provided in Dataset S1. (TIF) Click here for additional data file. Figure S3 Cherrington1976 Glucagon deficiency. Figure analog to Figure 4C. (top row) Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data [27] (center), and taking the experimental profile of hormones and glucose utilization into account (right). (middle row) Time course of HGP (red) and whole-body glucose utilization GU (black) for the respective hormone profiles. (bottom row) Resulting changes in plasma glucose as a consequence of the changes in HGP for the respective hormone profiles and HGP/HGU time courses. Experimental data is provided in Dataset S1. (TIF) Click here for additional data file. Figure S4 Cherrington1976 Insulin deficiency. Figure analog to Figure 4C. (top row) Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data [27] (center), and taking the experimental profile of hormones and glucose utilization into account (right). (middle row) Time course of HGP (red) and whole-body glucose utilization GU (black) for the respective hormone profiles. (bottom row) Resulting changes in plasma glucose as a consequence of the changes in HGP for the respective hormone profiles and HGP/HGU time courses. Experimental data is provided in Dataset S1. (TIF) Click here for additional data file. Figure S5 Cherrington1976 Somatostatin and insulin and glucagon restoration. Figure analog to Figure 4C. (top row) Insulin (blue) and glucagon (red) time courses for an idealized hormone perturbation (left), the experimental data [27] (center), and taking the experimental profile of hormones and glucose utilization into account (right). (middle row) Time course of HGP (red) and whole-body glucose utilization GU (black) for the respective hormone profiles. (bottom row) Resulting changes in plasma glucose as a consequence of the changes in HGP for the respective hormone profiles and HGP/HGU time courses. Experimental data is provided in Dataset S1. (TIF) Click here for additional data file. Figure S6 Krssak2004 Postprandial hepatic glycogen metabolism. Time courses of insulin, glucagon, glycogen and endogenous glucose production/net HGP for mixed meal tests (left) and hyperglycemic, hyperinsulinemic clamp tests (right) in Humans. Experimental data from [26] shown as black box with error bars (experimental data extracted from figures and tables). (I) Blue curves are model simulations with experimental glucose, insulin and glucagon time courses as model input without using hormone dose response curves. (II) Red curves model predictions under given glucose time course with predicted insulin and glucagon time courses via the respective hormone dose response curves as model input. In case of identical simulation results for (I) and (II) only the blue curves are shown. Initial glycogen in the mixed meal simulations are set to 275 mM glycogen and 220 mM glycogen in the clamp simulation. Grey error estimations are based on model simulations taking the errors in insulin and glucagon into account. Predicted insulin, glucagon and glycogen time courses are in agreement with the experimentally observed time courses. Predicted time courses of glycogen and EGP are very similar for (I) and (II). The glycogen synthesis rate in the clamp experiment is overestimated by the model. Imperfect simulation of glycogen synthesis in the mixed-meal and clamp simulations may be due to the fact that throughout the model the phosphorylation state of all regulatory enzymes was the same at given concentration of glucagon and insulin. This assumption is a simplification which could be the reason why the activation of the glycogen synthase was overestimated in our simulations. The measured endogenous glucose production (EGP) and the net hepatic glucose production (HGP) are substantially different. The presented model is a model describing the net hepatic output of the liver, not taking into account possible simultaneous production and utilization of glucose within the liver, for instance in periportal and perivenious regions. (TIF) Click here for additional data file. Figure S7 Fast time course simulation: model concentrations. Time course of model metabolites during perturbations in plasma glucose from initial 5 mM to 7 mM at t = 0 min, from 7 mM to 3 mM glucose at t = 10 min and from 3 mM back to the initial 5 mM glucose at t = 20 min. Respective fluxes are depicted in Figure S8. Normal model simulations are shown in black. After perturbation the model reaches the new quasi-steady state (only slow change in glycogen) within minutes, even after strong perturbations in blood glucose. To analyze the effect of the hormonal regulation on the glycogen metabolism and to distinguish the effects due to fast allosteric regulations from the hormonal regulation, the model was simulated without changing the phosphorylation state of the glycogen synthase (GS) and glycogen phosphorylase (GP) during the glucose perturbations (red model simulations). The reduced glycogen metabolism leads to an accumulation of the glycolytic intermediates during hyperglycemia due to an inability of the system to store the glucose. If both simulations yield equal results only the red curve is visible. (TIF) Click here for additional data file. Figure S8 Fast time course simulation: model fluxes. Time course of model fluxes during glucose disturbance profiles. The description of the glucose profile and the corresponding concentration time courses are given in Figure S7. Even without hormonal regulation the glycogen system is able to switch direction between glycogen synthesis and glycogenolysis under hypoglycemia as a consequence of the futile cycle in glycogen metabolism. Hormonal regulation supports the inherent switching of directions increasing the flux in the respective directions. GP decreased in hypoglycemia and strongly increased in hypoglycemia. (TIF) Click here for additional data file. Figure S9 Metabolic control analysis of HGP. Metabolic response coefficients of the HGP/HGU flux (top), gluconeogenesis/glycolysis flux (middle) and glycogenolysis/glycogenesis flux (bottom) for hypoglycemic conditions of 3 mM (left) and hyperglycemic conditions of 8 mM (right) for various glycogen concentrations. Analyzed were the effects of changes in Vmax values of the enzymes of hepatic glucose metabolism on the steady state fluxes under the given conditions. The main interesting effects are: (i) The key enzymes depend strongly on the glucose concentration, different enzymes change the HGP/HGU, gluconeogenesis/glycolysis and glycogen synthesis/glycogenolysis under hypoglycemia and hyperglycemia. (ii) Different Vmax show the main response on HGP/HGU, gluconeogenesis/glycolysis and the glycogen metabolism (iii) The interconvertible enzymes show strong responses to changes in the Vmax values (iv) Only a selected number of enzymes has a strong response, the change in Vmax of many enzymes results in almost no change in key glucose fluxes. (TIF) Click here for additional data file. Text S1 Overview metabolites. The full metabolite name, the used short name in the model, the and the respective compartment are depicted. The Initial concentrations with the respective references used for all simulations are either from human or rat liver/hepatocytes. The concentrations are given in more detailed table format in Dataset S1. In the case that no compartment-specific data for mitochondrion was available, the respective metabolite concentrations for cytosol were used. The glycogen concentration is given in glucose equivalents. (PDF) Click here for additional data file. Text S2 Overview reactions and transporters. The full reaction/transporter name, the used short name in the model, the and the respective compartment are depicted. The rate equations of the kinetic model are given in Text S3. The given references are the references for the kinetic parameters. Furthermore, the fitted Vmax values used for simulations are given. (PDF) Click here for additional data file. Text S3 Kinetic rate equations of the model. Detailed rate equations of all processes of the model of human hepatic glucose metabolism. (PDF) Click here for additional data file. Text S4 Overview allosteric regulations. The respective rate equations are given (Text S3). Allosterically regulated enzymes characteristic for either HGP or HGU are marked respectively. Allosteric regulators can function as activators are inhibitors. The references stating the roles of the allosteric effectors and the parameters in the rate equations are given. (PDF) Click here for additional data file. Text S5 Overview interconvertible enzymes of human hepatic glucose metabolism. Overview over the alterations in kinetic properties as a consequence of changed phosphorylation state between phosphorylated and dephosphorylated state of key interconvertible enzymes of HGP and HGU. (PDF) Click here for additional data file.
Author and article information
Journal
Journal ID (nlm-ta): CPT Pharmacometrics Syst Pharmacol
Journal ID (iso-abbrev): CPT Pharmacometrics Syst Pharmacol
Title:
CPT: Pharmacometrics & Systems Pharmacology
Publisher:
Nature Publishing Group
ISSN
(Electronic):
2163-8306
Publication date
(Print):
July
2013
Publication date
(Electronic):
10
July
2013
Publication date PMC-release: 1
July
2013
Volume: 2
Issue: 7
Page: e54
Affiliations
[1
]BioModels Group, EMBL - European Bioinformatics Institute, Wellcome Trust Genome Campus , Hinxton, Cambridge, UK
Author notes
Article
Publisher Item ID: psp201330DOI: 10.1038/psp.2013.30
PMC ID: 3731829
PubMed ID: 23842097
SO-VID: 41c277da-1a75-440c-8f4e-6778a3cec122
Copyright © Copyright © 2013 American Society for Clinical Pharmacology and Therapeutics
License:
CPT: Pharmacometrics and Systems Pharmacology is an open-access journal published by Nature Publishing Group. This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/
History
Date
received
: 02
October
2012
Date
accepted
: 18
April
2013
Categories
Subject:
Review
Data availability: