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 h_base and maximal hormone concentration h_max, 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.

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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.

https://doi.org/10.1371/journal.pcbi.1002577.g002

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Table 1. Fit functions and parameters for the insulin, glucagon and epinephrine GHR.

https://doi.org/10.1371/journal.pcbi.1002577.t001

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.

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Table 2. Experimental data for HGP, gluconeogenesis (GNG), glycogenolysis (GLY).

https://doi.org/10.1371/journal.pcbi.1002577.t002

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 ().

Hormonal regulation of glucose metabolism

The plasma concentrations of the gluco-regulatory hormones change with changes in plasma glucose concentration. With increasing glucose the insulin level increases (Figure 2C), whereas the levels of glucagon (Figure 2A) and epinephrine (Figure 2B) decrease. Consequently, the phosphorylation state of the interconvertible enzymes (Figure 2D, Equation 5) changes from phosphorylated (94% at 2 mM) to dephosphorylated (5% at 14 mM) with increasing blood glucose. Changes in the phosphorylation state are accompanied by changes of the kinetic properties of the respective enzymes (see Equation 4). Due to the higher activity of the interconvertible enzymes of HGP (GP and FBP2) in the phosphorylated form and the increased activity of the enzymes of HGU (GS, PFK2, PDH, PK) in their dephosphorylated form (Text S5), the hepatic glucose metabolism is shifted from a glucose producing phenotype (HGP) under low glucose concentrations to a glucose consuming phenotype (HGU) at hyperglycemia. This change from an anabolic metabolism of glucose production (HGP) to a catabolic mode of glucose utilization (HGU) is a short term adaptation via changes in the kinetic properties of crucial enzymes of glucose metabolism. By adapting the HGP/HGU, gluconeogenesis/glycolysis and glycogen metabolism to the current hormonal signals and blood glucose concentration, the liver is able to fulfill its important role in glucose homeostasis in a variety of physiological states ranging from hypoglycemic states in overnight and short term fasting to hyperglycemic states postprandial.

HGP, gluconeogenesis and glycogenolysis in short term fasting

The liver is the main glucose supplier in overnight fasting and short term fasting. The produced hepatic glucose (HGP, Figure 3A) results either from de novo synthesis via gluconeogenesis (GNG, Figure 3B) or from degradation of hepatic glycogen via glycogenolysis (GLY, Figure 3C). The relative contributions of these two processes to HGP change over the time course of fasting with gluconeogenesis becoming more and more important, whereas the share of glycogenolysis to HGP decreases (GNG/HGP, Figure 3D).

The simulations for short term fasting under constant glucose concentrations between 5 mM (green) and 3 mM (red), the normal range of fasting glucose concentration [22], [23], are in agreement with the experimental data (Figure 3, Table 2). Taking into account the gradual decrease in blood glucose concentration during fasting from 5 mM to 3.6 mM (blue), an agreement between simulation and experimental data is observed.

HGP decreases with ongoing fasting to a constant basal rate of 7–8 at around 40 h (Table 3D, Table 3F). With increasing blood glucose HGP decreases. Gluconeogenesis rate is constant for given blood glucose concentrations (see [2] for review). Taking into account the gradual decrease in blood glucose over fasting the rate of gluconeogenesis increases gradually (blue). In contrast, glycogenolysis decreases sharply during fasting due to the emptying glycogen stores (see also Figure 4A). As a consequence, the fractional contributions of glycogenolysis and glycogen synthesis to HGP shift from initially equal contribution of both processes to glucose finally completely synthesized de novo. Whereas after an overnight fast (∼10 h) half of the HGP comes from glycogenolysis [2], after 40 h fasting only 10% of HGP result from glycogen, 90% from gluconeogenesis.

Glycogenolysis in short term fasting

Liver glycogen has an important role as a short term glucose buffer for glucose homeostasis. At low blood glucose concentrations, like in the fasting state or during extensive muscle activity, glucose is released from glycogen (Figure 4A), whereas during periods of high blood glucose like postprandial glycogen is synthesized from glucose (Figure 4B).

Over 40–50 h short term fasting the glycogen stores are emptied. During an overnight fast glycogen is utilized and the resulting glucose from glycogenolysis is exported resulting in half-filled glycogen stores after ∼16 h (Table 3E). The rate of glycogenolysis is almost constant and decreases only at low glycogen concentrations (Table 3E).

The simulations for glycogen depletion under hypoglycemia (Figure 4A) for glucose concentrations between 3.6 mM (red) and 5 mM (green) are in agreement with experimental data [22], [23]. The rate of glycogenolysis depends on the blood glucose concentration (see also Figure 3C). With decreasing blood glucose concentration, the rate of glycogenolysis increases, glycogen stores are emptied faster. As a consequence of the drop in glucose concentration over fasting, simulations at low glucose concentration overestimate the decrease in glycogen, simulations at high glucose concentrations underestimate the depletion. When taking the gradual drop of blood glucose from 5 to 3.6 mM into account (blue), the concordance of simulations experimental data further improved.

Glycogen synthesis postprandial

Blood glucose levels are elevated postprandial and glycogen is stored via glycogen synthesis (Figure 4B). The simulations for glycogen synthesis under hyperglycemia for different glucose concentrations between 5.5 mM (red) and 8.0 mM (green) (normal range of postprandial glucose concentrations) are in good agreement with the experimental data [8], [24], [25], especially the simulation at 7 mM (blue) representing a normal blood glucose value postprandial. With increasing blood glucose the rate of glycogen synthesis increases and the hepatic glycogen stores are filled faster. For medium filled glycogen stores between 200 and 300 mM the rate of glycogen synthesis is constant for a given blood glucose concentration.

To further evaluate the predictive capacity of our model, we performed time-course simulations of experimentally determined glycogen levels in dogs monitored under conditions of hyperglycemic, hyperinsulinemic clamps and administration of a mixed meal diet [26], as can be seen from the respective figures in the supplement (Figure S1, S2, S3, S4, S5, S6), our model simulation were in good agreement with the measured time courses of insulin, glucagon and glycogen. Of note, in these simulations none of the data was used for the calibration of the model. Moreover, successful simulation of these experiments under conditions where the two hormones insulin and glucagon could be varied independently from each other while in vivo their levels are coupled by the glucose level of the blood demonstrates the validity of the phenomenological glucose-hormone response functions (Equation 2 and 3) and interconversion –versus-hormone function γ (Equation 5) used in our model.

Role of insulin and glucagon for HGP

To analyze the effects of insulin and glucagon on HGP classical hormone perturbation experiments of hepatic glucose metabolism were simulated [27]. In these experiments a deficiency in either insulin or glucagon or both was achieved by somatostatin infusion, an inhibitor of insulin and glucagon secretion, in combination with hormone replacement infusions. Figure 4C depicts exemplarily the effect of a transient drop of insulin which is characterized by a marked increase in HGP and a consequent rise in plasma glucose concentration. HGP and plasma glucose return to normal after the perturbation. Simulations of other cases (saline control, insulin and glucagon depletion, glucagon depletion and somatostatin in combination with insulin and glucagon restoration) are depicted in Figure S1, S2, S3 and S5, respectively. For all hormone perturbations the time courses of HGP as well as glucose are in good agreement with the experimentally observed changes.

Predicted changes in basal glucose production (HGP) are very similar to the experimentally observed changes (Figure 4C). Insulin and glucagon depletion or glucagon depletion alone reduce the HGP to around 70% of basal values, insulin depletion increases HGP to around 130%. Insulin and glucagon have a dramatic effect on the human hepatic glucose metabolism, with basal glucagon being responsible for about 30% of HGP and basal insulin preventing increased HGP as a consequence of an unrestrained glucagon action [27].

Glucose homeostasis by the liver at rapidly changing blood glucose levels

The special role of the liver for glucose homeostasis results from the ability to switch between an anabolic glucose producing mode (HGP) to a catabolic glucose utilizing mode (HGU) depending on the blood glucose concentration and hormonal signals. In this process the hepatic metabolism is altered from glucose production via gluconeogenesis and glycogenolysis at hypoglycemia to glucose utilization via glycolysis and glycogen synthesis at hyperglycemia – a short term switch between metabolic pathways occurring in the range of minutes.

To analyze the temporal response of hepatic glucose metabolism to rapid variations in the external glucose concentrations we performed time-dependent simulations with a step-wise constant concentration profile of blood glucose and constant internal glycogen concentrations. (Figure S7 and S8). To evaluate the impact of hormone induced fast changes in the phosphorylation state of key regulatory enzymes on metabolic regulation we performed these simulations also in the absence of this mode of regulation in the glycogen metabolism, i.e. frozen phosphorylation state of glycogen synthase and glycogen phosphorylase and regulation of these enzyme activities only by allosteric effects. We found clear differences in the simulated time-courses (black and red curves in Figure S7 and S8) indicating that hormonal regulation contributes substantially in the rapid adaptation of the network to abrupt changes of blood glucose. In both cases (i.e. presence or absence of hormonal control) the simulation revealed that even after changing the external glucose concentrations abruptly by 2 mM and more a new steady state was reached within several minutes.

Glucose homeostasis by the liver at slowly changing blood glucose levels

Based on the above finding that the a new metabolic steady-state (except for glycogen) is achieved within a few minutes after abrupt changes of the blood glucose level it appeared justified to treat the metabolic network as being in a quasi-steady state as long blood glucose changes are slow and the glycogen pool is quasi constant.

Hence, the quasi-steady state approximation was applied to simulate HGP/HGU at varying concentrations of blood glucose and cellular glycogen (Figure 5). In these calculations, the external glucose concentration was varied between constant values of 2 to 14 mM and the glycogen concentration kept at constant values between 0 and 500 mM. For each couple of glucose/glycogen values the resulting HGP/HGU (Figure 5A), the contribution of gluconeogenesis/glycolysis to HGP/HGU (Figure 5C), the contribution of glycogenolysis/glycogen synthesis to HGP/HGU (Figure 5D) and the ability of the liver to respond to changes in blood glucose expressed through the response coefficient HGRC (Figure 5B) in this quasi steady-state where analyzed. The fluxes depicted Figure 5 for a given constant glucose and glycogen concentrations are the reached steady state fluxes for this glucose and glycogen concentrations.

The hepatic glucose metabolism switches between anabolic HGP and catabolic HGU at blood glucose concentrations of 6.6 mM for half-filled glycogen stores (Figure 5A). The set point above the normal glucose concentration of around 5–5.5 mM [1], [28] is in line with the role of the liver as glucose producer under normoglycemic conditions [1], [29]. The liver contributes to glucose homeostasis by exporting glucose below the HGP/HGU set point (red) and importing glucose above this threshold (green) (Table 3A). With increasing blood glucose levels, HGP decreases and HGU increases, in accordance with reported suppression of glucose production and increase in HGU with increasing postprandial glucose level [8]. Glycogen has almost no influence on HGU whereas for low blood glucose concentrations HGP strongly depends on glycogen due to increased contribution of glycogenolysis to HGP with increasing glycogen content (Figure 5D). For low glucose concentrations the increase in glycogenolysis with increasing glycogen is strongest for low glycogen levels, more moderate for glycogen concentrations above 200 mM. HGP is maximal for filled glycogen store and low blood glucose.

The switch between gluconeogenesis and glycolysis occurs at 8.5 mM glucose for half-filled glycogen stores (Figure 5C). Gluconeogenesis and glycolysis rate are mainly determined by the prevalent blood glucose and only marginally depend on glycogen. Below 8 mM blood glucose gluconeogenesis is remarkably constant [2] (see also Figure 3B).

For plasma glucose levels below 5.1 mM glucose is released from glycogen stores via glycogenolysis, above 5.1 mM glycogen is synthesized for half-filled glycogen stores (Figure 5D, Table 3B) with rates of glycogenesis and cumulative glycogen as reported (Table 3C). In contrast to gluconeogenesis/glycolysis, glycogen metabolism is markedly affected by the glycogen content. Glycogenolysis is almost constant for partially filled glycogen stores and decreases only for low glycogen concentrations (Table 3H). Glycogenesis increases with increasing glucose concentration (Figure 5D), which is in accordance with the view that blood glucose determines the maximal rate of glycogenesis [8].

Most interestingly the switching behavior for glycogenesis/glycogenolysis and glycolysis/gluconeogenesis is very distinct, with different set points and different dependencies on glucose. Gluconeogenesis is almost constant for low blood glucose and glycolysis increases with increasing glucose. In contrast glycogenolysis is almost constant for high blood glucose and increases with decreasing blood glucose at low blood glucose. In combination of the two processes a HGP/HGU output of the liver is generated with a set point at normal blood glucose concentrations which is able to react over the whole range of physiological glucose concentrations (Figure 5A).

Direct and indirect glycogen synthesis

As a consequence of the different set points for gluconeogenesis/glycolysis and glycogenesis/glycogenolysis, hepatic glycogen can be synthesized via two alternative pathways: the direct pathway, in which glucose taken from the blood (HGU) is directly stored as glycogen above the gluconeogenesis/glycolysis set point of ∼8.5–8.8 mM; the indirect pathway, in which glucose synthesized via gluconeogenesis, is stored as glycogen for blood glucose concentrations between 5 mM and the HGP/HGU set point of ∼6.6–7 mM; In the intermediate region between the set points of HGP/HGU and gluconeogenesis/glycolysis glycogen is synthesized via the direct and indirect pathway with varying contributions [8] (Table 3G).

Response coefficients

The effects of changes of values on HGP/HGU, gluconeogenesis/glycolysis and glycogenolysis/gluconeogenesis were analyzed via response coefficients [30] (Figure S9, Equation 7). The important findings of this analysis are as follows: (i) The effect of changes in the is strongly dependent on the external concentration of glucose and to a much lesser extent on the glycogen level. (ii) Under hypoglycemic conditions main control of hepatic glucose metabolism is exerted by a completely different set of reactions than in hyperglycemia. (iii) The response of the opposing pathway couples gluconeogenesis/glycolysis is very different to the glycogenolysis/gluconeogenesis. (iv) A small group of enzymes catalyzing irreversible reaction steps (GK, G6PASE, GP, PFK1, FBPASE1, PFK2, and FBPASE2) have a significant influence on the hepatic glucose metabolism whereas the majority of enzymes have only marginal effects. This control analysis is of particular interest for gene expression studies of hepatic glucose metabolism like for instance studies of circadian or feeding induced changes in hepatic expression [31].

The liver – a glucose homeostate (HGRC)

The hepatic counter-regulatory capacity to changes in blood glucose was evaluated using the HGRC (Equation 6, Figure 5B), a measurement of the ability of the liver to react to changes in the blood glucose with changes in HGP or HGU, respectively. Our simulations showed that the liver is able to respond over the whole range of physiological blood glucose concentrations with being the lower limit, i.e. a change of 0.01 mM in blood glucose results in a change of HGP/HGU of at least . Intriguingly, our analysis suggest the counter-regulatory response of the liver to variations in the external glucose concentration to be particularly pronounced around the HGP/HGU set point at ∼6.6 mM which is flanked by a strong counter-regulatory response to hypoglycemia and a weaker response to elevated blood glucose levels. The strong counter-regulation to a decrease of blood glucose below 5 mM results in an effective increase in HGP whereby the rise of HGP depends on the glycogen content, with most effective counter-regulation for filled glycogen stores. The liver counteracts falling blood glucose levels to avoid hypoglycemia with a strong response. Furthermore, an increased response to elevated blood glucose levels of above 7.5 mM is seen enabling the hepatocyte to react efficiently to elevated blood glucose levels as occurring postprandially (6–10 mM). The hepatic glucose metabolism has ideal regulatory properties to react to the typical physiological challenges to glucose homeostasis: counter regulation to hypoglycemia in fasting and under extensive muscle activity and counter regulation to postprandial increase in blood glucose.